Exact Satisfiability Problem Research Articles

Improved algorithms for the general exact satisfiability problem

The Exact Satisfiability problem asks if we can find a satisfying assignment to each clause such that exactly one literal in each clause is assigned 1, while the rest are all assigned 0. We can generalise this problem further by defining that a Cj clause is solved iff exactly j of the literals in the clause are 1 and all others are 0. We now introduce the family of Generalised Exact Satisfiability problems called GiXSAT as the problem to check whether a given instance consisting of Cj clauses with j∈{0,1,…,i} for each clause has a satisfying assignment. In this paper, we present faster exact polynomial space algorithms, using a nonstandard measure, to solve GiXSAT, for i∈{2,3,4}, in O(1.3674n) time, O(1.5687n) time and O(1.6545n) time, respectively, using polynomial space, where n is the number of variables. This improves the current state of the art for polynomial space algorithms from O(1.4203n) time for G2XSAT by Zhou, Jiang and Yin and from O(1.6202n) time for G3XSAT by Dahllöf and from O(1.6844n) time for G4XSAT which was by Dahllöf as well. In addition, we present faster exact algorithms solving G2XSAT, G3XSAT and G4XSAT in O(1.3188n) time, O(1.3407n) time and O(1.3536n) time respectively at the expense of using exponential space.

Faster than classical quantum algorithm for dense formulas of exact satisfiability and occupation problems

We present an exact quantum algorithm for solving the Exact Satisfiability problem, which belongs to the important NP-complete complexity class. The algorithm is based on an intuitive approach that can be divided into two parts: the first step consists in the identification and efficient characterization of a restricted subspace that contains all the valid assignments of the Exact Satisfiability; while the second part performs a quantum search in such restricted subspace. The quantum algorithm can be used either to find a valid assignment (or to certify that no solution exists) or to count the total number of valid assignments. The query complexities for the worst-case are respectively bounded by and , where n is the number of variables and the number of linearly independent clauses. Remarkably, the proposed quantum algorithm results to be faster than any known exact classical algorithm to solve dense formulas of Exact Satisfiability. As a concrete application, we provide the worst-case complexity for the Hamiltonian cycle problem obtained after mapping it to a suitable Occupation problem. Specifically, we show that the time complexity for the proposed quantum algorithm is bounded by for 3-regular undirected graphs, where n is the number of nodes. The same worst-case complexity holds for -regular bipartite graphs. As a reference, the current best classical algorithm has a (worst-case) running time bounded by . Finally, when compared to heuristic techniques for Exact Satisfiability problems, the proposed quantum algorithm is faster than the classical WalkSAT and Adiabatic Quantum Optimization for random instances with a density of constraints close to the satisfiability threshold, the regime in which instances are typically the hardest to solve. The proposed quantum algorithm can be straightforwardly extended to the generalized version of the Exact Satisfiability known as Occupation problem. The general version of the algorithm is presented and analyzed.

FROM XSAT TO SAT BY EXHIBITING BOOLEAN FUNCTIONS

Given a Boolean formula in conjunctive normal form (CNF), the Exact Satisfiability problem (XSAT), a variant of the Satisfiability problem (SAT), consists in finding an assignment to the variables such that each clause contains exactly one satisfied literal. Best algorithms to solve this problem run in [Formula: see text] ([Formula: see text] for X3SAT). Another possibility is to transform each clause in a set of equivalent clauses for the Satisfiability problem and to use modern and powerful solvers (zChaff, Berkmin, MiniSat, RSat etc.) to find such truth assignment. In this paper we introduce three new encodings from XSAT instances to SAT instances that lead to a lot of structural information (equivalency gates and and gates) which is naturally hidden in the pairwise transformation. Some solvers (lsat,march_dl,eqsatz) can take into account this kinds of structural information to make simplifications as pretreatment and speed-up the resolution. Then we show the interest of dealing with the XSAT formalism by introducing an encoding of binary CSP and graph coloring problem into XSAT instances. Preliminary results on real-world binary CSP and graph coloring problem show the importance of exhibiting equivalencies for the XSAT problem.

Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings

We present exact algorithms with exponential running times for variants of n-element set cover problems, based on divide-and-conquer and on inclusion–exclusion characterizations. We show that the Exact Satisfiability problem of size l with m clauses can be solved in time 2 m lO(1) and polynomial space. The same bounds hold for counting the number of solutions. As a special case, we can count the number of perfect matchings in an n-vertex graph in time 2nn O(1) and polynomial space. We also show how to count the number of perfect matchings in time O(1.732n) and exponential space. We give a number of examples where the running time can be further improved if the hypergraph corresponding to the set cover instance has low pathwidth. This yields exponential-time algorithms for counting k-dimensional matchings, Exact Uniform Set Cover, Clique Partition, and Minimum Dominating Set in graphs of degree at most three. We extend the analysis to a number of related problems such as TSP and Chromatic Number.

New algorithms for Exact Satisfiability

The Exact Satisfiability problem is to determine if a CNF-formula has a truth assignment satisfying exactly one literal in each clause; Exact 3-Satisfiability is the version in which each clause contains at most three literals. In this paper, we present algorithms for Exact Satisfiability and Exact 3-Satisfiability running in time O ( 2 0.2325 n ) and O ( 2 0.1379 n ) , respectively. The previously best algorithms have running times O ( 2 0.2441 n ) for Exact Satisfiability (Methods Oper. Res. 43 (1981) 419–431) and O ( 2 0.1626 n ) for Exact 3-Satisfiability (Annals of Mathematics and Artificial Intelligence 43 (1) (2005) 173–193 and Zapiski nauchnyh seminarov POMI 293 (2002) 118–128). We extend the case analyses of these papers and observe that a formula not satisfying any of our cases has a small number of variables, for which we can try all possible truth assignments and for each such assignment solve the remaining part of the formula in polynomial time.

Maximum Exact Satisfiability: NP-completeness Proofs and Exact Algorithms

Inspired by the Maximum Satisfiability and Exact Satisfiability problems we present two Maximum Exact Satisfiability problems. The first problem called Maximum Exact Satisfiability is: given a formula in conjunctive normal form and an integer k, is there an assignment to all variables in the formula such that at least k clauses have exactly one true literal. The second problem called Restricted Maximum Exact Satisfiability has the further restriction that no clause is allowed to have more than one true literal. Both problems are proved NP-complete restricted to the versions where each clause contains at most two literals. In fact Maximum Exact Satisfiability is a generalisation of the well-known NP-complete problem MaxCut. We present an exact algorithm for Maximum Exact Satisfiability where each clause contains at most two literals with time complexity O(poly(L) . 2^{m/4}), where m is the number of clauses and L is the length of the formula. For the second version we give an algorithm with time complexity O(poly(L) . 1.324718^n) , where n is the number of variables. We note that when restricted to the versions where each clause contains exactly two literals and there are no negations both problems are fixed parameter tractable. It is an open question if this is also the case for the general problems.

Algorithms for four variants of the exact satisfiability problem

We present four polynomial space and exponential time algorithms for variants of the E XACT S ATISFIABILITY problem. First, an O(1.1120 n ) (where n is the number of variables) time algorithm for the NP-complete decision problem of E XACT 3-S ATISFIABILITY , and then an O(1.1907 n ) time algorithm for the general decision problem of E XACT S ATISFIABILITY . The best previous algorithms run in O(1.1193 n ) and O(1.2299 n ) time, respectively. For the #P-complete problem of counting the number of models for E XACT 3-S ATISFIABILITY we present an O(1.1487 n ) time algorithm. We also present an O(1.2190 n ) time algorithm for the general problem of counting the number of models for E XACT S ATISFIABILITY ; presenting a simple reduction, we show how this algorithm can be used for computing the permanent of a 0/1 matrix.

Algorithms for four variants of the exact satisfiability problem

We present four polynomial space and exponential time algorithms for variants of the E XACT S ATISFIABILITY problem. First, an O(1.1120 n ) (where n is the number of variables) time algorithm for the NP-complete decision problem of E XACT 3-S ATISFIABILITY, and then an O(1.1907 n ) time algorithm for the general decision problem of E XACT S ATISFIABILITY. The best previous algorithms run in O(1.1193 n ) and O(1.2299 n ) time, respectively. For the #P-complete problem of counting the number of models for E XACT 3-S ATISFIABILITY we present an O(1.1487 n ) time algorithm. We also present an O(1.2190 n ) time algorithm for the general problem of counting the number of models for E XACT S ATISFIABILITY; presenting a simple reduction, we show how this algorithm can be used for computing the permanent of a 0/1 matrix.

New Algorithms for Exact Satisfiability

The Exact Satisfiability problem is to determine if a CNF-formula has a truth assignment satisfying exactly one literal in each clause; Exact 3-Satisfiability is the version in which each clause contains at most three literals. In this paper, we present algorithms for Exact Satisfiability and Exact 3-Satisfiability running in time O(2^{0.2325n}) and O(2^{0.1379n}), respectively. The previously best algorithms have running times O(2^{0.2441n}) for Exact Satisfiability (Monien, Speckenmeyer and Vornberger (1981)) and O(2^{0.1626n}) for Exact 3-Satisfiability (Kulikov and independently Porschen, Randerath and Speckenmeyer (2002)). We extend the case analyses of these papers and observe, that a formula not satisfying any of our cases has a small number of variables, for which we can try all possible truth assignments and for each such assignment solve the remaining part of the formula in polynomial time.

Exact satisfiability, a natural extension of set partition, and its average case behavior

The problem of determining whether a Boolean formula in conjunctive normal form is satisfiable in such a way that in each clause exactly one literal is set true, and all the other literals are set false is called the exact satisfiability problem. The exact satisfiability problem is well known to be NP-complete [5] and it contains the well known set partitioning problem as a special case. We study here the average time complexity of a simple backtracking strategy for solving the exact satisfiability problem under two probability models, the constant density model and the constant degree model. For both models we present results sharply separating classes of instances solvable in low degree polynomial time in the average from classes for which superpolynomial or exponential time is needed in the average.