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Proofs of unsatisfiability via semidefinite programming

Proofs of unsatisfiability via semidefinite programming
Proofs of unsatisfiability via semidefinite programming
The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there is a truth assignment to the variables such that the formula is satisfied. It is well known that SAT is in general NP-complete, although several important special cases can be solved in polynomial time. Semidefinite programming (SDP) refers to the class of optimization problems where a linear function of a matrix variable $X$ is maximized (or minimized) subject to linear constraints on the elements of $X$ and the additional constraint that $X$ be positive semidefinite. In this paper, we focus on the application of SDP to obtain proofs of unsatisfiability. Using a new semidefinite programming relaxation for SAT, we obtain proofs of unsatisfiability for some hard instances with up to 260 variables and over 400 clauses. In particular, we were able to prove the unsatisfiability of the smallest unsatisfiable instance that remained unsolved during the SAT Competition 2003. This shows that the SDP relaxation is competitive with the top solvers in the competition, and that this technique has the potential to complement existing techniques for SAT.
3540214453
308-316
Springer Science+Business Media
Anjos, Miguel F.
0103b24d-cab5-4109-9924-dc9eebd04257
Ahr, D.
Fahrion, R.
Oswald, M.
Reinelt, G.
Anjos, Miguel F.
0103b24d-cab5-4109-9924-dc9eebd04257
Ahr, D.
Fahrion, R.
Oswald, M.
Reinelt, G.

Anjos, Miguel F. (2003) Proofs of unsatisfiability via semidefinite programming. Ahr, D., Fahrion, R., Oswald, M. and Reinelt, G. (eds.) In Operations Research Proceedings 2003: Selected papers of the International Conference on Operations Research (OR 2003). vol. XVI, Springer Science+Business Media. pp. 308-316 .

Record type: Conference or Workshop Item (Paper)

Abstract

The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there is a truth assignment to the variables such that the formula is satisfied. It is well known that SAT is in general NP-complete, although several important special cases can be solved in polynomial time. Semidefinite programming (SDP) refers to the class of optimization problems where a linear function of a matrix variable $X$ is maximized (or minimized) subject to linear constraints on the elements of $X$ and the additional constraint that $X$ be positive semidefinite. In this paper, we focus on the application of SDP to obtain proofs of unsatisfiability. Using a new semidefinite programming relaxation for SAT, we obtain proofs of unsatisfiability for some hard instances with up to 260 variables and over 400 clauses. In particular, we were able to prove the unsatisfiability of the smallest unsatisfiable instance that remained unsolved during the SAT Competition 2003. This shows that the SDP relaxation is competitive with the top solvers in the competition, and that this technique has the potential to complement existing techniques for SAT.

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More information

Published date: 2003
Venue - Dates: Operations Research Proceedings, Conference 2003, 2003-09-03 - 2003-09-05
Organisations: Operational Research

Identifiers

Local EPrints ID: 29677
URI: https://eprints.soton.ac.uk/id/eprint/29677
ISBN: 3540214453
PURE UUID: 36ae470c-a8f4-4342-842e-fb5575b413c7

Catalogue record

Date deposited: 09 Jun 2006
Last modified: 17 Jul 2017 15:57

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