dart_csp 2.2.1

A comprehensive Constraint Satisfaction Problem (CSP) solver with backtracking, AC-3, GAC, and string constraint parsing.

dart_csp

A powerful, general-purpose library for modeling and solving Constraint Satisfaction Problems (CSPs) in Dart. Built with intelligent backtracking search, consistency-checking algorithms, and smart heuristics to efficiently solve complex logic puzzles.

This library offers three intuitive ways to define constraints: string expressions for natural syntax, a high-level Problem builder for fast development, and a manual CspProblem class for direct control over the underlying structure.

Note: An early version of the solver core in this repository was a Dart port of csp.js by Prajit Ramachandran. That code has since been replaced by a clean-room rewrite written from textbook references with no access to the upstream source. See NOTICE for the history.

What is a Constraint Satisfaction Problem?

A CSP is a mathematical problem where you need to find values for variables that satisfy a set of constraints. Every CSP consists of three components:

Component	Description	Example (Sudoku)
Variables	The unknowns you need to solve for	The 81 empty squares
Domains	Possible values each variable can take	Numbers 1-9 for each square
Constraints	Rules that restrict variable assignments	No repeats in rows/columns/blocks

Classic CSP Examples

• Sudoku: Variables are grid squares, domains are numbers 1-9, constraints prevent duplicates in rows/columns/blocks
• Map Coloring: Variables are regions, domains are colors, constraints prevent adjacent regions from having the same color
• N-Queens: Variables are queen positions, domains are board squares, constraints prevent queens from attacking each other

Features

This solver goes beyond brute-force search with the following implemented algorithms:

Core Algorithms

• Backtracking Search: Intelligent depth-first search that backtracks when constraints are violated
• AC-3 Algorithm: Enforces arc consistency for binary constraints (two-variable rules)
• Generalized Arc Consistency (GAC): Handles n-ary constraints (multi-variable rules)
• Multiple Solution Finding: Stream-based API to find all possible solutions efficiently

Smart Heuristics

• Minimum Remaining Values (MRV): Chooses the most constrained variable to assign next ("fail-first" principle)
• Least Constraining Value (LCV): Selects values that preserve the most options for other variables

Built-in Constraint Library

• 20+ Pre-built Constraints: Common constraint patterns like allDifferent(), exactSum(), ascending(), etc.
• Optimized Performance: Built-in constraints are faster than equivalent lambda functions
• Extension Methods: Fluent API methods like addAllDifferent() for cleaner code

FlatZinc frontend

• Drop-in MiniZinc compatibility: parse and solve .fzn files produced by mzn2fzn (or any other MiniZinc compiler).
• CLI binary bin/dart_csp_fzn.dart reads from a file path or stdin and emits the standard FlatZinc output format (name = value;, ----------, ==========), so it plugs straight into the MiniZinc solver-configuration pipeline.
• Builtin coverage: every primitive in the M1–M4 plan (int/bool comparisons, linear arithmetic, SAT-style clauses, bool2int), every standard global (all_different, circuit, cumulative, disjunctive, diffn, inverse, regular, table_int, lex_*, nvalue, global_cardinality, bin_packing_load, value_precede_chain_int, array_int_element), and every reified variant (int_*_reif, int_lin_*_reif, bool_eq_reif, bool_clause_reif).
• Set variables: bounded var set of L..U declarations and the set constraint family (set_in, set_card, set_eq/set_ne, set_subset/set_superset, set_union/set_intersect/set_diff/set_symdiff, plus _reif variants) map onto the set-variable layer; solutions render as FlatZinc set literals.
• Conflict explanation traces back to FlatZinc: every lowered constraint carries a <fzn_name>#<counter> label so MUS output points straight at the source .fzn.

See doc/flatzinc.md for the full reference.

String Constraint Parsing

• Natural Language Syntax: Write constraints as strings like "A + B == 10" or "A != B != C"
• Advanced Expression Support: Complex expressions like "5 <= A + B <= 7" and "A * B + C == 15"
• Variable Equations: Support for "A + B == C" where one variable equals an expression of others
• Set Membership: Constraints like "A in [1, 3, 5]" for allowed value sets
• Comprehensive Parser: Handles arithmetic, comparisons, ranges, and chained operations

Developer Features

• Modular Architecture: Clean separation of concerns across multiple focused modules
• Comprehensive Test Suite: Full test coverage with test cases covering all functionality
• Multiple APIs: Choose between string constraints, builder pattern, or manual construction
• Fluent Builder API: An intuitive Problem class to easily define your CSP
• Rich Examples: Complete demo showcasing all constraint types and problem-solving techniques
• Debugging Tools: Problem validation, summary printing, and step-by-step visualization
• Type Safety: Full Dart type system integration with proper error handling
• Async Support: Non-blocking solving with Future-based API

Quick Start

1. Installation

Add this package to your pubspec.yaml:

dependencies:
  dart_csp: ^2.0.0

Then run:

dart pub get

2. Import and Use

import 'package:dart_csp/dart_csp.dart';

3. Your First Problem - Map Coloring

Future<void> main() async {
  final p = Problem();
  
  // Variables: Australian states, Domain: Colors
  p.addVariables(['WA', 'NT', 'SA', 'Q', 'NSW', 'V'], ['red', 'green', 'blue']);
  
  // Constraints using string expressions (easiest way!)
  p.addStringConstraints([
    'WA != SA',
    'NT != SA', 
    'Q != SA',
    'NSW != SA',
    'V != SA',
    'WA != NT',
    'NT != Q',
    'Q != NSW',
    'NSW != V'
  ]);
  
  // Get first solution
  final solution = await p.getSolution();
  print(solution); // {WA: red, NT: blue, SA: green, ...}
  
  // Or find all solutions
  print('All possible colorings:');
  await for (final solution in p.getSolutions()) {
    print(solution);
  }
}

4. Run the Comprehensive Demo

The library includes a complete demo showcasing all features:

dart run example/demo.dart

This runs several constraint satisfaction problems demonstrating:

• Sudoku Solving
• Magic Square Generation
• Resource Allocation
• Class Scheduling
• And more!

Map-coloring and N-queens demos are currently being rewritten clean-room (see REWRITE-DEMOS.md); they'll be back in a follow-up commit.

How to Use dart_csp

Method 1: String Constraints (Recommended)

The most intuitive way - write constraints as natural expressions:

final p = Problem();

// 1. Add variables and domains
p.addVariables(['A', 'B', 'C'], [1, 2, 3, 4, 5]);

// 2. Add constraints using natural string syntax
p.addStringConstraints([
  'A != B != C',           // All different (chained)
  'A + B == 7',            // Exact sum
  'A < B < C',             // Strict ordering
  '5 <= A + B <= 8',       // Range constraint
  'A * B >= 6',            // Minimum product
  'A + B == C'             // Variable equation
]);

// 3. Solve
final solution = await p.getSolution();

Supported String Constraint Syntax:

Pattern	Description	Example
Equality/Inequality	Variable comparisons	"A == B", "A != B"
Chained Operations	Multiple comparisons	"A != B != C", "A < B < C"
Arithmetic Equality	Exact sums/products	"A + B == 10", "A * B == 12"
Arithmetic Inequality	Min/max constraints	"A + B >= 5", "A * B <= 20"
Range Constraints	Bounded values	"5 <= A + B <= 10"
Variable Equations	Inter-variable relations	"A + B == C", "A * B == D"
Complex Expressions	Mixed operations	"2*A + 3*B == 15", "A*B + C >= 10"
Set Membership	Allowed values	"A in [1, 3, 5]"
Single Variable	Constant comparisons	"A > 5", "B != 3"

Method 2: The Problem Builder

The Problem class provides a clean, step-by-step builder pattern with built-in constraints:

final p = Problem();

// 1. Add variables and domains
p.addVariables(['A', 'B', 'C'], [1, 2, 3, 4, 5]);

// 2. Use built-in constraints (highly optimized)
p.addAllDifferent(['A', 'B', 'C']);      // All different values
p.addExactSum(['A', 'B'], 7);            // A + B = 7
p.addAscending(['A', 'B', 'C']);         // A ≤ B ≤ C

// 3. Or define custom constraints with lambda functions
p.addConstraint(['A', 'B'], (a, b) => a * b <= 10);

// 4. Solve
final solution = await p.getSolution();

Method 3: Manual CspProblem Construction

Direct access to underlying data structures for programmatic generation:

var variables = <String, List<dynamic>>{
  'A': [1, 2, 3],
  'B': [1, 2, 3, 4],
  'C': [3, 4, 5],
};

var binaryConstraints = <BinaryConstraint>[
  BinaryConstraint('A', 'B', (a, b) => a < b),
  BinaryConstraint('B', 'A', (b, a) => b > a),
];

final problem = CspProblem(
  variables: variables,
  constraints: binaryConstraints,
);

final solution = await CSP.solve(problem);

Built-in Constraint Library

The library provides optimized, reusable constraint functions for common patterns:

Equality/Inequality Constraints

// All variables must have different values
p.addAllDifferent(['A', 'B', 'C', 'D']);

// All variables must have the same value  
p.addAllEqual(['X', 'Y', 'Z']);

Arithmetic Constraints

// Sum constraints
p.addExactSum(['A', 'B', 'C'], 15);           // A + B + C = 15
p.addSumRange(['A', 'B'], 5, 10);             // 5 ≤ A + B ≤ 10

// Weighted sums
p.addExactSum(['A', 'B'], 20, multipliers: [3, 4]); // 3*A + 4*B = 20

// Product constraints  
p.addExactProduct(['X', 'Y'], 12);            // X * Y = 12
p.addConstraint(['A', 'B', 'C'], minProduct(8));     // A * B * C ≥ 8

Set Membership Constraints

// Variables must be from allowed set
p.addInSet(['A', 'B'], {2, 3, 5, 7});        // Only prime numbers

// Variables cannot be from forbidden set
p.addNotInSet(['X', 'Y'], {2, 4, 6});        // No even numbers

// At least N variables must be from set
p.addConstraint(['A', 'B', 'C'], someInSet({1, 3, 5}, 2)); // ≥2 odd numbers

Ordering Constraints

// Ordering (preserves variable sequence)
p.addAscending(['A', 'B', 'C']);             // A ≤ B ≤ C
p.addStrictlyAscending(['X', 'Y', 'Z']);     // X < Y < Z  
p.addDescending(['P', 'Q', 'R']);            // P ≥ Q ≥ R

Symmetry-Breaking

Two kinds of symmetry recur in CSP modeling and dart_csp has a primitive for each.

Variable (or sequence) symmetry — when two or more parts of your problem are interchangeable (rows of a matrix, identical workers), every solution has a dual obtained by swapping them. addLexLeq / addLexLt between two equal-length variable lists keeps a single canonical representative:

// Two interchangeable workers W1, W2 each picking two days.
// Keep only the assignment where W1's pair is lex-smaller than W2's.
p.addLexLt(['W1Day1', 'W1Day2'], ['W2Day1', 'W2Day2']);

Cuts the symmetric solution set exactly in half. Comparison uses Comparable, so it works on int, double, String, etc. For breaking row-permutation symmetry across more than two interchangeable rows in one call, use addLexChain(rows) — sugar that posts lexLeq between every consecutive pair (or lexLt with strict: true to forbid equal rows).

// Three interchangeable rows. addLexChain keeps one canonical
// row-order per orbit; lex-leq is transitive on Comparable so
// the consecutive chain is equivalent to the full pairwise set.
p.addLexChain([
  ['r0a', 'r0b'],
  ['r1a', 'r1b'],
  ['r2a', 'r2b'],
]);

Value symmetry — when the labels assigned to interchangeable alternatives (colors, agents, machine labels, bin IDs) can be permuted without changing the solution structure, every solution has k! permutations under relabeling. addValuePrecedence enforces a canonical ordering: the first occurrence of each value must follow the canonical sequence:

// Triangle K3, three interchangeable colors. Without symmetry
// breaking: 6 colorings (3! permutations of r,g,b). With:
// exactly 1 canonical representative (r,g,b).
final p = Problem()
  ..addVariables(['a', 'b', 'c'], ['r', 'g', 'b'])
  ..addAllDifferent(['a', 'b', 'c'])
  ..addValuePrecedence(['a', 'b', 'c'], ['r', 'g', 'b']);

Posts one n-ary precedence constraint per consecutive pair in the canonical value list. Values outside the canonical list are unconstrained. Reduces the search space by up to k! where k = values.length.

Using Factory Functions Directly

You can also use the constraint factory functions directly:

// Using factory functions with addConstraint()
p.addConstraint(['A', 'B', 'C'], allDifferent());
p.addConstraint(['X', 'Y'], exactSumBinary(10)); // For 2 variables, more efficient

// Custom combinations
p.addConstraint(['A', 'B', 'C'], (assignment) {
  // Custom logic combining multiple constraint types
  return allDifferent()(assignment) && exactSum(12)(assignment);
});

Convenience Functions

For quick problem solving, use the top-level convenience functions:

// Quick all-different problem
final solution1 = await solveAllDifferent(
  variables: ['A', 'B', 'C'],
  domain: [1, 2, 3]
);

// Quick sum problem  
final solution2 = await solveSumProblem(
  variables: ['X', 'Y'],
  domain: [1, 2, 3, 4, 5],
  targetSum: 7
);

// General string constraint problem
final solution3 = await solveProblem(
  variables: {'A': [1, 2, 3, 4], 'B': [1, 2, 3, 4]},
  constraints: ['A != B', 'A + B >= 5']
);

Real-World Examples

Sudoku Solver with String Constraints

Future<void> solveSudoku(List<List<int>> puzzle) async {
  final p = Problem();
  
  // Add variables for each cell
  for (int r = 0; r < 9; r++) {
    for (int c = 0; c < 9; c++) {
      final key = '$r-$c';
      if (puzzle[r][c] != 0) {
        p.addVariable(key, [puzzle[r][c]]);  // Clue
      } else {
        p.addVariable(key, [1, 2, 3, 4, 5, 6, 7, 8, 9]);  // Empty cell
      }
    }
  }
  
  // Add all-different constraints using built-in methods
  // Rows
  for (int r = 0; r < 9; r++) {
    final row = List.generate(9, (c) => '$r-$c');
    p.addAllDifferent(row);
  }
  
  // Columns
  for (int c = 0; c < 9; c++) {
    final col = List.generate(9, (r) => '$r-$c');
    p.addAllDifferent(col);
  }
  
  // 3x3 blocks
  for (int br in [0, 3, 6]) {
    for (int bc in [0, 3, 6]) {
      final block = <String>[];
      for (int r = br; r < br + 3; r++) {
        for (int c = bc; c < bc + 3; c++) {
          block.add('$r-$c');
        }
      }
      p.addAllDifferent(block);
    }
  }
  
  final solution = await p.getSolution();
  // Print solved puzzle...
}

Magic Square with String Constraints

Future<void> generateMagicSquare() async {
  final p = Problem();
  
  // 3x3 grid with numbers 1-9
  final positions = ['A1', 'A2', 'A3', 'B1', 'B2', 'B3', 'C1', 'C2', 'C3'];
  p.addVariables(positions, [1, 2, 3, 4, 5, 6, 7, 8, 9]);
  
  // Each number appears exactly once
  p.addAllDifferent(positions);
  
  // All rows, columns, and diagonals sum to 15 using string constraints
  p.addStringConstraints([
    // Rows
    'A1 + A2 + A3 == 15',
    'B1 + B2 + B3 == 15', 
    'C1 + C2 + C3 == 15',
    // Columns
    'A1 + B1 + C1 == 15',
    'A2 + B2 + C2 == 15',
    'A3 + B3 + C3 == 15',
    // Diagonals
    'A1 + B2 + C3 == 15',
    'A3 + B2 + C1 == 15'
  ]);
  
  final solution = await p.getSolution();
  // Display magic square...
}

Resource Allocation with Mixed Constraints

Future<void> allocateResources() async {
  final p = Problem();
  
  // Teams get 3-10 resources each  
  p.addVariables(['TeamA', 'TeamB', 'TeamC'], [3, 4, 5, 6, 7, 8, 9, 10]);
  
  // Use string constraints for clarity
  p.addStringConstraints([
    'TeamA + TeamB + TeamC == 20',  // Total budget
    'TeamA >= TeamB',               // Priority constraint
    'TeamA >= 3',                   // Minimum allocation
    'TeamB >= 3',
    'TeamC >= 3'
  ]);
  
  final solution = await p.getSolution();
  print('Team A: ${solution['TeamA']} resources');
  print('Team B: ${solution['TeamB']} resources'); 
  print('Team C: ${solution['TeamC']} resources');
}

Finding Multiple Solutions

Most problems have more than one valid assignment. dart_csp exposes several ways to get at them without searching the full tree twice.

// Stream solutions one at a time; break out when you've seen enough.
await for (final s in p.getSolutions()) {
  print(s);
}

// Or collect them all (only if the set fits in memory).
final all = await p.getAllSolutions();

// Just want to know how many?
final n = await p.countSolutions();

// Just want to know if the answer is unique?
final unique = !(await p.hasMultipleSolutions());

// Bounded sample — stops after N.
final firstFive = await p.getFirstNSolutions(5);

The top-level convenience functions (solveAllProblems, countAllSolutions, hasMultipleSolutions, getFirstNSolutions, solveAllDifferentMultiple, solveSumProblemMultiple) wrap the same operations without building a Problem first.

See doc/multi-solutions.md for guidance on which API to pick and how the underlying stream generator behaves.

Min-Conflicts (Local Search) Solver

For very large problems where you only need some solution quickly, the library also ships a Min-Conflicts local-search solver:

final p = Problem();
// ... set up an N-queens, scheduling, or map-coloring problem ...

final solution = await p.solveWithMinConflicts(maxSteps: 1000);
if (solution is Map<String, dynamic>) {
  print(solution);
} else {
  print('Local search gave up (FAILURE) — try more steps or a restart loop.');
}

Min-Conflicts is incomplete: it can return 'FAILURE' even when a solution exists (local optima). It also cannot enumerate solutions. Default to the backtracking solver and reach for this one only when the default isn't fast enough and you don't need uniqueness or completeness.

See doc/min-conflicts.md for tradeoffs, sizing maxSteps, restart strategies, and current limitations.

Optimization (Branch-and-Bound)

If your problem has more than one feasible solution, minimize / maximize return the one that minimizes / maximizes a named numeric variable:

// Choose three distinct values from [1..5] whose sum S is as small
// as possible.
final p = Problem()
  ..addVariables(['A', 'B', 'C'], [1, 2, 3, 4, 5])
  ..addAllDifferent(['A', 'B', 'C'])
  ..addVariable('S', [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15])
  ..addStringConstraint('A + B + C == S');

final best = await p.minimize('S');
// best = {A: 1, B: 2, C: 3, S: 6} (or a permutation of A,B,C)

Returns the optimal assignment, or 'FAILURE' if the problem is infeasible. Throws ArgumentError if the named variable doesn't exist or its domain isn't numeric.

Implementation is integrated branch-and-bound inside the backtracking engine. Each strictly-improving leaf becomes the new incumbent and the objective's domain is permanently pruned to values that still improve over the current bound (every existing trail snapshot is re-filtered in place so rollback can't reintroduce stale values). Search continues from the same point, avoiding the per-improvement restart cost that a classic bound-tightening loop pays. Problem.lastStats is populated by minimize / maximize.

Large Neighborhood Search (LNS)

Problem.lnsMinimize / Problem.lnsMaximize find a near-optimal solution by decomposing the optimization into a sequence of small focused sub-solves. LNS finds an initial feasible solution, then iteratively "destroys" a random subset of variables (frees them while pinning every other variable to its incumbent value) and re-solves the smaller sub-problem; improving candidates replace the incumbent. The result is not guaranteed optimal — it's a metaheuristic — but on hard problems LNS can be an order of magnitude faster than plain minimize. See doc/lns.md for the full design, the four shipped destroy policies (random, window, related, combined) and the two acceptance strategies (improving, simulatedAnnealing).

final result = await problem.lnsMinimize(
  'maxLoad',
  policy: LnsPolicy.random(fraction: 0.5),
  iterationBudget: 50,
  seed: 17,
);
final best = result.solution as Map<String, dynamic>;
print('best maxLoad = ${best["maxLoad"]}');
print(result.stats);

LNS is experimental (STABILITY.md). The surface may change across minor versions until ALNS / late-acceptance / parallel exploration land.

Lazy Clause Generation (LCG, preview)

Problem.solveWithLcg is the entry point for the in-progress Lazy Clause Generation feature — the conflict-driven nogood-learning technique that gives CP-SAT, Chuffed, and similar modern solvers orders-of-magnitude speedups on hard structured problems. M1+M2 + M3 complete (M3a–M3g): the engine learns conflict clauses on boolean-clause-propagator failures and on every specialised propagator — allDifferent / global-cardinality (addGcc) / regular (addRegular) / cumulative (addCumulative) / diff_n (addDiffN) / circuit (addCircuit / addSubcircuit) conflicts, posts them dynamically (so they prune future branches), and backtracks non-chronologically by default. On the classic pigeonhole-CNF UNSAT proofs the decision count drops by an order of magnitude (~10× on 7-in-6, ~30× on 8-in-7). For the matching-based propagators, a synthetic AtomInScc "bridge" atom (M3-tighten) collapses each tight Hall set into one resolvable literal so the first-UIP analyser converges, and the Hall set / GCC capacity cut is extracted by closing forward reachability in the residual graph (the alternating-path Régin / Quimper-Walsh construction, capacity-aware for GCC): e.g. Inkala's "World's Hardest Sudoku" learns ~25 clauses, and the exact-counts addGcc encoding learns identically to addAllDifferent. For regular (M3d), a value pruned at one position is explained by the other positions' value removals, collapsed through the same AtomInScc bridge; because regular is GAC-strong, this learning surfaces on UNSAT grids. For cumulative (M3e) and diff_n (M3f), a pruned coordinate is explained by the compulsory-part / forbidden-region bounds of the tasks / rectangles responsible — the consumers of the implication trail's bound atoms (AtomGe/AtomLe); neither propagator is GAC, so even satisfiable scheduling / packing instances search and learn. For circuit (M3g), a prune is explained by the fixed successor edges (AtomEq) that force it. With M3g the last propagator is explained; only conflicts flowing through the generic binary/predicate path now fall back to chronological backtrack. The search is sound + complete under any picker — pass useVsids: / useDomWdeg: / seed: to drive it under an activity/weighted-degree picker.

Non-chronological backjumping (the default engine). solveWithLcg runs on an iterative trail-based CDCL engine (LCG_PLAN.md §M4) that performs sound non-chronological backjumping — the real LCG search-tree speedup. It backjumps on short boolean/CNF clauses (pigeonhole 7-in-6 drops to ~240 decisions with 70+ real backjumps), and posts-but-backtracks-chronologically on the wide, weak clauses that allDifferent / GCC produce (so hard sudoku still converges fast). Opaque conflicts and non-integer-domain problems fall back automatically. Pass useIterativeCdcl: false for the original recursive chronological-backtracking-with-learning path (still sound + complete, without the backjump speedup). Learned clauses are run through recursive (self-subsuming) clause minimisation (Sörensson & Eén 2009) before posting, which strengthens them and deepens the backjumps (SolverStats.lcgMinimisedLiterals counts the literals removed). When useVsids / useDomWdeg is on, the engine applies the canonical CDCL rule of bumping the activity of every variable in the learned clause (not just the detecting constraint), so the picker tracks the learned structure. useRestarts: true adds Luby restarts that drop the search tree back to the root while retaining the learned-clause pool, the activity tables, and the per-variable saved phase (phase saving) — on heavy-tailed satisfiable instances this cuts decisions ~2×. LCG_PLAN.md has the full milestone roadmap.

final p = Problem()
  ..addVariable('a', [0, 1])
  ..addVariable('b', [0, 1])
  ..addVariable('c', [0, 1])
  ..addClause(positive: ['a', 'b', 'c'])
  ..addClause(negative: ['a', 'b']);
final solution = await p.solveWithLcg();
// Returns a satisfying boolean assignment; LCG bookkeeping is
// transparent — same return contract as p.getSolution().
print(CSP.lastStats!.learnedClauses);  // # clauses learned (0 on easy
                                       // SAT problems; grows on hard
                                       // UNSAT proofs)

The learnedClauseCap: kwarg (default 1000) bounds the learned- clause pool; when it overflows the oldest half are dropped via FIFO forget. CSP.lastStats carries learnedClauses and forgottenClauses counters alongside the existing backjumps / backjumpLevelsSkipped (which LCG bumps too).

solveWithLcg, the Atom hierarchy, the DomainView interface, the ImplicationReason types, the AnalysisResult / firstUipAnalyse analyser, the learnedClauseCap: kwarg, and the new SolverStats.learnedClauses / SolverStats.forgottenClauses fields are all experimental (STABILITY.md) and will evolve as M3 lands.

Reified Constraints (b ⇔ C)

A reified constraint introduces a 0/1 boolean variable whose value tracks whether some underlying relation holds:

p.addVariable('X', [1, 2, 3, 4, 5]);
p.addReifiedEquals('bX3', 'X', 3);   // bX3 ⇔ (X == 3)
p.addStringConstraint('bX3 == 1');   // forces X = 3

The killer use case is counting: once each sub-constraint is reified to a bool, you can use the existing arithmetic parser to express "at least k of these hold":

p.addReifiedEquals('b1', 'X1', 1);
p.addReifiedEquals('b2', 'X2', 1);
p.addReifiedEquals('b3', 'X3', 1);
p.addStringConstraint('b1 + b2 + b3 >= 2');   // at least 2 hold

The full set of helpers in the ReifiedConstraints extension:

Helper	Meaning
addReifiedEquals(b, X, c)	b ⇔ (X == c)
addReifiedNotEquals(b, X, c)	b ⇔ (X != c)
addReifiedLessThan(b, X, c)	b ⇔ (X < c)
addReifiedLessOrEqual(b, X, c)	b ⇔ (X <= c)
addReifiedGreaterThan(b, X, c)	b ⇔ (X > c)
addReifiedGreaterOrEqual(b, X, c)	b ⇔ (X >= c)
addReifiedInSet(b, X, {...})	b ⇔ (X ∈ set)
addReifiedEqualsVar(b, X, Y)	b ⇔ (X == Y)
addReified(b, vars, pred)	generic — b ⇔ predicate(assignment)

b is auto-added with domain [0, 1] if you haven't already added it.

Logical Combinators

The LogicalConstraints extension layers natural logical combinations on top of boolean variables — combine with reified constraints to express full propositional logic over your problem:

Helper	Meaning
addAtLeast([b1, b2, ...], k)	at least k of the bools are 1
addAtMost([b1, b2, ...], k)	at most k are 1
addExactly([b1, b2, ...], k)	exactly k are 1
addImplies(a, c)	a == 1 → c == 1
addReifiedAnd(b, [...])	b ⇔ all are 1
addReifiedOr(b, [...])	b ⇔ any is 1
addReifiedNot(b, other)	b ⇔ ¬other

// "X = 1 implies Y = 1" via two reified equalities + implies.
p.addReifiedEquals('bX', 'X', 1);
p.addReifiedEquals('bY', 'Y', 1);
p.addImplies('bX', 'bY');

SAT-style clauses (addClause)

For CNF-style modeling, addClause(positive: [...], negative: [...]) posts a disjunction of boolean literals — at least one positive variable must be 1, OR at least one negative variable must be 0. Tagged internally with a ClauseSpec so the engine dispatches to a unit-propagation propagator: when every literal but one has been falsified and none satisfied, the remaining literal is forced to its satisfying value.

// CNF for XOR: (a ∨ b) ∧ (¬a ∨ ¬b)
final p = Problem()
  ..addVariable('a', [0, 1])
  ..addVariable('b', [0, 1])
  ..addClause(positive: ['a', 'b'])
  ..addClause(negative: ['a', 'b']);
// Solutions: (a=0, b=1) and (a=1, b=0).

Compose with reified equalities for CNF over data variables:

// "x == 1 ∨ y == 5"
p.addReifiedEquals('bX1', 'x', 1);
p.addReifiedEquals('bY5', 'y', 5);
p.addClause(positive: ['bX1', 'bY5']);

The propagator uses the classical two-watched-literal scheme (Moskewicz et al., "Chaff", DAC 2001): each clause keeps two watcher pointers into its literal list that are updated lazily across propagation calls. Per-call work is O(1) amortized once the watchers are initialized, instead of O(literals) for a full scan.

The seeding side is matched to the watchers: once a clause's watchers are set, the engine wakes the propagator only when one of the two watched literals' variables is reduced. Reductions to other variables in the clause's scope cannot falsify the watched literals (the invariant is monotone under backtrack), so they're filtered out before the propagator is even enqueued. Width-2 clauses (e.g., "at most one" pairwise encodings) skip this filter — both literals are always watched, so the check could never fire and would just be overhead. The user-visible pruning behavior is the standard unit-propagation rule shown above; the watchers and the seeding filter are implementation details.

Global Constraints

The GlobalConstraints extension adds two structured n-ary constraints that recur across CSP modeling:

// Element: list[idxVar] == valueVar  (lookup / indirection)
final costs = [50, 20, 70, 10, 40];
p.addElement('item', costs, 'cost');
final cheapest = await p.minimize('cost');
// → picks item index 3, cost 10

// Table: (vars[0], vars[1], ...) must equal one of the tuples
p.addTable(['color', 'size'], [
  ['red', 'small'],
  ['blue', 'large'],
]);

element for lookup tables and indirection ("the cost of the chosen item is X"). table for arbitrary relations that don't have a clean closed-form predicate — compatibility matrices, FSM transitions, etc.

Counting Constraints: among, nvalue, gcc

The GlobalConstraints extension also covers the three classic counting constraints, each in both a count-variable form (lets you optimize over the count) and a fixed-k form.

// among: how many of vars take a value in the set?
p.addAmong(['s1', 's2', 's3'], {'morning'}, 'nMornings');
p.addAmongExactly(['s1', 's2', 's3'], {'morning'}, 2);

// nvalue: how many distinct values are used across vars?
p.addNvalue(['p1', 'p2', 'p3'], 'nColors');
final fewest = await p.minimize('nColors');   // chromatic-number-style
p.addNvalueExactly(['p1', 'p2', 'p3'], 2);

// gcc: each value must appear the specified number of times.
p.addGcc(slots, {'morning': 3, 'afternoon': 2, 'night': 1});
// or ranged:
p.addGccRanges(slots, {
  'morning': (min: 0, max: 3),
  'night':   (min: 1, max: 4),
});

Helper	Meaning
addAmong(vars, values, countVar)	countVar = #vars whose value ∈ values
addAmongExactly(vars, values, k)	exactly k of vars ∈ values
addNvalue(vars, countVar)	countVar = number of distinct values in vars
addNvalueExactly(vars, k)	exactly k distinct values used
addGcc(vars, counts)	each value → count must hold exactly
addGccRanges(vars, ranges)	each value → (min, max) must hold inclusively

gcc generalizes allDifferent (the case where every value of interest has count 1). It dispatches to a Régin-style network-flow propagator: a bipartite matching with value multiplicity finds a feasible assignment in polynomial time, then SCC analysis on the residual graph prunes any variable→value edge that is on no max matching. Upper-bound constraints receive full GAC. Lower-bound constraints receive conservative GAC: when the matching's distribution doesn't certify the lower bounds but the leaf isn't reached yet, the propagator declines to prune rather than risk a false-positive infeasibility verdict — the leaf check then catches any real violation. See doc/global-cardinality.md for modeling patterns (shift rosters, palette restriction, chromatic number).

Sequencing & Packing: circuit, bin_packing

// circuit: vars[i] is the successor of position i; the successor
// function must form a single Hamiltonian cycle through every
// position. Use for TSP-like routing, single-tour sequencing.
p.addVariables(['n0', 'n1', 'n2', 'n3'], [0, 1, 2, 3]);
p.addCircuit(['n0', 'n1', 'n2', 'n3']);

// bin_packing: each items[i] is the bin assignment for item i;
// binLoads[b] equals the sum of sizes of items in bin b. Constrain
// the load vars separately (with <=, ranges, or minimize) for
// capacity / balancing requirements.
p.addVariables(['it0', 'it1', 'it2', 'it3'], [0, 1]);   // 2 bins
p.addVariables(['load0', 'load1'], [0, 1, 2, ..., 10]);
p.addBinPacking(['it0', 'it1', 'it2', 'it3'],
                [4, 3, 2, 1],
                ['load0', 'load1']);
p.addStringConstraints(['load0 <= maxLoad', 'load1 <= maxLoad']);
final balanced = await p.minimize('maxLoad');

Helper	Meaning
addCircuit(vars)	the successor function vars[i] forms a single Hamiltonian cycle
addSubcircuit(vars)	same as addCircuit, but vars[i] = i is allowed and means position i is skipped; the non-skipped positions still form a single cycle (or there is no cycle at all)
addBinPacking(items, sizes, binLoads)	each binLoads[b] equals the sum of sizes of items assigned to bin b
addInverse(forward, inverse)	channelling: forward[i] = j ⇔ inverse[j] = i for all i, j in 0..n-1
addDiffN(xs, ys, widths, heights)	2D rectangle non-overlap (diff_n); each rect has a registered (x, y) corner and constant (w, h) size

addSubcircuit is the standard CP primitive for vehicle routing with optional stops, "visit some subset of cities in one tour", and any sequencing problem where the visited set itself is part of the decision. The propagator is shared with addCircuit: it tracks singleton-edge chains, rejects strict sub-cycles among non-skipped positions, prunes intermediate chain nodes from the tail's domain, and forces every non-cycle position to self-loop when a pure cycle closes early. The empty subcircuit (every position skipped) is a valid solution. Posting addAllDifferent(vars) alongside is redundant — the permutation property is enforced inherently (each value, including the self-loop slots, is used at most once).

// Decide *which* of three optional stops to visit in a single
// tour, plus the order. n=3, full domains. Either skip all three,
// pick a 2-stop tour, or take the full 3-stop loop.
final p = Problem()
  ..addVariables(['n0', 'n1', 'n2'], [0, 1, 2])
  ..addSubcircuit(['n0', 'n1', 'n2']);
// 6 valid assignments: 1 empty + 2 three-cycles + 3 (skip one,
// swap the other two).

2D Non-Overlap: diff_n

addDiffN generalises addNoOverlap from a unary (1D time) resource to two dimensions: given n axis-aligned rectangles described by (xs[i], ys[i]) corners (registered variables) and constant (widths[i], heights[i]) sizes, enforces that no two rectangles overlap. Standard CP primitive for rectangle packing, floor planning, 2D scheduling, and tile-placement puzzles.

// Pack four 1×1 squares into a 2×2 grid — exactly 4! = 24 tilings.
final p = Problem();
for (var i = 0; i < 4; i++) {
  p..addVariable('x$i', [0, 1])
   ..addVariable('y$i', [0, 1]);
}
p.addDiffN(
  [for (var i = 0; i < 4; i++) 'x$i'],
  [for (var i = 0; i < 4; i++) 'y$i'],
  List<int>.filled(4, 1),
  List<int>.filled(4, 1),
);

Half-open box semantics: a rectangle at (x, y) with size (w, h) occupies [x, x+w) × [y, y+h), so two rectangles that touch at an edge (xi + wi == xj) do not overlap.

Tags the constraint with a DiffNSpec so the engine dispatches to a forbidden-region sweep propagator (Beldiceanu & Carlsson, "Sweep as a generic pruning technique applied to the non-overlapping rectangles constraint", CP 2001). For each rectangle and each dimension the propagator aggregates the forbidden-position intervals induced by every other rectangle whose compulsory part in the orthogonal dimension forces an overlap, then filters the current domain in a single pass. The constraint scopes all 2n coordinate variables in the order [xs..., ys...] and propagates once per change to any rectangle — substantially less work than the pairwise GAC support search the previous decomposition required. Zero-area rectangles (width == 0 or height == 0) are excluded from the constraint — they never conflict with anything.

Channelling Inverse Maps: inverse

addInverse(forward, inverse) posts the standard channelling constraint between two equal-length variable lists. For every i, j ∈ 0..n-1:

forward[i] = j   ⇔   inverse[j] = i

Use when modelling the same relation in both directions is more natural than picking one — assignment problems where you want both "which machine runs task i?" and "which task runs on machine j?", scheduling models that switch between "what is in slot t?" and "when is event e?", or any permutation where some constraints are cleaner on the forward map and others on the inverse.

// Tasks ↔ Machines bijection. Forbid task 0 → machine 0.
final p = Problem()
  ..addVariables(['task0','task1','task2','task3'], [0,1,2,3])
  ..addVariables(['machine0','machine1','machine2','machine3'], [0,1,2,3])
  ..addInverse(['task0','task1','task2','task3'],
               ['machine0','machine1','machine2','machine3']);
// Once forbidden via the forward side, the inverse side gets
// pruned automatically:
p.addStringConstraint('task0 != 0');

Decomposes into n² binary constraints so AC-3 propagates them effectively. The channelling implies both lists are partial permutations of 0..n-1 — you don't need to add addAllDifferent separately for either side.

circuit dispatches to a cycle-detection propagator that maintains the partial chains formed by singleton-domain edges, prunes values that would close a premature sub-cycle, and enforces successor uniqueness (no value can be the successor of more than one variable). Composing with addAllDifferent adds Régin's hyper-arc-consistent pruning on top of the uniqueness already enforced by the circuit propagator, giving even stronger early filtering on hard instances. bin_packing has no built-in capacity notion; constrain the binLoads variables (or run minimize over them) to express per-bin limits and balancing objectives.

DFA-checked Sequences: regular

The regular constraint takes a sequence of variables and a Dfa (deterministic finite automaton) and accepts only sequences the DFA accepts. Useful for any sequencing rule expressible as a finite automaton: pattern matching, run-length bounds, at-most-k counting, alternation requirements.

// "At most 2 morning shifts across the week."
// State = how many M seen so far; state 3 is the trap.
final dfa = Dfa(
  numStates: 4,
  start: 0,
  accepting: {0, 1, 2},
  transitions: {
    0: {'M': 1, 'A': 0, 'N': 0},
    1: {'M': 2, 'A': 1, 'N': 1},
    2: {'M': 3, 'A': 2, 'N': 2},
    3: {'M': 3, 'A': 3, 'N': 3},
  },
);

final p = Problem()
  ..addVariables(['mon', 'tue', 'wed', 'thu', 'fri'], ['M', 'A', 'N'])
  ..addRegular(['mon', 'tue', 'wed', 'thu', 'fri'], dfa);

The Dfa value type is exposed from the top-level library import. Symbols are dynamic to accommodate any variable value type (int, String, custom objects).

Some constraints can be expressed either as a regular DFA or with the cardinality helpers (addAmongExactly, addGcc). Prefer the specialized helper when one applies — it's clearer and as fast or faster; reach for regular when the rule has positional structure (adjacency, run-length, alternation) that cardinality counts can't capture.

The engine dispatches addRegular to a partial-state propagator (Pesant 2004): per-position forward + backward reachable state sets are computed from the DFA and the current domains, and any value whose transition lies on no accepting path is pruned. This achieves generalized arc consistency on the regular constraint — pruning happens during search rather than only at leaves, so DFAs that forbid an early symbol class detect infeasibility immediately instead of after exhaustive enumeration.

Soft Constraints / MaxCSP

When some constraints are preferences rather than hard requirements, declare them soft with a non-negative integer weight and call maximizeSatisfaction to find the feasible assignment that maximizes the total weight of satisfied soft constraints:

final p = Problem()..addVariable('X', [1, 2, 3]);
p.addReifiedEquals('b1', 'X', 1);
p.addReifiedEquals('b2', 'X', 2);
p.addReifiedEquals('b3', 'X', 3);
p.declareSoft('b1', 1);   // weight 1 if X = 1
p.declareSoft('b2', 5);   // weight 5 if X = 2  (highest)
p.declareSoft('b3', 1);   // weight 1 if X = 3

final result = await p.maximizeSatisfaction();
// → X = 2 (the weight-5 soft wins)

The one-step helper bundles reification:

final boolName = p.addSoftConstraint(
  10,                            // weight
  ['X', 'Y'],                    // vars the predicate reads
  (a) => (a['X'] as int) + (a['Y'] as int) == 4,
);

Hard constraints (added the usual way) must still hold; 'FAILURE' is returned if no feasible assignment exists at all. The original problem is not mutated.

Luby Restart Strategy

On hard instances where chronological backtracking gets stuck in an early-doomed subtree, restarting from the root with a different value ordering finds a solution dramatically faster. getSolutionWithRestarts uses the universal Luby schedule (Luby, Sinclair & Zuckerman, 1993):

final result = await p.getSolutionWithRestarts(
  seed: 42,           // reproducibility
  scale: 100,         // backtracks per Luby unit
  maxRestarts: 50,    // cap total effort
  useDomWdeg: true,   // optional: pair with dom/wdeg
);

Each attempt has a scale × luby(i) backtrack budget. The Luby sequence is 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 4, 8, ... — a provably-optimal universal restart policy. Returns 'FAILURE' if any attempt completes the tree without finding a solution (proves infeasibility) or if maxRestarts is exhausted.

dom/wdeg Variable Heuristic

getSolutionWithDomWdeg() swaps the default MRV heuristic for dom/wdeg (Boussemart, Hemery, Lecoutre, Sais, 2004), which biases variable selection toward variables touching constraints that have recently caused failures. Often outperforms MRV on structured industrial problems. Composes naturally with restarts via the useDomWdeg: true flag on getSolutionWithRestarts.

VSIDS-Style Variable Activity

getSolutionWithActivity() uses a VSIDS-style per-variable activity heuristic (Moskewicz, Madigan, Zhao, Zhang, Malik, 2001 — originally the Chaff SAT solver), adapted to CSPs. On every propagation conflict, each variable in the failing constraint's scope gets a bump; bump magnitudes grow multiplicatively per conflict, so recent conflicts dominate the score (the standard MiniSat trick — mathematically equivalent to uniformly decaying every existing score, but O(1) per conflict instead of O(|vars|)).

Variable selection minimizes dom_size / (1 + activity), mirroring dom/wdeg's dom_size / wdeg shape — before any conflicts the ratio reduces to MRV, and as conflicts accumulate the picker gravitates toward variables that have been near recent failures.

final sol = await p.getSolutionWithActivity();

Composes with restarts (useVsids: true on getSolutionWithRestarts), forward checking, and conflict-directed backjumping. When dom/wdeg and VSIDS are both enabled, VSIDS wins the picker; both bump tables are still updated. Useful complement to dom/wdeg for SAT-style instances and for problems where the "guilty" structure shifts over the course of search — VSIDS's decaying bumps react faster than dom/wdeg's monotone weights.

Impact-Based Search (IBS)

getSolutionWithImpact() uses Impact-Based Search (Refalo, 2004 — "Impact-Based Search Strategies for Constraint Programming", CP 2004). After every decision the engine measures the impact of pinning (variable, value): the fraction of the joint search space (product of remaining domain sizes) that propagation eliminated. A failed propagation contributes impact 1.0 (the entire branch is gone); a successful one contributes 1 - exp(logP_after - logP_before), clamped to [0, 1]. Per-(var, value) running means are stored, so IBS learns from every decision — not just failures, as dom/wdeg and VSIDS do.

Variable selection minimizes dom_size / (1 + Σ_a I(v, a)) where the sum is over values currently in v's domain. Pre-observation this reduces to MRV; as impacts accumulate the picker gravitates toward variables whose values historically prune the largest fraction of the search space.

final sol = await p.getSolutionWithImpact();

Composes with restarts (useImpact: true on getSolutionWithRestarts), forward checking, SAC preprocessing, and conflict-directed backjumping. When all of useImpact, useVsids, and useDomWdeg are set, the picker dispatch order is impact → VSIDS → dom/wdeg; the other heuristics' bump tables continue to update so the picker choice is independent of which conflicts were observed.

IBS is the most informative of the three conflict-driven heuristics on problems with a wide spread of per-decision pruning — its score combines both how often a decision leads to failure (via the impact- 1.0 contribution) and how strongly successful decisions prune. The canonical comparison surface is the same one VSIDS competes on: structured combinatorial problems where MRV's tie-breaking is arbitrary.

Last-Conflict Heuristic

getSolutionWithLastConflict() layers Last-Conflict reasoning (Lecoutre 2009 — "Reasoning from last conflict(s) in constraint programming", Artificial Intelligence 173) on top of whichever underlying picker you choose. After every propagation failure, the engine records the variable being pinned at the failure point. The next time the picker runs, if that variable is still unassigned, the picker returns it directly — focusing search on the conflict cause — instead of consulting the underlying heuristic. When the recorded variable becomes assigned (via propagation or via the decision pin), the picker falls through.

LC is a wrapper, not a sibling heuristic: it modifies the variable choice without changing the score functions. Pass useDomWdeg:, useVsids:, or useImpact: to choose the underlying picker. The canonical deployment from Lecoutre's experiments is LC + dom/wdeg:

final sol = await p.getSolutionWithLastConflict(useDomWdeg: true);

Composes with restarts (useLastConflict: true on getSolutionWithRestarts), forward checking, SAC preprocessing, and conflict-directed backjumping. On the in-repo benchmark, LC+dom/wdeg wins wall-clock on UNSAT pigeonhole CNF and matches dom/wdeg on feasible n-queens — see dart run benchmark/benchmark.dart for the full comparison (the "heuristic comparisons" section runs MRV / dom/wdeg / VSIDS / IBS / LC+dom-wdeg head-to-head on five problems with a 5-rep-warm-up + 25-rep-median harness).

Conflict-Directed Backjumping (CBJ)

Pass enableConflictBackjumping: true to any backtracking solver entry point to opt into CBJ (Prosser, 1993) instead of chronological backtracking. After all candidate values for a decision variable have failed propagation, the engine jumps directly to the deepest earlier variable that participated in some propagation failure for the current variable — skipping intermediate decisions that couldn't have mattered to those failures.

final sol = await p.getSolution(enableConflictBackjumping: true);
print(p.lastStats!.backjumps);            // how many times the
                                          // engine jumped on a
                                          // conflict
print(p.lastStats!.backjumpLevelsSkipped); // total decision levels
                                          // skipped past chrono BT

CBJ is sound and complete; only the choice of backtrack target differs from the default. Composes with forward checking, restarts, dom/wdeg, VSIDS activity, IBS, LC, and the optimization solvers (minimize, maximize). Off by default because plain chronological backtracking has zero per-decision overhead — CBJ adds a coarse trail walk on each propagation failure and a small set per decision frame, paying for itself only on problems where the failure cause is genuinely far from the most recent decision.

See doc/cbj.md for the algorithm, the conflict-cause approximation, the integration with each search variant, and worked-example benchmarks showing when CBJ helps and when it doesn't.

Linear Arithmetic Constraints

For arithmetic constraints of the form Σ coeffs[i] · vars[i] ∘ bound (with ∘ ∈ {==, ≤, ≥}), use the dedicated linear API. The engine dispatches these to a bounds-consistency propagator that computes each variable's interval contribution and prunes values inconsistent with the constraint's overall bounds — much stronger than predicate-only n-ary encoding, especially when the free neighborhood exceeds the engine's GAC work bound.

final p = Problem()
  ..addVariables(['A', 'B', 'C'], [1, 2, 3, 4, 5])
  ..addLinearEquals(['A', 'B', 'C'], [1, 1, 1], 9);     // A + B + C = 9
final s = await p.getSolution();

// Weighted equalities:
p.addLinearEquals(['X', 'Y'], [2, 3], 12);              // 2X + 3Y = 12

// Inequalities:
p.addLinearLeq(['A', 'B', 'C'], [1, 1, 1], 10);         // A + B + C ≤ 10
p.addLinearGeq(['A', 'B', 'C'], [1, 1, 1], 5);          // A + B + C ≥ 5

// Mixed-sign coefficients (works for reformulating equations):
p.addLinearEquals(['A', 'B'], [2, -1], 3);              // 2A - B = 3

Coefficients may be positive, negative, or zero. All involved variables must have numeric domains; this is checked at constraint registration time. The propagator handles bounds consistency on integer and floating-point domains alike.

Why it matters. Predicate-only encodings of large arithmetic constraints (such as the classic SEND + MORE = MONEY cryptarithmetic) bail out of GAC support search when the free neighborhood is too big to enumerate, so the engine effectively falls back to pure search. The linear propagator computes interval bounds in O(n) per call and prunes immediately. On the SEND + MORE benchmark, the linear form solves in ~1 ms vs ~1.8 s for the predicate-only encoding (~1800× speedup).

Consistency Level

Every backtracking entry point accepts an optional consistency: ConsistencyLevel parameter that selects how strongly the engine propagates after each decision:

• ConsistencyLevel.arcConsistency (default) — full AC-3 on binary constraints and GAC on n-ary constraints. Each domain change requeues the constraints reachable through the changed variable and runs to a fixed point. Strongest pruning per decision, almost always a net win on structured problems.
• ConsistencyLevel.forwardChecking — revise each constraint touching the just-assigned variable once. A revise that merely narrows a neighbor's domain does NOT trigger further work; a revise that assigns a neighbor (reduces its domain to a singleton) does cascade once, so newly-deduced assignments still have their constraints checked. Cheapest per decision; helpful on problems with loose constraint graphs or where each constraint is already strong on its own.
• ConsistencyLevel.singletonArcConsistency — runs AC-3 / GAC during search just like arcConsistency, plus an extra preprocessing pass at the top of search (Debruyne & Bessière, 1997 — algorithm SAC-1). For every (variable, value) pair currently in some domain, the engine tentatively pins the variable to that value, runs propagation, and prunes the value if any domain wipes; it repeats the whole pass until a fixpoint. SAC pruning is strictly stronger than AC and catches root infeasibility that AC alone misses (e.g. x == y ∧ y == z ∧ x != z over {1, 2, 3} is AC-consistent but SAC-empty). More expensive at the root than arcConsistency and adds nothing to per-decision cost. Useful when the root domains contain a lot of dead-end values that survive AC but die quickly under any single-variable pin.

// Single solution with FC.
final p = Problem()
  ..addVariables(['A', 'B', 'C'], [1, 2, 3])
  ..addAllDifferent(['A', 'B', 'C']);
final s = await p.getSolution(
  consistency: ConsistencyLevel.forwardChecking,
);

// Composes with optimization, dom/wdeg, restarts, and the streaming API:
await p.minimize('A', consistency: ConsistencyLevel.forwardChecking);
await p.getSolutionWithDomWdeg(
  consistency: ConsistencyLevel.forwardChecking,
);
await p.getSolutionWithRestarts(
  seed: 1,
  consistency: ConsistencyLevel.forwardChecking,
);
await for (final s in p.getSolutions(
  consistency: ConsistencyLevel.forwardChecking,
)) { /* ... */ }

The choice is purely a propagation knob — both modes are sound and complete; they only differ in how much pruning they do per decision. SolverStats.binaryRevises / naryRevises let you compare the work done by each mode on your problem.

Conflict Explanation (MUS)

When a model is infeasible the solver returns the literal 'FAILURE' — sound but uninformative. For a model with dozens of constraints, finding the conflict by inspection is a debugging nightmare. Problem.findMinimalUnsatisfiableSubset (in the ConflictExplanation extension) identifies a minimal unsatisfiable subset (MUS): a subset of the posted constraints that's still infeasible, and from which removing any single constraint yields a satisfiable subproblem.

final p = Problem();
p.addVariables(['a', 'b', 'c'], [1, 2]);
p.addConstraint(['a', 'b'], (dynamic a, dynamic b) => a != b);
p.addConstraint(['b', 'c'], (dynamic b, dynamic c) => b != c);
p.addConstraint(['a', 'c'], (dynamic a, dynamic c) => a != c);
p.addConstraint(['a', 'b'],
    (dynamic a, dynamic b) => (a as num) < (b as num)); // redundant

final mus = await p.findMinimalUnsatisfiableSubset();
if (mus == null) {
  print('Satisfiable — no explanation needed.');
} else {
  for (final ref in mus) {
    print(ref); // binary(a, b), binary(b, c), binary(a, c)
  }
}

The triangle has no 2-coloring; all three inequality edges are in the MUS. The redundant a < b is dropped automatically. Returns null when the problem has at least one solution.

Algorithm: classic deletion-based MUS (Bakker et al. 1993, Junker 2001) — linearly probe each posted constraint, drop it permanently if the residual stays unsat, otherwise restore. O(n) calls to CSP.solve where n is the number of user-posted constraints; each solve runs ordinary AC-3 search. Returned MUS is locally minimal (no single deletion preserves infeasibility), not necessarily the smallest unsat subset (NP-hard in general).

Pass cancelToken: to bound the work. Cancellation in step 1 (the initial satisfiability check) returns null; cancellation in step 2 (the deletion loop) returns the current kept set — still unsatisfiable but possibly non-minimal. Pass consistency: to control propagation strength of the internal solves.

For larger models with small MUSes, the sibling findMinimalUnsatisfiableSubsetQuickXplain runs the divide-and- conquer QuickXplain algorithm (Junker 2004) instead: O(k · log(n/k)) calls to CSP.solve where k is the MUS size and n is the total constraint count, vs the deletion pass's O(n). Same return shape and same ConstraintRef semantics; different MUSes may surface (both are locally minimal but not the smallest possible). QuickXplain cancellation always returns null — its recursion has no sound mid-flight kept set to surface.

final mus = await p.findMinimalUnsatisfiableSubsetQuickXplain();

Every primary constraint helper on Problem accepts an optional label: parameter (any String) that surfaces on ConstraintRef.label and in toString. Without it, refs print as linearLeq(w0, w1, w2); with it, linearLeq[max-load](w0, w1, w2). The label is for display only — equality on ConstraintRef is keyed by id alone — but it makes MUS output dramatically more legible on models where the same kind of constraint appears many times:

p.addLinearLeq(['w0', 'w1', 'w2'], [1, 1, 1], 3, label: 'max-load');
p.addLinearGeq(['w0', 'w1', 'w2'], [1, 1, 1], 10, label: 'min-load');
final mus = await p.findMinimalUnsatisfiableSubset();
print(mus);
// [linearLeq[max-load](w0, w1, w2), linearGeq[min-load](w0, w1, w2)]

Decomposed helpers (addInverse, addLexChain, addValuePrecedence, binary addAllEqual) propagate the label to every decomposed piece.

See doc/conflict-explanation.md for a worked example, notes on granularity / decomposition, and the "which algorithm to call" guidance.

Cancellation and Timeouts

Every backtracking solver and the min-conflicts runner accept an optional cancelToken: CancellationToken parameter. A cancelled token aborts the search at the next checkpoint and the entry point returns the literal 'FAILURE' — the same shape as an unsatisfiable problem, so distinguish cancel from infeasibility by inspecting token.isCancelled after the call.

import 'dart:async';

final token = CancellationToken();
// Cancel after a deadline; this works for any solver entry point.
Timer(Duration(seconds: 5), token.cancel);
final result = await p.getSolution(cancelToken: token);
if (result == 'FAILURE' && token.isCancelled) {
  print('Timed out.');
}

The engine cooperatively yields to the event loop on every ~100 decisions (and the min-conflicts runner on every ~200 iterations), which is also what lets a wrapping Future.timeout(...) actually fire on an otherwise CPU-bound solve:

try {
  await p.getSolution().timeout(Duration(seconds: 5));
} on TimeoutException {
  print('Took longer than 5 seconds.');
}

Both forms work without spawning an isolate — the search stays on the main isolate but yields often enough that timers and stream listeners get their turns. The fast path is one integer compare per decision; the yield itself amortizes to well under 1% of search wall-clock on every benchmark in benchmark/benchmark.dart. For cases where you also want to free the main isolate's CPU budget, see "Solving on a worker isolate" below.

When cancelling an optimization (minimize / maximize), the result is 'FAILURE' regardless of whether an improving incumbent was found before the cancel. The current API doesn't expose the last-seen incumbent; if you need it, run an enumerating getSolutions and track the best yourself.

Solving on a worker isolate

Cooperative cancellation makes a CPU-bound solve responsive to timers and .timeout(...), but the search still runs on the calling isolate — every yield is short and CPU pressure stays on the main thread. For workloads where you want to free the main isolate entirely (e.g. a Flutter app whose UI must stay smooth, or a server running multiple solves), the runner in isolate_runner.dart spawns the solve on a fresh worker isolate.

import 'package:dart_csp/dart_csp.dart';

// Top-level builder so the closure is sendable. The builder runs
// inside the worker, not on the caller's isolate.
Problem buildMySchedule() {
  final p = Problem();
  // ... addVariable / addConstraint / ...
  return p;
}

Future<void> main() async {
  final solution = await solveInIsolate(
    buildMySchedule,
    timeout: Duration(seconds: 30),
  );
  // CSP.lastStats has been populated with the worker's counters.
  print(solution);
}

Four entry points are exported alongside their in-process twins:

In-process	Worker-isolate
Problem.getSolution	solveInIsolate
Problem.getSolutions	solveAllInIsolate
Problem.minimize	minimizeInIsolate
Problem.maximize	maximizeInIsolate
Problem.solveWithLcg	solveWithLcgInIsolates (portfolio of N)

All four take a Problem Function() builder instead of a constructed Problem — predicate closures attached to a Problem generally aren't sendable across an isolate boundary, but a top-level builder closure is. The builder runs inside the worker to construct the problem fresh there.

Common options across all four:

• consistency: — same ConsistencyLevel enum as the in-process solvers.
• cancelToken: — main-isolate CancellationToken; cancelling it forwards a cancel signal to the worker, which aborts its search at the next checkpoint. The returned future / stream resolves with 'FAILURE' shortly after.
• timeout: — built-in deadline. When it fires the runner sends the worker a cancel signal, waits a brief grace window for the worker to flush, then hard-kills the isolate. Prefer this over wrapping await solveInIsolate(...).timeout(...) — the built-in path actually terminates the worker; a wrapping .timeout() would leave it running until natural completion.

The worker's CSP.lastStats is shipped back to the main isolate on normal completion, so the documented "stats populated when the future resolves" contract still holds. Stats are not copied when the runner short-circuits on a pre-cancelled token or when the hard-kill grace fires (the worker had no chance to flush).

If the builder itself throws, the exception surfaces on the main isolate as an IsolateRunnerException carrying the message and a stringified stack trace from inside the worker. The original exception object isn't recoverable across the boundary.

The runner is not available on Dart Web — dart:isolate doesn't exist there. The test file is marked @TestOn('vm') for the same reason. All other dart_csp surface continues to work on every platform Dart supports.

Parallel LCG portfolio + cooperative clause sharing

solveWithLcgInIsolates runs Problem.solveWithLcg across workerCount worker isolates as a portfolio — each worker gets a distinct seeds entry (and defaults to useVsids, so the seeds actually diversify the search). The first worker to find a solution wins and the rest are torn down; the aggregate is 'FAILURE' only once every worker has exhausted its search, so a cancelled or timed-out worker can never be mistaken for a genuine UNSAT proof. Same contract as solveInIsolate (Map on success, 'FAILURE' otherwise); the winner's SolverStats is copied into CSP.lastStats.

final solution = await solveWithLcgInIsolates(
  buildMyProblem,
  workerCount: 4,
  shareClauses: true,   // cooperative mode (default: portfolio only)
);

With shareClauses: true the portfolio becomes cooperative: each worker exports its short learned clauses (≤ maxSharedClauseLen, default 8 literals) to the parent, which re-broadcasts them to the siblings; every worker imports the others' clauses into its own learned-clause pool (SolverStats.importedClauses counts them). Sharing is always sound — every worker solves the same problem, so a clause learned by one is a valid nogood for all — and it never changes the verdict. Experimental, like parallel LNS.

Range-Domain Variables (Scheduling)

For variables whose domain is a contiguous integer range — for example the start time of a task in a schedule, where the value can be any minute in [0, horizon] — use addRangeVariable:

final p = Problem()
  ..addRangeVariable('start',    0, 100)   // start in [0, 100]
  ..addRangeVariable('duration', 1,   5)   // duration in [1, 5]
  ..addRangeVariable('end',      0, 100)   // end in [0, 100]
  ..addLinearEquals(['start', 'duration', 'end'], [1, 1, -1], 0);
final s = await p.minimize('end');

For range spans larger than 1024, the engine uses a compact (min, max) domain representation internally, with O(1) membership / length / bounds. Bounds-only reductions (e.g. from the linear arithmetic propagator) stay in the same form without allocating per-step domain lists. For smaller ranges, addRangeVariable is equivalent to addVariable(name, [for (var i = min; i <= max; i++) i]) and the engine uses the existing bitset representation.

Disjunctive scheduling: addNoOverlap

For unary-resource scheduling (one task at a time on a machine), use addNoOverlap(starts, durations). It posts, for every pair (i, j), the disjunction starts[i] + durations[i] <= starts[j] OR starts[j] + durations[j] <= starts[i], so the half-open intervals [start, start + duration) do not overlap. Durations are constants; for variable durations, model the end time as a separate variable and use the linear-arithmetic propagator to relate them.

// Three jobs on one machine, durations 4 / 3 / 2, horizon 20.
// Minimize makespan (the latest end time).
final p = Problem()
  ..addRangeVariable('s0', 0, 20)
  ..addRangeVariable('s1', 0, 20)
  ..addRangeVariable('s2', 0, 20)
  ..addRangeVariable('mk', 0, 20)
  ..addNoOverlap(['s0', 's1', 's2'], [4, 3, 2])
  ..addLinearGeq(['mk', 's0'], [1, -1], 4)  // mk >= s0 + 4
  ..addLinearGeq(['mk', 's1'], [1, -1], 3)  // mk >= s1 + 3
  ..addLinearGeq(['mk', 's2'], [1, -1], 2); // mk >= s2 + 2
final s = await p.minimize('mk');
// mk = 9 (sum of durations — they're forced to pack tight).

The current addNoOverlap posts O(n²) pairwise binary disjunctions. This is sound and compact; for stronger pruning with arbitrary integer capacity, use addCumulative (see below) — the unary case (capacity = 1, all demands = 1) is a strictly stronger encoding of the same constraint when paired with the time-table + energetic-reasoning propagators.

Cumulative resource scheduling

For a renewable resource with integer capacity that several tasks may share (machines with K slots, parallel workers, a shop with K looms, etc.), use addCumulative(starts, durations, demands, capacity). At every time t, the sum of demands[i] across tasks whose half-open interval [starts[i], starts[i] + durations[i]) covers t must not exceed capacity.

Tagged internally with a CumulativeSpec so the engine dispatches to two complementary filtering passes:

1. A time-table propagator (Beldiceanu & Carlsson 2002 style): compute each task's compulsory part — the interval the task must occupy in every feasible schedule — sum the compulsory parts into a global usage profile, and prune any start value that would push the profile above capacity at some time.
2. An energetic-reasoning pass (Baptiste, Le Pape & Nuijten 1999): over the relevant time windows, sum every task's minimum intersection energy; a window whose total exceeds capacity × width is infeasible, and the left-shift / right-shift slack bounds tighten earliest-start / latest-completion times that the time-table profile alone leaves loose. This catches energy conflicts in problems with slack (tight RCPSP-style schedules) well before the time-table would. The O(n³) pass is capped above 64 tasks (the time-table alone — still sound — covers those); under LCG learning the overload check still runs (an over-capacity window is a sound, explained conflict) while the bound adjustments are skipped (their prunes have no explanation companion). Pass useEnergeticReasoning: false to addCumulative to opt out of the pass entirely.

On tight RCPSP instances energetic reasoning more than pays back its per-propagation cost (plain backtracking, default MRV, 25-rep median — bench(cumulative)):

8-task cap-2 (UNSAT)	time-table only	+ energetic
energetic overload at root	871 dec / 43 ms	0 dec / 0.05 ms
energetic pruning in search	770 dec / 28 ms	112 dec / 8.8 ms

// Three tasks on a machine with 2 parallel slots.
// dur=2 dem=1, dur=2 dem=1, dur=2 dem=2; total resource-time = 8
// on capacity 2 ⇒ makespan lower bound 4 (the two dem=1 tasks run
// in parallel; the dem=2 task runs alone).
final p = Problem()
  ..addRangeVariable('s0', 0, 10)
  ..addRangeVariable('s1', 0, 10)
  ..addRangeVariable('s2', 0, 10)
  ..addRangeVariable('mk', 0, 10)
  ..addCumulative(
      ['s0', 's1', 's2'], [2, 2, 2], [1, 1, 2], 2)
  ..addLinearGeq(['mk', 's0'], [1, -1], 2)
  ..addLinearGeq(['mk', 's1'], [1, -1], 2)
  ..addLinearGeq(['mk', 's2'], [1, -1], 2);
final s = await p.minimize('mk');   // mk = 4

Composes naturally with multiple resource types: post one addCumulative per resource (RCPSP-style models). Setting capacity = 1 and every demand to 1 reduces addCumulative to disjunctive (unary) no-overlap, with stronger propagation than the pairwise-disjunction addNoOverlap encoding on tight schedules.

Set Variables

For variables that range over the subsets of a finite universe, use addSetVariable. Internally each universe element gets a 0/1 indicator variable, and the existing primitives (linear arithmetic for cardinality, generic n-ary for membership combinations) handle propagation. Solutions returned through every solve entry point expose set variables as Set<dynamic> of the included elements — the internal indicators are stripped from the result map.

const roster = ['alice', 'bob', 'carol', 'dave', 'erin'];
final p = Problem()
  ..addSetVariables(['Team', 'Bench'], universe: roster)
  ..addSetCardinality('Team', 3)
  ..addSetCardinality('Bench', 2)
  ..addSetDisjoint('Team', 'Bench')
  ..addRequiredInSet('Team', 'alice');

final result = await p.getSolution() as Map<String, dynamic>;
print(result['Team']);   // Set<dynamic> of 3 names including 'alice'
print(result['Bench']);  // Set<dynamic> of 2 names, disjoint from Team

Declaration

• addSetVariable(name, {universe, required, excluded}) — declares one set variable. required pins the listed elements in, excluded pins them out at declaration time.
• addSetVariables(names, {universe, required, excluded}) — convenience for declaring several set variables that share a universe and (optionally) pin sets.

Cardinality

• addSetCardinality(setName, k) — |setName| == k.
• addSetCardinalityRange(setName, minCard, maxCard) — minCard <= |setName| <= maxCard.
• addSetCardinalityVar(setName, countVar) — |setName| == countVar (compose with minimize / maximize to drive size optimization).

Element pins (post-declaration)

• addRequiredInSet(setName, element) — element ∈ setName.
• addExcludedFromSet(setName, element) — element ∉ setName.

Pairwise relations

• addSubset(subName, superName) — subName ⊆ superName.
• addSetEquals(a, b) — a == b (universes must match as sets).
• addSetDisjoint(a, b) — a ∩ b == ∅.

Ternary relations

• addSetUnion(a, b, result) — result == a ∪ b.
• addSetIntersection(a, b, result) — result == a ∩ b.
• addSetDifference(a, b, result) — result == a \ b.

All three set variables must share the same universe (as a set).

Escape hatch: memberIndicator

When you need to compose set membership with the reified, logical, linear, or arithmetic helpers, look up the underlying 0/1 indicator variable with memberIndicator(setName, element). For example, "if X assigns the value 7 then 7 must be in S":

p.addReifiedEquals('bX7', 'X', 7);
final ind = p.memberIndicator('S', 7);
p.addConstraint([ind, 'bX7'],
    (i, b) => !(b == 1 && i == 0));  // bX7 == 1 ⇒ ind == 1

See doc/set-variables.md for the indicator-decomposition model in full and for guidance on when the sugar is worth using vs. modeling the indicators directly.

Documentation

In-depth topical guides live in doc/:

• doc/algorithms.md — Backtracking, AC-3, GAC, MRV/LCV, and the cooperative-yield contract that lets .timeout() fire on a CPU-bound solve.
• doc/cbj.md — Conflict-directed backjumping: the algorithm, the coarse conflict-cause approximation, how to read backjumps / backjumpLevelsSkipped, when CBJ pays off and when it doesn't.
• doc/propagation-trace.md — The opt-in fine-grained propagation trace (onPropagation / solveWithTrace): the event schema, how a prune's cause mirrors MUS output, the maxEvents cap, zero-overhead-when-unset, and crossing the isolate boundary.
• doc/conflict-explanation.md — Conflict explanation via deletion-based MUS: the algorithm, how ConstraintRef granularity surfaces decomposed helpers, when MUS is useful (and when it's overkill), open follow-ups.
• doc/heuristics.md — Consolidated guide to the variable-ordering heuristics (MRV / dom-wdeg / VSIDS / IBS / LC): how each works, which to use when, picker dispatch order, composition rules, and a side-by-side bench snapshot.
• doc/cancellation.md — The unified story for CancellationToken, wrapping Future.timeout(...), and the worker-isolate runner — when to reach for each and how they compose.
• doc/string-constraints.md — Full grammar reference for the string parser, dispatch table to built-in factories, what's not supported.
• doc/multi-solutions.md — Decision tree for the streaming / counting / enumeration APIs.
• doc/min-conflicts.md — When (and when not) to use the local-search solver.
• doc/global-cardinality.md — among, nvalue, gcc and how to compose them for rostering, palette restriction, and chromatic-number-style optimization.
• doc/set-variables.md — Set-valued variables, the indicator-decomposition model, when the sugar pays off, and how to compose with the rest of the library.

See also STABILITY.md for the public-API stability tiers (stable vs experimental), the semver policy that governs each tier, and the documented gotchas (notably the single-static-slot nature of lastStats).

The CI workflow also runs dart doc . on every build to generate inline-API HTML to doc/api/.

Testing

The library includes a comprehensive test suite covering all functionality:

# Run all tests
dart test

# Run tests with coverage
dart pub global activate coverage
dart pub global run coverage:test_with_coverage

# Run specific test groups
dart test test/dart_csp_test.dart -n "Basic Problem Creation"
dart test test/dart_csp_test.dart -n "String Constraints"

# Run a specific test file
dart test test/builtin_and_parser_test.dart
dart test test/minconflicts_tests.dart
dart test test/multisolutions_tests.dart

Test Coverage

The test suite includes test cases covering:

• Basic Problem Creation: Variable addition, domain validation, constraint setup
• Binary Constraints: Two-variable relationships and consistency
• N-ary Constraints: Multi-variable constraints and complex relationships
• String Constraints: All parsing scenarios and edge cases
• Built-in Constraints: Every constraint factory function and extension method
• Complex Problems: Magic squares, N-Queens, Sudoku, map coloring
• Convenience Functions: Top-level utility functions
• Failure Cases: Over-constrained and unsolvable problems
• Problem Utilities: Validation, debugging, and introspection
• Edge Cases: Malformed constraints, missing variables, empty domains

Advanced Usage

Debugging and Problem Validation

final p = Problem();
p.addVariables(['A', 'B', 'C'], [1, 2, 3]);
p.addStringConstraint('A != B');

// Print problem summary
p.printSummary();

// Validate problem for common issues
final issues = p.validate();
if (issues.isEmpty) {
  print('Problem validation: ✓ No issues found');
} else {
  print('Problem validation issues:');
  for (final issue in issues) {
    print('  - $issue');
  }
}

Visualization and Monitoring

Monitor the solver's progress with callback functions:

void visualizer(
  Map<String, List<dynamic>> assigned,
  Map<String, List<dynamic>> unassigned,
) {
  print("\n--- Solver Step ---");
  print("Assigned: $assigned");
  print("Unassigned Domains: $unassigned");
}

final p = Problem();
// ... define problem ...
p.setOptions(
  timeStep: 100, // 100ms pause between steps
  callback: visualizer,
);

final solution = await p.getSolution();

Fine-grained propagation trace

For a step-by-step replay of propagation itself — which arc/constraint fired, which value left which domain, and whether the branch ended in a solution or a dead-end — register a PropagationObserver instead of (or alongside) the coarse callback above:

final trace = await p.solveWithTrace();
// trace.result    -> Map assignment, or 'FAILURE'
// trace.events    -> List<PropagationEvent> in emission order
// trace.truncated -> true if the maxEvents cap was hit

for (final e in trace.events) {
  switch (e.kind) {
    case PropagationEventKind.decision:
      print('decide ${e.variable} = ${e.value} @depth ${e.depth}');
    case PropagationEventKind.prune:
      print('prune ${e.variable}: -${e.removedValues} '
          '(${e.domainBefore} -> ${e.domainAfter}) by ${e.causeDescription}');
    case PropagationEventKind.domainWipeout:
      print('dead end: ${e.variable} emptied by ${e.causeDescription}');
    case PropagationEventKind.backtrack:
    case PropagationEventKind.backjump:
    case PropagationEventKind.solution:
      print(e);
  }
}

Each prune / domainWipeout carries the pruned variable, the removed value(s), the domain before→after, and the cause as causeKind + causeLabel + causeScope — the same kind / label / scope vocabulary the conflict-explanation MUS API uses, so a UI can render propagation causes exactly like conflict explanations. The observer is zero overhead when unset (every emission is guarded), is bounded by maxEvents (default 100000, with trace.truncated / CSP.lastTraceTruncated flagging truncation), and emits plain-map serializable events (toMap / fromMap) that cross the isolate boundary via solveInIsolateWithTrace(...) (and minimize / maximize / solveAll siblings). See doc/propagation-trace.md for the full event schema. The original coarse callback is unchanged and independent.

Performance Optimization

1. Use String Constraints: Often more readable and just as fast as built-ins
2. Use Built-in Constraints: addAllDifferent() is faster than custom lambdas
3. Restrict Domains Early: Smaller initial domains = faster solving
4. Strategic Constraint Ordering: Add most constraining constraints first
5. Consider Clues: For puzzles, pre-fill strategic positions to reduce search space

// Performance example: Magic square with a strategic clue.
//
// For a 3x3 magic square, the center cell must be 5. Pinning B2={5} and
// excluding 5 from every other cell's domain prunes the search by ~20x:
// with the clue the problem solves in ~5s; without it, ~100s.
//
// String constraints are NOT slower than the built-in helpers here. The
// parser dispatches 'X + Y + Z == 15' to the same `exactSum(15)` factory
// as `addExactSum`, so the choice is one of style, not performance.
final p = Problem();

// Strategic clue: the center cell of a 3x3 magic square is always 5.
p.addVariable('B2', [5]);

// Remaining variables exclude the clue value.
final otherPositions = ['A1', 'A2', 'A3', 'B1', 'B3', 'C1', 'C2', 'C3'];
p.addVariables(otherPositions, [1, 2, 3, 4, 6, 7, 8, 9]);

p.addAllDifferent(['A1', 'A2', 'A3', 'B1', 'B2', 'B3', 'C1', 'C2', 'C3']);

// Rows, columns, and diagonals all sum to 15. Either form works.
p.addStringConstraints([
  'A1 + A2 + A3 == 15',
  'B1 + B2 + B3 == 15',
  'C1 + C2 + C3 == 15',
  'A1 + B1 + C1 == 15',
  'A2 + B2 + C2 == 15',
  'A3 + B3 + C3 == 15',
  'A1 + B2 + C3 == 15',
  'A3 + B2 + C1 == 15',
]);

How string constraints relate to the built-in factories

When you write a recognized pattern like 'A + B + C == 15', the parser dispatches it to the corresponding built-in factory (exactSum(15) here) rather than a generic expression evaluator. The same applies to:

Pattern	Dispatches to
A + B + ... == k	exactSum(k)
A * B * ... == k	exactProduct(k)
A + B + ... >= k (/ >)	minSum(k) (k offset by 1e-3 for strict)
A + B + ... <= k (/ <)	maxSum(k)
A * B * ... >= k (/ >)	minProduct(k)
A * B * ... <= k (/ <)	maxProduct(k)
k <= A + B + ... <= m	sumInRange(k, m)
A != B != C ...	allDifferent()
A < B < C ... (/ <=)	strictlyAscendingInOrder() / ascendingInOrder()
A in [...] / A not in [...]	inSet() / notInSet()
A op N (single var vs constant)	specialized lambda
A op B (two vars, op ∈ <,<=,>,>=)	specialized binary lambda

Anything else falls through to a generic ExpressionEvaluator.evaluateBoolean on every check, which is substantially slower because it re-tokenizes the expression for each candidate assignment. If you find a constraint that's hot in your profile, try to phrase it so it hits one of the fast paths above, or call the factory directly.

API Reference

Problem Class (Builder) - Core Methods

Method	Parameters	Description
addVariable()	String name, List<dynamic> domain	Adds a single variable with its domain
addVariables()	List<String> names, List<dynamic> domain	Adds multiple variables sharing the same domain
addConstraint()	List<String> vars, Function predicate	Adds custom binary or n-ary constraint
addStringConstraint()	String constraint	Adds constraint from string expression
addStringConstraints()	List<String> constraints	Adds multiple string constraints
setOptions()	int? timeStep, CspCallback? callback	Sets visualization parameters
getSolution()	(none)	Builds and solves the problem

Problem Class - Built-in Constraint Extensions

Method	Parameters	Description
addAllDifferent()	List<String> variables	All variables have different values
addAllEqual()	List<String> variables	All variables have the same value
addExactSum()	List<String> vars, num sum, List<num>? multipliers	Variables sum to exact value
addSumRange()	List<String> vars, num min, num max, List<num>? multipliers	Sum within range
addExactProduct()	List<String> vars, num product	Variables multiply to exact value
addInSet()	List<String> vars, Set<dynamic> allowed	Variables must be from allowed set
addNotInSet()	List<String> vars, Set<dynamic> forbidden	Variables cannot be from forbidden set
addAscending()	List<String> variables	Variables in non-decreasing order
addStrictlyAscending()	List<String> variables	Variables in strictly increasing order
addDescending()	List<String> variables	Variables in non-increasing order

Problem Class - Debugging Extensions

Method	Returns	Description
printSummary()	void	Prints problem overview to console
validate()	List<String>	Returns list of potential issues
copy()	Problem	Creates a deep copy of the problem
clear()	void	Removes all variables and constraints
variableCount	int	Number of variables in problem
constraintCount	int	Number of constraints in problem

Built-in Constraint Factory Functions

Equality Constraints

• allDifferent() → NaryPredicate
• allDifferentBinary() → BinaryPredicate
• allEqual() → NaryPredicate
• allEqualBinary() → BinaryPredicate

Arithmetic Constraints

• exactSum(num target, {List<num>? multipliers}) → NaryPredicate
• minSum(num minimum, {List<num>? multipliers}) → NaryPredicate
• maxSum(num maximum, {List<num>? multipliers}) → NaryPredicate
• sumInRange(num min, num max, {List<num>? multipliers}) → NaryPredicate
• exactProduct(num target) → NaryPredicate
• minProduct(num minimum) → NaryPredicate
• maxProduct(num maximum) → NaryPredicate

Set Membership Constraints

• inSet(Set<dynamic> allowed) → NaryPredicate
• notInSet(Set<dynamic> forbidden) → NaryPredicate
• someInSet(Set<dynamic> values, int minimum) → NaryPredicate
• someNotInSet(Set<dynamic> values, int minimum) → NaryPredicate

Ordering Constraints

• ascendingInOrder(List<String> order) → NaryPredicate
• strictlyAscendingInOrder(List<String> order) → NaryPredicate
• descendingInOrder(List<String> order) → NaryPredicate

Note: Binary versions (suffix Binary) are available for 2-variable optimizations.

Convenience Functions

Function	Parameters	Description
solveProblem()	Map<String, List<dynamic>> variables, List<String> constraints	Solve with string constraints
solveAllDifferent()	List<String> variables, List<dynamic> domain	Quick all-different solver
solveSumProblem()	List<String> variables, List<dynamic> domain, num targetSum	Quick sum constraint solver

CspProblem Class (Manual Construction)

Property	Type	Required	Description
variables	Map<String, List<dynamic>>	✅	Variable names mapped to their domains
constraints	List<BinaryConstraint>	❌	Rules between pairs of variables
naryConstraints	List<NaryConstraint>	❌	Rules among groups of variables
cb	CspCallback?	❌	Visualization callback function
timeStep	int	❌	Delay in ms between visualization steps

Constraint Classes

BinaryConstraint

BinaryConstraint(
  String head,                           // First variable
  String tail,                           // Second variable  
  bool Function(dynamic, dynamic) predicate  // Validation function
)

NaryConstraint

NaryConstraint({
  required List<String> vars,                        // All involved variables
  required bool Function(Map<String, dynamic>) predicate  // Validation function
})

Type Definitions

typedef BinaryPredicate = bool Function(dynamic headVal, dynamic tailVal);
typedef NaryPredicate = bool Function(Map<String, dynamic> assignment);
typedef CspCallback = void Function(
  Map<String, List<dynamic>> assigned, 
  Map<String, List<dynamic>> unassigned
);

Project Structure

The library is organized into focused, modular components:

lib/
├── dart_csp.dart                 # Main library export and convenience functions
└── src/
    ├── types.dart                # Core type definitions and interfaces  
    ├── solver.dart               # CSP solver with backtracking, AC-3, GAC
    ├── problem.dart              # Problem builder class and extensions
    ├── builtin_constraints.dart  # Optimized constraint factory functions
    └── constraint_parser.dart    # String constraint parsing engine

example/
├── demo.dart                     # Comprehensive demo of all features
├── example.dart                  # Walkthrough of the three constraint APIs
├── multi_solutions.dart          # Enumerating all / first-N solutions
├── gencw.dart                    # Arithmetic crosswords puzzle generator
└── gensq.dart                    # Arithmetic square puzzle generator

test/
├── dart_csp_test.dart            # Core solver, builder, and parser tests
├── builtin_and_parser_test.dart  # Built-in constraint factories + expression eval
├── minconflicts_tests.dart       # Min-conflicts local search solver
└── multisolutions_tests.dart     # Streaming / multi-solution enumeration

doc/
├── algorithms.md                 # Backtracking, AC-3, GAC, MRV/LCV, min-conflicts
├── string-constraints.md         # Full string-constraint grammar reference
├── multi-solutions.md            # Enumeration / streaming API guide
└── min-conflicts.md              # Local search solver guide

Running the Demo

The comprehensive demo (example/demo.dart) showcases all library features with 11 different problems:

dart run example/demo.dart

Demo Contents

1. USA Map Coloring - Compares old vs new constraint methods
2. 8-Queens Problem - Classic backtracking demonstration
3. Sudoku Solver - Using built-in allDifferent constraints
4. All Different & Equal Constraints - Basic constraint examples
5. Sum Constraints - Arithmetic constraint varieties
6. Product Constraints - Multiplication-based rules
7. Set Membership - Value inclusion/exclusion constraints
8. Ordering Constraints - Sequential arrangement rules
9. Magic Square - Complex multi-constraint problem with random clues
10. Resource Allocation - Real-world optimization scenario
11. Class Scheduling - Timetabling with multiple constraint types

Each demo includes:

• Problem setup explanation
• Constraint definition examples
• Solution verification
• Performance timing

Performance Tips

1. Use String Constraints: Natural syntax with good performance
2. Use Built-in Constraints: Pre-optimized functions are faster than equivalent lambdas
3. Strategic Domain Reduction: Limit initial domains as much as possible
4. Constraint Ordering: Add most restrictive constraints first
5. Choose Appropriate Constraint Types: Use binary constraints for 2-variable rules
6. Consider Problem Structure: Add strategic "clues" to reduce search space

Performance Comparison

// Readable and fast: String constraints  
p.addStringConstraints(['A != B != C', 'A + B + C == 10']);

// Also fast: Built-in constraints
p.addAllDifferent(['A', 'B', 'C']);
p.addExactSum(['A', 'B', 'C'], 10);

// Slower but flexible: Custom lambda
p.addConstraint(['A', 'B', 'C'], (assignment) {
  final values = assignment.values.toSet();
  return values.length == assignment.length && 
         values.fold<num>(0, (sum, v) => sum + v) == 10;
});

// Fastest for 2 variables: Binary constraint
p.addConstraint(['A', 'B'], allDifferentBinary());

Contributing

Contributions are welcome! Please open an issue or pull request on the repository.

Development Setup

1. Clone the repository
2. Install dependencies: dart pub get
3. Run tests: dart test
4. Run the demo: dart run example/demo.dart
5. Run a walkthrough example: dart run example/example.dart

License

MIT — see LICENSE. See NOTICE for a short historical note on how the codebase reached its current MIT-clean state.

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