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Relaxations of the Satisfiability Problem using Semidefinite Programming
, 2000
"... We derive a semidefinite relaxation of the satisfiability (SAT) problem and discuss its strength. We give both the primal and dual formulation of the relaxation. The primal formulation is an eigenvalue optimization problem, while the dual formulation is a semidefinite feasibility problem. It is s ..."
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Cited by 14 (2 self)
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We derive a semidefinite relaxation of the satisfiability (SAT) problem and discuss its strength. We give both the primal and dual formulation of the relaxation. The primal formulation is an eigenvalue optimization problem, while the dual formulation is a semidefinite feasibility problem. It is shown that using the relaxation, the notorious pigeon hole and mutilated chessboard problems are solved in polynomial time. As a byproduct we find a new `sandwich' theorem that is similar to Lov'asz' famous #function. Furthermore, using the semidefinite relaxation 2SAT problems are solved. By adding an objective function to the dual formulation, a specific class of polynomially solvable 3SAT instances can be identified. We conclude with discussing how the relaxation can be used to solve more general SAT problems and some empirical observations. 1 Introduction The satisfiability problem of propositional logic (SAT) is the original NP complete problem [5]. Many algorithms for SAT have ...
An Improved Semidefinite Programming Relaxation for the Satisfiability Problem
, 2002
"... 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 ..."
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Cited by 13 (3 self)
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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 NPcomplete, 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. We are interested in the application of SDP to satisfiability problems, and in particular in how SDP can be used to detect unsatisfiability. In this paper we introduce a new SDP relaxation for the satisfiability problem. This SDP relaxation arises from the recently introduced paradigm of “higher liftings” for constructing semidefinite programming relaxations of discrete optimization problems.
Solving Satisfiability Problems Using Elliptic Approximations. A Note on Volumes and Weights
"... In this note we propose to use the volume of elliptic approximations of satisfiability problems as a measure for computing weighting coefficients of clauses of different lengths. For random 3Sat formula it is confirmed experimentally that, when applied in a DPLL algorithm with a branching strategy ..."
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Cited by 10 (3 self)
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In this note we propose to use the volume of elliptic approximations of satisfiability problems as a measure for computing weighting coefficients of clauses of different lengths. For random 3Sat formula it is confirmed experimentally that, when applied in a DPLL algorithm with a branching strategy that is based on the ellipsoids as well, the weight deduced yields better results than the weights that are used in previous studies.
Semidefinite optimization approaches for satisfiability and maximumsatisfiability problems
 J. Satisf. Bool. Model. Comput
"... Semidefinite optimization, commonly referred to as semidefinite programming, has been a remarkably active area of research in optimization during the last decade. For combinatorial problems in particular, semidefinite programming has had a truly significant impact. This paper surveys some of the res ..."
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Cited by 8 (3 self)
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Semidefinite optimization, commonly referred to as semidefinite programming, has been a remarkably active area of research in optimization during the last decade. For combinatorial problems in particular, semidefinite programming has had a truly significant impact. This paper surveys some of the results obtained in the application of semidefinite programming to satisfiability and maximumsatisfiability problems. The approaches presented in some detail include the groundbreaking approximation algorithm of Goemans and Williamson for MAX2SAT, the Gap relaxation of de Klerk, van Maaren and Warners, and strengthenings of the Gap relaxation based on the Lasserre hierarchy of semidefinite liftings for polynomial optimization problems. We include theoretical and computational comparisons of the aforementioned semidefinite relaxations for the special case of 3SAT, and conclude with a review of the most recent results in the application of semidefinite programming to SAT and MAXSAT.
On Semidefinite Programming Relaxations of (2+p)SAT
, 2000
"... Recently, De Klerk, Van Maaren and Warners [7] investigated a relaxation of 3SAT via semidefinite programming. Thus a 3SAT formula is relaxed to a semidefinite feasibility problem. If the feasibility problem is infeasible then a certificate of unsatisfiability of the formula is obtained. The autho ..."
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Cited by 3 (0 self)
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Recently, De Klerk, Van Maaren and Warners [7] investigated a relaxation of 3SAT via semidefinite programming. Thus a 3SAT formula is relaxed to a semidefinite feasibility problem. If the feasibility problem is infeasible then a certificate of unsatisfiability of the formula is obtained. The authors proved that this approach is exact for several polynomially solvable classes of logical formulae, including 2SAT, pigeonhole formulae and mutilated chessboard formulae. In this paper we further explore this approach, and investigate the strength of the relaxation on (2 + p)SAT formulae, i.e. formulae with a fraction p of 3clauses and a fraction (1  p) of 2clauses. In the first instance, we provide an empirical computational evaluation of our approach. Secondly, we establish approximation guarantees of randomized and deterministic rounding schemes when the semidefinite feasibility problem is feasible.
An Extended Semidefinite Relaxation for Satisfiability
, 2007
"... This paper proposes a new semidefinite programming relaxation for the satisfiability problem. This relaxation is an extension of previous relaxations arising from the paradigm of partial semidefinite liftings for 0/1 optimization problems. We show that the proposed relaxation is exact for the import ..."
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This paper proposes a new semidefinite programming relaxation for the satisfiability problem. This relaxation is an extension of previous relaxations arising from the paradigm of partial semidefinite liftings for 0/1 optimization problems. We show that the proposed relaxation is exact for the important class of Tseitin instances, meaning that a Tseitin instance is unsatisfiable if and only if the corresponding semidefinite programming relaxation is infeasible.
Improved SDO Relaxations for SAT The Tseitin Instances on Toroidal Grid Graphs
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Centrum voor Wiskunde en Informatica REPORTRAPPORT Report SENR9903
, 1999
"... and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of ..."
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and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of