SAT and CSP/CP Solvers [complete search]

CSP-SAT Workshop, June 2011

IBM Watson Research Center, 2011 © 2011 IBM Corporation

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

IBM Research

© 2011 IBM Corporation

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”

2 June 18, 2011 A General Nogood-Learning Framework

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© 2011 IBM Corporation3 June 18, 2011 A General Nogood-Learning Framework

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– Learn 600-1000 clauses/second (#clauses grows rapidly)– Restart every 1-2 seconds (aggressive restarts)

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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

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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

x31 = true x51 = trueC5

No more propagation,no conflict… really?

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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:


x11 = true

x22 = true

x31 = true x51 = true

x41 ≠ true

x32 ≠ true

C5

C2

C1

x43 = trueC4

x42 = trueC3

conflict

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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


x11 = true

x22 = true

x31 = true x51 = true

x41 ≠ true

x32 ≠ true

C5

C2

C1

x43 = trueC4

x42 = trueC3

conflict

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CMVSAT-1

Use UIV rather than UIP

Use variable inequations as thecore representation:X1 = 1, X2 = 2, X3 = 1represented as X1 ≠ 2, X1 ≠ 3, …

finer granularity reasoning!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 !


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This Work: Generalize This Framework

Core representation in implication graph

– CMVSAT-1: variable inequations (X4 ≠ 1)

– in general: primitive constraints of the solver

– example: linear inequalities (Y2 ≤ 5, Y3 ≥ 1)

Constraints

– CMVSAT-1: multi-valued clauses (X4 = 1 || X3 = 6 || X3 = 7)

– in general: secondary constraints supported 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


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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

backtracking must 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


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Sufficient Conditions

1. System distinguishes between primitive and secondary constraints

2. Secondary constraint propagators:– Entail new primitive constraints

– Efficiently provide a set of primitives constraints sufficient for the entailment

3. Can efficiently detect conflicting sets of primitives– and represent the disjunction of their negations (the “nogood”)

as a secondary constraint

4. Certain sets of negated primitives (e.g., those arising from propagation upon backtracking) succinctly representable as a set of primitives

5. Branching executed as the addition of one or more* primitives

Under these conditions, we can efficiently learn strong nogoods!


[details in AAAI-2011 paper]

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Abstract Implication Graph (under sufficient conditions)


Cp: primary constraints branched upon or entailed at various decision levelsLearned nogood: disjunction of negations of primitives in shaded nodes

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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:1. The part of the nogood that is unit upon backtracking

need not always regard the same variable! (e.g., when X ≤ Y is primitive)2. Neither is regarding the same variable sufficient for being a valid nogood!

(e.g., X ≤ 4 || X ≥ 11 wouldn’t be representable)

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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]


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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 (cik Xik) ≤ (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}


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Empirical Evaluation: Results

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


SAT solver CSP solver MIP solver

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Summary

A generalized framework for SAT-style nogood-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!


SAT and CSP/CP Solvers [complete search]

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