SAT and CSP/CP Solvers [complete search]

1. A General Nogood-Learning Frameworkfor Pseudo-Boolean Multi-Valued SAT*Siddhartha Jain Brown UniversityAshish Sabharwal IBM WatsonMeinolf Sellmann IBM Watson* to appear at AAAI-2011

2. SAT and CSP/CP Solvers [complete search] • CP Solvers: • work at a high level: multi-valued variables, linear inequalities, element constraints, global constraint (knapsack, alldiff, …) • have access to problem structure and specialized inference algorithms! • SAT: • problem crushed down to one, fixed, simple input format:CNF, Boolean variables, clauses • very simple inference at each node: unit propagation • yet, translating CSPs to SAT can be extremely powerful!E.g., SUGAR (winner of CSP Competitions 2008 & 2009) How do SAT solvers even come close to competing with CP? A key contributor: efficient and powerful “nogood learning”

3. SAT Solvers as Search Engines • Systematic SAT solvers have becomereally efficient at searching fast and learning from “mistakes” • E.g., on an IBM model checking instance from SAT Race 2006, with~170k variables, 725k clauses, solvers such as MiniSat and RSat roughly: • Make 2000-5000 decisions/second • Deduce 600-1000 conflicts/second • Learn600-1000 clauses/second (#clauses grows rapidly) • Restart every 1-2 seconds(aggressive restarts)

4. An Interesting Line of Work: SAT-X Hybrids • Goal: try to bring more structure to SAT and/or bring SAT-style techniques to CSP solvers • Examples: • Pseudo-Boolean solvers: inequalities over binary variables • SAT Module Theories (SMT): “attach” a theory T to a SAT solver • Lazy clause generation: record clausal reasons for domain changes in a CP solver • Multi-valued SAT: incorporate multi-valued semantics into SAT preserve the benefits of SAT solvers in a more general context strengthen unit propagation: unit implication variables (UIV) strengthen nogood-learning! [Jain-O’Mahony-Sellmann, CP-2010] Starting point for this work

5. Conflict Analysis: Example • Consider a CSP with 5 variables:X1, X2, X4 {1, 2, 3}X3  {1, …, 7}X5  {1, 2} • Pure SAT encoding: variables x11, x12, x13, x21, x22, x23, x31, …, x37, x51, x52 • What happens when we set X1 = 1, X2 = 2, X3 = 1? x11= true x22= true No more propagation, no conflict… really? C5 x31= true x51= true

6. Conflict Analysis: Incorporating UIVs • Unit implication variable (UIV):a multi-valued clause is “unit”as soon as all unassigned “literals”in it regard the same MV variable • SAT encoding, stronger propagation using UIVs: C1 x41 ≠ true x11= true C2 x22= true x32 ≠ true C5 C4 x31= true x51= true x43 = true conflict C3 x42 = true

7. What Shall We Learn? • Not a good idea to set x41 ≠ true and x31 = true learn the nogood(x41 = true || x31 = false) • Problem? When we backtrack and set, say, X3 to anything in {2, 3, 4, 5}, we end up performing exactly the same analysis again and again! • Solution: represent conflict graph nodes as variable inequations only C1 x41 ≠ true x11= true C2 x22= true x32 ≠ true C5 C4 x31= true x51= true x43 = true conflict C3 x42 = true

8. CMVSAT-1 • Use UIV rather than UIP • Use variable inequationsas thecorerepresentation:X1 = 1, X2 = 2, X3 = 1represented as X1 ≠ 2, X1 ≠ 3, … finer granularityreasoning!X3 doesn’t necessarily need to be 1for a conflict, it just cannot be 6 or 7 • Stronger learned multi-valued clause:(X4 = 1 || X3 = 6 || X3 = 7) • Upon backtracking, we immediatelyinfer X3 ≠ 1, 2, 3, 4, 5 !

9. This Work: Generalize This Framework • Core representation in implication graph • CMVSAT-1: variable inequations (X4 ≠ 1) • in general: primitive constraintsof the solver • example: linear inequalities (Y2 ≤ 5, Y3 ≥ 1) • Constraints • CMVSAT-1: multi-valued clauses (X4 = 1 || X3 = 6 || X3 = 7) • in general: secondary constraintssupported by the solver • example: (X1 = true || Y3≤ 4 ||X3 = Michigan) • Propagation of a secondary constraint Cs:entailment of a new primitive constraint from Cs and known primitives

10. This Work: Generalize This Framework • Learned nogoods • CMVSAT-1: multi-valued clauses (X4 = 1 || X3 = 6 || X3 = 7) • in general: disjunctions of negations of primitives(with certain desirable properties) • example: (X1 = true || Y3 ≤ 4 || X3 = Michigan) • Desirable properties of nogoods? • The “unit” part of the learned nogood that is implied upon backtrackingmust be representable as a set of primitives! • E.g., if Y3 is meant to become unit upon backtracking, then(X1 = true || Y3 ≤ 4 || Y3 ≥ 6 || X3 = Michigan) is NOT desirable • cannot represent Y3 ≤ 4 || Y3 ≥ 6 as a conjunction of primitives  •  upon backtracking, cannot propagate the learned nogood

11. Sufficient Conditions [details in AAAI-2011 paper] • System distinguishes between primitive and secondary constraints • Secondary constraint propagators: • Entail new primitive constraints • Efficiently provide a set of primitives constraints sufficient for the entailment • Can efficiently detect conflicting sets of primitives • and represent the disjunction of their negations (the “nogood”)as a secondary constraint • Certain sets of negated primitives (e.g., those arising from propagation upon backtracking) succinctly representable as a set of primitives • Branching executed as the addition of one or more* primitives Under these conditions, we can efficiently learn strong nogoods!

12. Abstract Implication Graph (under sufficient conditions) Cp: primary constraintsbranched upon or entailed at various decision levels Learned nogood: disjunction of negations of primitives in shaded nodes

13. General Framework: Example • X1 {true, false}X3  {NY, TX, FL, CA}X5 {r, g, b}X2, X4, X6, X7  {1, …, 100} • Branch on X1 ≠ true, X2 ≤ 50: Learn: (X3 = FL || X4 ≥ 31) Notes: The part of the nogood that is unit upon backtrackingneed not always regard the same variable! (e.g., when X ≤ Y is primitive) Neither is regarding the same variable sufficient for being a valid nogood!(e.g., X ≤ 4 || X ≥ 11 wouldn’t be representable)

14. Empirical Evaluation • Ideas implemented in CMVSAT-2 • Currently supported: • usual domain variables, and range variables • linear inequalities, e.g. (X1 + 5 X2 – 13 X3 ≤ 6) • disjunctions of equations and range constraintse.g. (X1 [5…60] || X2  [37…74] || X5 = b || X10 ≤ 15) • Comparison against: • SAT solver: Minisat[Een-Sorensson 2004, version 2.2.0] • CSP solver: Mistral[Hebrard 2008] • MIP solver: SCIP[Achterberg 2004, version 2.0.1] Encodings generated using Numberjack[Hebrard et al 2010, version 0.1.10-11-24]

15. Empirical Evaluation • Benchmark domains (100 instances of each) • QWH-C: weighted quasi-group with holes / Latin square completion • random cost cik {1, …, 10} assigned to cell (i,k) • cost constraint: sumik (cikXik) ≤ (sum of all costs) / 2 • size 25 x 25, 40% filled entries, all satisfiable • MSP-3: market split problem • notoriously hard for constraint solvers • partition 20 weighted items into 3 equal sets, 10% satisfiable • NQUEENS-W: weighted n-queens, 30x30 • average weight of occupied cells ≥ 70% of weightmax • size 30 x 30, random weights  {1,…,10}

16. Empirical Evaluation: Results SAT solver CSP solver MIP solver • CMVSAT-2 shows good performance across a variety of domains • Solved all 300 instances in < 4 sec on average • MiniSat not suitable for domains like QWH-C • Encoding (even “compact” ones like in Sugar) too large to generate or solve • 20x slower on MSP-3 • Mistral explores a much larger search space than needed • Lack of nogood learning becomes a bottleneck:e.g.: 3M nodes for QWH-C (36% solved), compared to 231 nodes for CMVSAT-2 • SCIP takes 100x longer on QWH-c, 10x longer on NQUEENS-W

17. Summary • A generalized framework for SAT-stylenogood-learning • extends CMVSAT-1 • low-overhead process, retains efficiency of SAT solvers • sufficient conditions: • primitive constraints • secondary constraints, propagation as entailment of primitives • valid cutsets in conflict graphs: facilitate propagation upon backtracking • other efficiency / representation criteria • CMVSAT-2: robust performance across a variety of problem domains • compared to a SAT solver, a CSP solver, a MIP solver • open: more extensive evaluation and comparison against, e.g., lazy clause generation, pseudo-Boolean solvers, SMT solvers • nonetheless, a promising and fruitful direction!