SAT by MaxSAT From NP to Beyond NP and Back Again Joao - - PowerPoint PPT Presentation

SAT by MaxSAT

From NP to Beyond NP and Back Again

Joao Marques-Silva

(joint work with A. Ignatiev and A. Morgado)

University of Lisbon, Portugal

RTiKC Dagstuhl Seminar Schloss Dagstuhl, Germany

September 2017

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Why “SAT by MaxSAT”?

Knowledge Compilation Empowerment Horn LUBs Non-Clausal Primes Hitting Set Dualization Maximum Satisfiability (MaxSAT) Horn MaxSAT Other Problems MaxHS SAT to Horn MaxSAT Polynomial Time Refutations for PHP MaxSAT Resolution Core- Guided MaxSAT 2 / 33

Outline

Maximum Satisfiability Pigeonhole Formulas From SAT to Horn MaxSAT Polynomial Time Refutations of PHP Experimental Results ?

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Outline

Maximum Satisfiability Pigeonhole Formulas From SAT to Horn MaxSAT Polynomial Time Refutations of PHP Experimental Results ?

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What is Maximum Satisfiability (MaxSAT)?

x6 ∨ x2 ¬x6 ∨ x2 ¬x2 ∨ x1 ¬x1 ¬x6 ∨ x8 x6 ∨ ¬x8 x2 ∨ x4 ¬x4 ∨ x5 x7 ∨ x5 ¬x7 ∨ x5 ¬x5 ∨ x3 ¬x3

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What is Maximum Satisfiability (MaxSAT)?

x6 ∨ x2 ¬x6 ∨ x2 ¬x2 ∨ x1 ¬x1 ¬x6 ∨ x8 x6 ∨ ¬x8 x2 ∨ x4 ¬x4 ∨ x5 x7 ∨ x5 ¬x7 ∨ x5 ¬x5 ∨ x3 ¬x3

• Given unsatisfiable formula

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What is Maximum Satisfiability (MaxSAT)?

x6 ∨ x2 ¬x6 ∨ x2 ¬x2 ∨ x1 ¬x1 ¬x6 ∨ x8 x6 ∨ ¬x8 x2 ∨ x4 ¬x4 ∨ x5 x7 ∨ x5 ¬x7 ∨ x5 ¬x5 ∨ x3 ¬x3

• Given unsatisfiable formula, find largest satisfiable subset of clauses

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What is Maximum Satisfiability (MaxSAT)?

x6 ∨ x2 ¬x6 ∨ x2 ¬x2 ∨ x1 ¬x1 ¬x6 ∨ x8 x6 ∨ ¬x8 x2 ∨ x4 ¬x4 ∨ x5 x7 ∨ x5 ¬x7 ∨ x5 ¬x5 ∨ x3 ¬x3

• Given unsatisfiable formula, find largest satisfiable subset of clauses

MaxSAT Variants Hard Clauses? No Yes Weights? No Plain Partial Yes Weighted Weighted Partial

• Many practical applications

[e.g. SZGN17] 5 / 33

Many algorithms for practical MaxSAT

MaxSAT Algorithms

Branch & Bound Iterative Core Guided Iterative MHS Model Guided

No unit prop; No cl. learning All cls relaxed Relax cls given unsat cores Iterative MHS & SAT Relax cls given models

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Many algorithms for practical MaxSAT

MaxSAT Algorithms

Branch & Bound Iterative Core Guided Iterative MHS Model Guided

No unit prop; No cl. learning All cls relaxed Relax cls given unsat cores Iterative MHS & SAT Relax cls given models

• Most effective in practice: iterative MHS & core-guided [Many refs...]

– Solving formulas with tens/hundreds of thousands of clauses, and more – Instantiations of problem solving with SAT oracles

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A core-guided algorithm (MSU3)

x6 ∨ x2 ¬x6 ∨ x2 ¬x2 ∨ x1 ¬x1 ¬x6 ∨ x8 x6 ∨ ¬x8 x2 ∨ x4 ¬x4 ∨ x5 x7 ∨ x5 ¬x7 ∨ x5 ¬x5 ∨ x3 ¬x3 Example CNF formula

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A core-guided algorithm (MSU3)

x6 ∨ x2 ¬x6 ∨ x2 ¬x2 ∨ x1 ¬x1 ¬x6 ∨ x8 x6 ∨ ¬x8 x2 ∨ x4 ¬x4 ∨ x5 x7 ∨ x5 ¬x7 ∨ x5 ¬x5 ∨ x3 ¬x3 Formula is UNSAT; OPT ≤ |ϕ| − 1; Get unsat core

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A core-guided algorithm (MSU3)

x6 ∨ x2 ¬x6 ∨ x2 ¬x2 ∨ x1∨r1 ¬x1∨r2 ¬x6 ∨ x8 x6 ∨ ¬x8 x2 ∨ x4∨r3 ¬x4 ∨ x5∨r4 x7 ∨ x5 ¬x7 ∨ x5 ¬x5 ∨ x3∨r5 ¬x3∨r6 6

i=1 ri ≤ 1

Add relaxation variables and AtMostk, k = 1, constraint

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A core-guided algorithm (MSU3)

x6 ∨ x2 ¬x6 ∨ x2 ¬x2 ∨ x1∨r1 ¬x1∨r2 ¬x6 ∨ x8 x6 ∨ ¬x8 x2 ∨ x4∨r3 ¬x4 ∨ x5∨r4 x7 ∨ x5 ¬x7 ∨ x5 ¬x5 ∨ x3∨r5 ¬x3∨r6 6

i=1 ri ≤ 1

Formula is (again) UNSAT; OPT ≤ |ϕ| − 2; Get unsat core

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A core-guided algorithm (MSU3)

x6 ∨ x2∨r7 ¬x6 ∨ x2∨r8 ¬x2 ∨ x1∨r1 ¬x1∨r2 ¬x6 ∨ x8 x6 ∨ ¬x8 x2 ∨ x4∨r3 ¬x4 ∨ x5∨r4 x7 ∨ x5∨r9 ¬x7 ∨ x5∨r10 ¬x5 ∨ x3∨r5 ¬x3∨r6 10

i=1 ri ≤ 2

Add new relaxation variables and update AtMostk, k=2, constraint

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A core-guided algorithm (MSU3)

x6 ∨ x2∨r7 ¬x6 ∨ x2∨r8 ¬x2 ∨ x1∨r1 ¬x1∨r2 ¬x6 ∨ x8 x6 ∨ ¬x8 x2 ∨ x4∨r3 ¬x4 ∨ x5∨r4 x7 ∨ x5∨r9 ¬x7 ∨ x5∨r10 ¬x5 ∨ x3∨r5 ¬x3∨r6 10

i=1 ri ≤ 2

Instance is now SAT

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A core-guided algorithm (MSU3)

x6 ∨ x2∨r7 ¬x6 ∨ x2∨r8 ¬x2 ∨ x1∨r1 ¬x1∨r2 ¬x6 ∨ x8 x6 ∨ ¬x8 x2 ∨ x4∨r3 ¬x4 ∨ x5∨r4 x7 ∨ x5∨r9 ¬x7 ∨ x5∨r10 ¬x5 ∨ x3∨r5 ¬x3∨r6 10

i=1 ri ≤ 2

MaxSAT solution is |ϕ| − I = 12 − 2 = 10

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A core-guided algorithm (MSU3)

x6 ∨ x2∨r7 ¬x6 ∨ x2∨r8 ¬x2 ∨ x1∨r1 ¬x1∨r2 ¬x6 ∨ x8 x6 ∨ ¬x8 x2 ∨ x4∨r3 ¬x4 ∨ x5∨r4 x7 ∨ x5∨r9 ¬x7 ∨ x5∨r10 ¬x5 ∨ x3∨r5 ¬x3∨r6 10

i=1 ri ≤ 2

MaxSAT solution is |ϕ| − I = 12 − 2 = 10

AtMostk/PB constraints used Relaxed soft clauses become hard

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A core-guided algorithm (MSU3)

x6 ∨ x2∨r7 ¬x6 ∨ x2∨r8 ¬x2 ∨ x1∨r1 ¬x1∨r2 ¬x6 ∨ x8 x6 ∨ ¬x8 x2 ∨ x4∨r3 ¬x4 ∨ x5∨r4 x7 ∨ x5∨r9 ¬x7 ∨ x5∨r10 ¬x5 ∨ x3∨r5 ¬x3∨r6 10

i=1 ri ≤ 2

MaxSAT solution is |ϕ| − I = 12 − 2 = 10

AtMostk/PB constraints used Relaxed soft clauses become hard Some clauses not relaxed

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A core-guided algorithm (MSU3)

x6 ∨ x2∨r7 ¬x6 ∨ x2∨r8 ¬x2 ∨ x1∨r1 ¬x1∨r2 ¬x6 ∨ x8 x6 ∨ ¬x8 x2 ∨ x4∨r3 ¬x4 ∨ x5∨r4 x7 ∨ x5∨r9 ¬x7 ∨ x5∨r10 ¬x5 ∨ x3∨r5 ¬x3∨r6 10

i=1 ri ≤ 2

MaxSAT solution is |ϕ| − I = 12 − 2 = 10

AtMostk/PB constraints used Relaxed soft clauses become hard Some clauses not relaxed

Note: # of SAT oracle calls grows linear with solution cost!

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = ∅

• Find MHS of K:

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = ∅

• Find MHS of K: ∅

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = ∅

• Find MHS of K: ∅
• SAT(F \ ∅)?

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = ∅

• Find MHS of K: ∅
• SAT(F \ ∅)? No

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = ∅

• Find MHS of K: ∅
• SAT(F \ ∅)? No
• Core of F: {c1, c2, c3, c4}

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}}

• Find MHS of K: ∅
• SAT(F \ ∅)? No
• Core of F: {c1, c2, c3, c4}. Update K

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}}

• Find MHS of K:

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}}

• Find MHS of K: E.g. {c1}

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}}

• Find MHS of K: E.g. {c1}
• SAT(F \ {c1})?

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}}

• Find MHS of K: E.g. {c1}
• SAT(F \ {c1})? No

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}}

• Find MHS of K: E.g. {c1}
• SAT(F \ {c1})? No
• Core of F: {c9, c10, c11, c12}

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}, {c9, c10, c11, c12}}

• Find MHS of K: E.g. {c1}
• SAT(F \ {c1})? No
• Core of F: {c9, c10, c11, c12}. Update K

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}, {c9, c10, c11, c12}}

• Find MHS of K:

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}, {c9, c10, c11, c12}}

• Find MHS of K: E.g. {c1, c9}

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}, {c9, c10, c11, c12}}

• Find MHS of K: E.g. {c1, c9}
• SAT(F \ {c1, c9})?

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}, {c9, c10, c11, c12}}

• Find MHS of K: E.g. {c1, c9}
• SAT(F \ {c1, c9})? No

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}, {c9, c10, c11, c12}}

• Find MHS of K: E.g. {c1, c9}
• SAT(F \ {c1, c9})? No
• Core of F: {c3, c4, c7, c8, c11, c12}

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}, {c9, c10, c11, c12}, {c3, c4, c7, c8, c11, c12}}

• Find MHS of K: E.g. {c1, c9}
• SAT(F \ {c1, c9})? No
• Core of F: {c3, c4, c7, c8, c11, c12}. Update K

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}, {c9, c10, c11, c12}, {c3, c4, c7, c8, c11, c12}}

• Find MHS of K:

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}, {c9, c10, c11, c12}, {c3, c4, c7, c8, c11, c12}}

• Find MHS of K: E.g. {c4, c9}

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}, {c9, c10, c11, c12}, {c3, c4, c7, c8, c11, c12}}

• Find MHS of K: E.g. {c4, c9}
• SAT(F \ {c4, c9})?

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}, {c9, c10, c11, c12}, {c3, c4, c7, c8, c11, c12}}

• Find MHS of K: E.g. {c4, c9}
• SAT(F \ {c4, c9})? Yes,

e.g. x1 = x2 = 1, x3 = x4 = x5 = x6 = x7 = x8 = 0

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}, {c9, c10, c11, c12}, {c3, c4, c7, c8, c11, c12}}

• Find MHS of K: E.g. {c4, c9}
• SAT(F \ {c4, c9})? Yes,

e.g. x1 = x2 = 1, x3 = x4 = x5 = x6 = x7 = x8 = 0

• Terminate & return 2

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MaxSAT with Minimum Hitting Sets (MHS)

c1 = x6 ∨ x2 c2 = ¬x6 ∨ x2 c3 = ¬x2 ∨ x1 c4 = ¬x1 c5 = ¬x6 ∨ x8 c6 = x6 ∨ ¬x8 c7 = x2 ∨ x4 c8 = ¬x4 ∨ x5 c9 = x7 ∨ x5 c10 = ¬x7 ∨ x5 c11 = ¬x5 ∨ x3 c12 = ¬x3

K = {{c1, c2, c3, c4}, {c9, c10, c11, c12}, {c3, c4, c7, c8, c11, c12}}

• Find MHS of K: E.g. {c4, c9}
• SAT(F \ {c4, c9})? Yes,

e.g. x1 = x2 = 1, x3 = x4 = x5 = x6 = x7 = x8 = 0

• Terminate & return 2

Possibly many MHSes, with one SAT oracle call for each MHS!

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Recap Horn MaxSAT

• What is Horn MaxSAT?

– All soft clauses are Horn

◮ Most often, unit soft clauses

– All hard clauses are Horn

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Recap Horn MaxSAT

• What is Horn MaxSAT?

– All soft clauses are Horn

◮ Most often, unit soft clauses

– All hard clauses are Horn

• How hard is Horn MaxSAT?

– Max-2SAT (and so MaxSAT) is NP-hard

[GJS76]

– Decision K-SAT is NP-complete – Horn MaxSAT is NP-hard

[JS87]

– Decision K-HornSAT is NP-complete

[JS87] ◮ By definition, any problem in NP is reducible to K-HornSAT ◮ But ... 9 / 33

Why use Horn MaxSAT?

• Practical perspective:

– MaxSAT with MHSes is very efficient in practice – For Horn MaxSAT, we can replace SAT call (worst-case exponential) with LTUR call (worst-case linear)

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Why use Horn MaxSAT?

• Practical perspective:

– MaxSAT with MHSes is very efficient in practice – For Horn MaxSAT, we can replace SAT call (worst-case exponential) with LTUR call (worst-case linear)

• Theoretical perspective:

– Reducing SAT to Horn MaxSAT & applying a MaxSAT algorithm yields a new proof system

◮ MaxSAT resolution ◮ Core-guided algorithm(s) ◮ MaxHS-like algorithms ◮ ...

– Reducing PHP to SAT and then to Horn MaxSAT admits polynomial time refutations for some MaxSAT algorithms

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What can we solve with Horn MaxSAT?

[IMMS17,MSIM17]

SAT ≤P Horn MaxSAT and so CSP, ASP, SMT*, ... CSP ≤P Horn MaxSAT besides CSP ≤P SAT PHP ≤P Horn MaxSAT besides PHP ≤P SAT MaxClique ≤P Horn MaxSAT and so MinVC, MaxIS MinHS ≤P Horn MaxSAT and so MaxSP MinDS ≤P Horn MaxSAT

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What can we solve with Horn MaxSAT?

[IMMS17,MSIM17]

SAT ≤P Horn MaxSAT and so CSP, ASP, SMT*, ... CSP ≤P Horn MaxSAT besides CSP ≤P SAT PHP ≤P Horn MaxSAT besides PHP ≤P SAT MaxClique ≤P Horn MaxSAT and so MinVC, MaxIS MinHS ≤P Horn MaxSAT and so MaxSP MinDS ≤P Horn MaxSAT

• Most encodings of cardinality constraints are Horn

– Sequential counters; totalizers; sorting networks; (pairwise) (cardinality networks); bitwise (for AtMost1)

[S05,ES06,ANORC11,...] 11 / 33

What can we solve with Horn MaxSAT?

[IMMS17,MSIM17]

SAT ≤P Horn MaxSAT and so CSP, ASP, SMT*, ... CSP ≤P Horn MaxSAT besides CSP ≤P SAT PHP ≤P Horn MaxSAT besides PHP ≤P SAT MaxClique ≤P Horn MaxSAT and so MinVC, MaxIS MinHS ≤P Horn MaxSAT and so MaxSP MinDS ≤P Horn MaxSAT

• Most encodings of cardinality constraints are Horn

– Sequential counters; totalizers; sorting networks; (pairwise) (cardinality networks); bitwise (for AtMost1)

[S05,ES06,ANORC11,...]

• Some encodings of pseudo-Boolean constraints are Horn

– Local polynomial watchdog (LPW)

[BBR09] 11 / 33

What can we solve with Horn MaxSAT?

[IMMS17,MSIM17]

SAT ≤P Horn MaxSAT and so CSP, ASP, SMT*, ... CSP ≤P Horn MaxSAT besides CSP ≤P SAT PHP ≤P Horn MaxSAT besides PHP ≤P SAT MaxClique ≤P Horn MaxSAT and so MinVC, MaxIS MinHS ≤P Horn MaxSAT and so MaxSP MinDS ≤P Horn MaxSAT

• Most encodings of cardinality constraints are Horn

– Sequential counters; totalizers; sorting networks; (pairwise) (cardinality networks); bitwise (for AtMost1)

[S05,ES06,ANORC11,...]

• Some encodings of pseudo-Boolean constraints are Horn

– Local polynomial watchdog (LPW)

[BBR09]

Knapsack ≤P Horn MaxSAT

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What can we solve with Horn MaxSAT?

[IMMS17,MSIM17]

SAT ≤P Horn MaxSAT and so CSP, ASP, SMT*, ... CSP ≤P Horn MaxSAT besides CSP ≤P SAT PHP ≤P Horn MaxSAT besides PHP ≤P SAT MaxClique ≤P Horn MaxSAT and so MinVC, MaxIS MinHS ≤P Horn MaxSAT and so MaxSP MinDS ≤P Horn MaxSAT

• Most encodings of cardinality constraints are Horn

– Sequential counters; totalizers; sorting networks; (pairwise) (cardinality networks); bitwise (for AtMost1)

[S05,ES06,ANORC11,...]

• Some encodings of pseudo-Boolean constraints are Horn

– Local polynomial watchdog (LPW)

[BBR09]

Knapsack ≤P Horn MaxSAT

• Apparently, we live in a (mostly) Horn world!

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Outline

Maximum Satisfiability Pigeonhole Formulas From SAT to Horn MaxSAT Polynomial Time Refutations of PHP Experimental Results ?

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Pigeonhole formulas I

• Pigeonhole principle:

– Typical: if m + 1 pigeons are distributed by m holes, then at least

• ne hole contains more than one pigeon

– Alternative: there exists no injective function mapping from {1, 2, ..., m + 1} to {1, 2, ..., m}, for m ≥ 1

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Pigeonhole formulas I

• Pigeonhole principle:

– Typical: if m + 1 pigeons are distributed by m holes, then at least

• ne hole contains more than one pigeon

– Alternative: there exists no injective function mapping from {1, 2, ..., m + 1} to {1, 2, ..., m}, for m ≥ 1

• Propositional formulation:

Does there exist assignment such that the m + 1 pigeons can be placed into m holes?

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Pigeonhole formulas I

• Pigeonhole principle:

– Typical: if m + 1 pigeons are distributed by m holes, then at least

• ne hole contains more than one pigeon

– Alternative: there exists no injective function mapping from {1, 2, ..., m + 1} to {1, 2, ..., m}, for m ≥ 1

• Propositional formulation:

Does there exist assignment such that the m + 1 pigeons can be placed into m holes?

• Encoding: xij variables

1 1

pigeons holes

m + 1 i j m

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Pigeonhole formulas II – propositional encoding PHPm+1

m

• Variables:

– xij = 1 iff the ith pigeon is placed in the jth hole, 1 ≤ i ≤ m + 1, 1 ≤ j ≤ m

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Pigeonhole formulas II – propositional encoding PHPm+1

m

• Variables:

– xij = 1 iff the ith pigeon is placed in the jth hole, 1 ≤ i ≤ m + 1, 1 ≤ j ≤ m

• Constraints:

– Each pigeon must be placed in at least one hole, and each hole must not have more than one pigeon

m+1

i=1 AtLeast1(xi1, . . . , xim) ∧ m j=1 AtMost1(x1j, . . . , xm+1 j) 14 / 33

Pigeonhole formulas II – propositional encoding PHPm+1

m

• Variables:

– xij = 1 iff the ith pigeon is placed in the jth hole, 1 ≤ i ≤ m + 1, 1 ≤ j ≤ m

• Constraints:

– Each pigeon must be placed in at least one hole, and each hole must not have more than one pigeon

m+1

i=1 AtLeast1(xi1, . . . , xim) ∧ m j=1 AtMost1(x1j, . . . , xm+1 j)

• Example encoding, with pairwise encoding for AtMost1 constraint:

Constraint Clause(s) ∧m+1

i=1 AtLeast1(xi1, . . . , xim)

(xi1 ∨ . . . ∨ xim) ∧m

j=1AtMost1(x1j, . . . , xm+1 j)

∧m+1

r=2 ∧r−1 s=1 (¬xrj ∨ ¬xsj) 14 / 33

Outline

Maximum Satisfiability Pigeonhole Formulas From SAT to Horn MaxSAT Polynomial Time Refutations of PHP Experimental Results ?

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SAT reduces to Horn MaxSAT

F (x1 ∨ ¬x2 ∨ x3) ∧ (x2 ∨ x3) ∧ (¬x1 ∨ ¬x3)

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SAT reduces to Horn MaxSAT

F (x1 ∨ ¬x2 ∨ x3) ∧ (x2 ∨ x3) ∧ (¬x1 ∨ ¬x3)

• For each xi, create new variables pi and ni
• pi and ni cannot both be assigned 1:

– Add hard clause (¬pi ∨ ¬ni)

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SAT reduces to Horn MaxSAT

F (x1 ∨ ¬x2 ∨ x3) ∧ (x2 ∨ x3) ∧ (¬x1 ∨ ¬x3)

• For each xi, create new variables pi and ni
• pi and ni cannot both be assigned 1:

– Add hard clause (¬pi ∨ ¬ni)

• Reencode original clauses (as hard clauses):

– Literal xi replaced by ¬ni – Literal ¬xi replaced by ¬pi

• Goal is to assign value 1 to each variable, if possible:

– Add soft clauses (pi) and (ni)

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SAT reduces to Horn MaxSAT

F (x1 ∨ ¬x2 ∨ x3) ∧ (x2 ∨ x3) ∧ (¬x1 ∨ ¬x3)

• For each xi, create new variables pi and ni
• pi and ni cannot both be assigned 1:

– Add hard clause (¬pi ∨ ¬ni)

• Reencode original clauses (as hard clauses):

– Literal xi replaced by ¬ni – Literal ¬xi replaced by ¬pi

• Goal is to assign value 1 to each variable, if possible:

– Add soft clauses (pi) and (ni)

• All clauses are Horn

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SAT reduces to Horn MaxSAT

F (x1 ∨ ¬x2 ∨ x3) ∧ (x2 ∨ x3) ∧ (¬x1 ∨ ¬x3)

• For each xi, create new variables pi and ni
• pi and ni cannot both be assigned 1:

– Add hard clause (¬pi ∨ ¬ni)

• Reencode original clauses (as hard clauses):

– Literal xi replaced by ¬ni – Literal ¬xi replaced by ¬pi

• Goal is to assign value 1 to each variable, if possible:

– Add soft clauses (pi) and (ni)

• All clauses are Horn
• Original formula is satisfiable iff Horn MaxSAT formula can satisfy

n soft clauses (and the hard clauses)

– I.e., if there is an assignment to the n variables consistent with the

• riginal clauses

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SAT reduces to Horn MaxSAT (Cont.)

F (x1 ∨ ¬x2 ∨ x3) ∧ (x2 ∨ x3) ∧ (¬x1 ∨ ¬x3)

• Example:

– New variables: p1, p2, p3, n1, n2, n3 – Filter impossible assignments: {(¬p1 ∨ ¬n1), (¬p2 ∨ ¬n2), (¬p3 ∨ ¬n3)} – Original clauses reencoded: (¬n1 ∨ ¬p2 ∨ ¬n3) ∧ (¬n2 ∨ ¬n3) ∧ (¬p1 ∨ ¬p3) – Soft clauses: {(p1), (p2), (p3), (n1), (n2), (n3)}

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SAT reduces to Horn MaxSAT (Cont.)

F (x1 ∨ ¬x2 ∨ x3) ∧ (x2 ∨ x3) ∧ (¬x1 ∨ ¬x3)

• Example:

– New variables: p1, p2, p3, n1, n2, n3 – Filter impossible assignments: {(¬p1 ∨ ¬n1), (¬p2 ∨ ¬n2), (¬p3 ∨ ¬n3)} – Original clauses reencoded: (¬n1 ∨ ¬p2 ∨ ¬n3) ∧ (¬n2 ∨ ¬n3) ∧ (¬p1 ∨ ¬p3) – Soft clauses: {(p1), (p2), (p3), (n1), (n2), (n3)}

• Encoding is a variant of the dual-rail encoding, used since the mid

80s

[BBBCS87] 17 / 33

PHP as Horn MaxSAT

• New variables nij and pij, for each xij, 1 ≤ i ≤ m + 1, 1 ≤ j ≤ m
• The soft clauses S, with |S| = 2m(m + 1), are given by

{ (n11), . . . , (n1m), . . . , (nm+1 1), . . . , (nm+1 m), (p11), . . . , (p1m), . . . , (pm+1 1), . . . , (pm+1 m) }

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PHP as Horn MaxSAT

• New variables nij and pij, for each xij, 1 ≤ i ≤ m + 1, 1 ≤ j ≤ m
• The soft clauses S, with |S| = 2m(m + 1), are given by

{ (n11), . . . , (n1m), . . . , (nm+1 1), . . . , (nm+1 m), (p11), . . . , (p1m), . . . , (pm+1 1), . . . , (pm+1 m) }

• Clauses in P: P = {(¬nij ∨ ¬pij) | 1 ≤ i ≤ m + 1, 1 ≤ j ≤ m}

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PHP as Horn MaxSAT

• New variables nij and pij, for each xij, 1 ≤ i ≤ m + 1, 1 ≤ j ≤ m
• The soft clauses S, with |S| = 2m(m + 1), are given by

{ (n11), . . . , (n1m), . . . , (nm+1 1), . . . , (nm+1 m), (p11), . . . , (p1m), . . . , (pm+1 1), . . . , (pm+1 m) }

• Clauses in P: P = {(¬nij ∨ ¬pij) | 1 ≤ i ≤ m + 1, 1 ≤ j ≤ m}
• AtLeast1 constraints encoded as Li, 1 ≤ i ≤ m + 1
• AtMost1 constraints encoded as Mj, 1 ≤ j ≤ m

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PHP as Horn MaxSAT

• New variables nij and pij, for each xij, 1 ≤ i ≤ m + 1, 1 ≤ j ≤ m
• The soft clauses S, with |S| = 2m(m + 1), are given by

{ (n11), . . . , (n1m), . . . , (nm+1 1), . . . , (nm+1 m), (p11), . . . , (p1m), . . . , (pm+1 1), . . . , (pm+1 m) }

• Clauses in P: P = {(¬nij ∨ ¬pij) | 1 ≤ i ≤ m + 1, 1 ≤ j ≤ m}
• AtLeast1 constraints encoded as Li, 1 ≤ i ≤ m + 1
• AtMost1 constraints encoded as Mj, 1 ≤ j ≤ m
• Full reduction of PHP to Horn MaxSAT

H, S =

• ∧m+1

i=1 Li ∧ ∧m j=1Mj ∧ P, S

• 18 / 33

PHP as Horn MaxSAT

• New variables nij and pij, for each xij, 1 ≤ i ≤ m + 1, 1 ≤ j ≤ m
• The soft clauses S, with |S| = 2m(m + 1), are given by

{ (n11), . . . , (n1m), . . . , (nm+1 1), . . . , (nm+1 m), (p11), . . . , (p1m), . . . , (pm+1 1), . . . , (pm+1 m) }

• Clauses in P: P = {(¬nij ∨ ¬pij) | 1 ≤ i ≤ m + 1, 1 ≤ j ≤ m}
• AtLeast1 constraints encoded as Li, 1 ≤ i ≤ m + 1
• AtMost1 constraints encoded as Mj, 1 ≤ j ≤ m
• Full reduction of PHP to Horn MaxSAT

H, S =

• ∧m+1

i=1 Li ∧ ∧m j=1Mj ∧ P, S

• No more than m(m + 1) clauses can be satisfied, due to P
• PHPm+1

m

is satisfiable iff there exists an assignment that satisfies the hard clauses H and m(m + 1) soft clauses from S

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PHP as Horn MaxSAT II

• Clauses in each Li and in each Mj, with pairwise encoding

Original Constraint Encoded To Clauses ∧m+1

i=1 AtLeast1(xi1, . . . , xim)

Li (¬ni1 ∨ . . . ∨ ¬nim) ∧m

j=1AtMost1(x1j, . . . , xm+1,j)

Mj ∧m+1

r=2 ∧r−1 s=1 (¬prj ∨ ¬psj) 19 / 33

PHP as Horn MaxSAT II

• Clauses in each Li and in each Mj, with pairwise encoding

Original Constraint Encoded To Clauses ∧m+1

i=1 AtLeast1(xi1, . . . , xim)

Li (¬ni1 ∨ . . . ∨ ¬nim) ∧m

j=1AtMost1(x1j, . . . , xm+1,j)

Mj ∧m+1

r=2 ∧r−1 s=1 (¬prj ∨ ¬psj)

• Note: constraints with key structural properties:

Constraint Variables Li (¬ni1 ∨ . . . ∨ ¬nim) Lk (¬nk1 ∨ . . . ∨ ¬nkm) Mj ∧m+1

r=2 ∧r−1 s=1 (¬prj ∨ ¬psj)

Ml ∧m+1

r=2 ∧r−1 s=1 (¬prl ∨ ¬psl)

– Variables in each Li disjoint from any other Lk and Mj, k = i – Variables in each Mj disjoint from any other Ml, l = j

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Outline

Maximum Satisfiability Pigeonhole Formulas From SAT to Horn MaxSAT Polynomial Time Refutations of PHP Experimental Results ?

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Main claims

Claim 1

Core-guided MaxSAT (e.g. MSU3) produces a lower bound on the number of falsified clauses ≥ m(m + 1) + 1 in polynomial time

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Main claims

Claim 1

Core-guided MaxSAT (e.g. MSU3) produces a lower bound on the number of falsified clauses ≥ m(m + 1) + 1 in polynomial time

Claim 2

MaxSAT resolution produces a lower bound on the number of falsified clauses ≥ m(m + 1) + 1 in polynomial time

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Main claims

Claim 1

Core-guided MaxSAT (e.g. MSU3) produces a lower bound on the number of falsified clauses ≥ m(m + 1) + 1 in polynomial time

Claim 2

MaxSAT resolution produces a lower bound on the number of falsified clauses ≥ m(m + 1) + 1 in polynomial time

Remark

Horn MaxSAT encoding enables polynomial time refutations of the unsatisfiability of PHP instances, using CDCL SAT solvers

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Proof of claim 1 – outline

• 1. Assume MSU3 MaxSAT algorithm

– Note: Suffices to analyze disjoint sets separately

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Proof of claim 1 – outline

• 1. Assume MSU3 MaxSAT algorithm

– Note: Suffices to analyze disjoint sets separately

• 2. Relate soft clauses with each Li and each Mj

– Each constraint disjoint from the others (but not from P)

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Proof of claim 1 – outline

• 1. Assume MSU3 MaxSAT algorithm

– Note: Suffices to analyze disjoint sets separately

• 2. Relate soft clauses with each Li and each Mj

– Each constraint disjoint from the others (but not from P)

• 3. Derive large enough lower bound on # of falsified clauses:
• Constr. type

# falsified cls # constr In total Li 1 i = 1, . . . , m + 1 m + 1 Mj m j = 1, . . . , m m · m m(m + 1) + 1

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Proof of claim 1 – outline

• 1. Assume MSU3 MaxSAT algorithm

– Note: Suffices to analyze disjoint sets separately

• 2. Relate soft clauses with each Li and each Mj

– Each constraint disjoint from the others (but not from P)

• 3. Derive large enough lower bound on # of falsified clauses:
• Constr. type

# falsified cls # constr In total Li 1 i = 1, . . . , m + 1 m + 1 Mj m j = 1, . . . , m m · m m(m + 1) + 1

• 4. Each increase in the value of the lower bound obtained by unit

propagation (UP)

– In total: polynomial number of (linear time) UP runs

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Proof of claim 1 – unit propagation steps I

Constr Hard cls Soft cls Relaxed clauses Updated AtMostk constr LB incr Li (¬ni1 ∨ . . . ∨ ¬nim) (ni1), . . . , (nim) (sil ∨ ni1), 1 ≤ l ≤ m m

l=1 sil ≤ 1

1 Mj (¬p1j ∨ ¬p2j) (p1j), (p2j) (r1j ∨ p1j), (r2j ∨ p2j) 2

l=1 rlj ≤ 1

1 Mj (¬p1j ∨ ¬p3j), (¬p2j ∨ ¬p3j), (r1j ∨ p1j), (r2j ∨ p2j), 2

l=1 rlj ≤ 1

(p3j) (r3j ∨ p3j) 3

l=1 rlj ≤ 2

1 · · · Mj (¬p1j ∨¬pm+1j), . . ., (¬pmj ∨ ¬pm+1j), (r1j ∨ p1j), . . ., (rmj ∨ pmj), m

l=1 rlj ≤ m − 1

(pm+1j) (rm+1j ∨ pm+1j) m+1

l=1 rlj ≤ m

1

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Proof of claim 1 – unit propagation steps II

Clauses Unit Propagation (pk+1 j) pk+1 j = 1 (¬p1j ∨¬pk+1 j), . . . , (¬pkj ∨¬pk+1 j) p1j = . . . = pkj = 0 (r1j ∨ p1j), . . . , (rkj ∨ pkj) r1j = . . . = rkj = 1 k

l=1 rlj ≤ k − 1

k

l=1 rlj ≤ k − 1

• ⊢1 ⊥
• Key points:

– Clauses in P never used – For each Li, UP raises LB by 1 – For each Mj, UP raises LB by m – In total, UP raises LB by m(m + 1) + 1 – PHPm+1

m

is unsatisfiable

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Outline

Maximum Satisfiability Pigeonhole Formulas From SAT to Horn MaxSAT Polynomial Time Refutations of PHP Experimental Results ?

25 / 33

Experimental setup

• Instances:

– PHP-pw (46), PHP-sc (46), Urquhart (84), Comb (96) – All in CNF; goal was to analyze CNF formulas

• Solvers:

SAT SAT+ IHS MaxSAT CG MaxSAT MRes MIP OPB BDD

minisat glucose lgl crypto maxhs lmhs mscg wbo wpm3 eva lp cc sat4j∗ zres

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Results on PHP instances: pw vs. sc

10 20 30 40 50 60 instances 10−3 10−2 10−1 100 101 102 103 CPU time (s) lp-cnf lp-wcnf maxhs lmhs mscg eva lgl lmhs-nes zres glucose lgl-nocard cc-cnf cc-opb 10 20 30 40 50 instances 10−3 10−2 10−1 100 101 102 103 CPU time (s) lp-cnf lp-wcnf maxhs lmhs mscg lmhs-nes eva glucose lgl-nocard lgl zres cc-cnf cc-opb

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Effect of dropping P clauses

20 40 60 80 100 instances 10−3 10−2 10−1 100 101 102 103 CPU time (s) mscg (no P) maxhs lmhs wbo (no P) mscg eva (no P) eva lmhs-nes (no P) lmhs-nes wbo

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Effect of dropping P clauses on mscg and wbo

10−3 10−2 10−1 100 101 102 103 104 PHP-nop 10−3 10−2 10−1 100 101 102 103 104 PHP

1800 sec. timeout 1800 sec. timeout

10−3 10−2 10−1 100 101 102 103 104 PHP-nop 10−3 10−2 10−1 100 101 102 103 104 PHP

1800 sec. timeout 1800 sec. timeout

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Results on Urquhart & combined instances

10 20 30 40 50 60 70 80 90 instances 10−2 10−1 100 101 102 103 CPU time (s) zres maxhs lmhs lmhs-nes lgl lp-wcnf mscg lgl-nogauss glucose eva cc-cnf 20 40 60 80 100 instances 10−1 100 101 102 103 CPU time (s) lmhs lmhs-nes maxhs lgl-nocard zres lgl lp-wcnf glucose mscg eva cc-cnf

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Conclusions & research directions

• Initial motivation: optimize MaxHS-like algorithms

– E.g. by exploiting Horn MaxSAT & LTUR

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Conclusions & research directions

• Initial motivation: optimize MaxHS-like algorithms

– E.g. by exploiting Horn MaxSAT & LTUR

• Simple reduction from SAT to Horn MaxSAT

– Many other simple reductions to Horn MaxSAT

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Conclusions & research directions

• Initial motivation: optimize MaxHS-like algorithms

– E.g. by exploiting Horn MaxSAT & LTUR

• Simple reduction from SAT to Horn MaxSAT

– Many other simple reductions to Horn MaxSAT

• (Horn) MaxSAT solvers can solve (in polynomial time) hard

instances for resolution

– If equipped with the right reduction

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Conclusions & research directions

• Initial motivation: optimize MaxHS-like algorithms

– E.g. by exploiting Horn MaxSAT & LTUR

• Simple reduction from SAT to Horn MaxSAT

– Many other simple reductions to Horn MaxSAT

• (Horn) MaxSAT solvers can solve (in polynomial time) hard

instances for resolution

– If equipped with the right reduction

• Where to go with Horn MaxSAT?

– And of course, what about the new proof system(s)?

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Thank You!

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References to this work

• A. Ignatiev, A. Morgado, J. Marques-Silva:

On Tackling the Limits of Resolution in SAT Solving. SAT 2017: 164-183

• J. Marques-Silva, A. Ignatiev, A. Morgado:

Horn Maximum Satisfiability: Reductions, Algorithms and Applications. EPIA 2017: 681-694

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