Nadia Creignou

Complexity of generalized satisfiability counting problems

Abstract The class of generalized satisfiability problems, introduced in 1978 by Schaefer, presents a uniform way of studying the complexity of satisfiability problems with special conditions. The complexity of each decision and counting problem in this class depends on the involved logical relations. In 1979, Valiant defined the class #P, the class of functions definable as the number of accepting computations of a polynomial-time nondeterministic Turing machine. Clearly, all satisfiability counting problems belong to this class #P. We prove a Dichotomy Theorem for generalized satisfiability counting problems. That is, if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time, otherwise this function is #P-complete. This gives us a comparison between decision and counting generalized satisfiability problems. We can determine exactly the polynomial satisfiability decision problems whose number of solutions can be computed in polynomial time and also the polynomial satisfiability decision problems whose counting counterparts are already #P-complete. Moreover, taking advantage of a similar dichotomy result proved in 1978 by Schaefer for generalized satisfiability decision problems, we get as a corollary the implication that the counting counterpart of each NP-complete generalized satisfiability decision problem is #P-complete.

A dichotomy theorem for maximum generalized satisfiability problems

We study the complexity of an infinite class of optimization satisfiability problems. Each problem is represented through a finite set, S, of logical relations (generalizing the notion of clauses of bounded length). We prove the existence of a dichotomic classification for optimization satisfiability problems Max-Sat(S). We exhibit a particular infinite set of logical relations L, such that the following holds: If every relation in S is 0-valid (respectively 1-valid) or if even/relation in S belongs to L, then Max-Sat(S) is solvable in polynomial time, otherwise it is MAX SNP-complete. Therefore, Max-Sat(S) either is in P or has some ϵ-approximation algorithm with ϵ < 1 although not a polynomial-time approximation scheme, unless P = NP: L = {Posn, Negn, Spidern,p,q, Complete-Bipartiten,p:n, p, q ∈ N}, where Posn(x1, ..., xn) ≡ (x1 ∧ ··· ∧ xn),Negn(x1, ..., xn) ≡ (¬x1 ∧ ··· ∧ ¬xn),Spider n,p,q(x1, ..., xn, y1, ..., yp, z1 ..., zq) ≡ Λni=1 (xi → y1) ∧ Λpi=1 (y1≡yi ∧ Λqi=1 (y1 → zi), andComplete-Bipartiten,p(x1, ..., xn, y1, ..., yp) ≡ Λni=1 Λpj=1 (xi → yj).

On generating all solutions of generalized satisfiability problems

We examine whether all solutions of Generalized Satisfiability problems can be generated efficiently. By refining Schaefers [11] result we show that there exists a class G of problems such that for every problem in there exists a polynomial delay generating algorithm and for every Generalized Satisfiability problem not in G such an algorithm does not exist unless P = NP. The class G is made up of the problems equivalent to the satisfiability problem for conjunction of Horn clauses, anti-Horn clauses, 2-clauses or XOR-clauses.

Structure identification of Boolean relations and plain bases for co-clones

We give a quadratic algorithm for the following structure identification problem: given a Boolean relation R and a finite set S of Boolean relations, can the relation R be expressed as a conjunctive query over the relations in the set S? Our algorithm is derived by first introducing the concept of a plain basis for a co-clone and then identifying natural plain bases for every co-clone in Posts lattice. In the process, we also give a quadratic algorithm for the problem of finding the smallest co-clone containing a Boolean relation.

The class of problems that are linearly equivalent to Satisfiability or a uniform method for proving NP-completeness

Abstract We widely extend the class of problems that are linearly equivalent to Satisfiability. We show that many natural combinatorial problems are linear-time equivalent to Satisfiability (SAT-equivalent). We prove that the following problems are SAT-equivalent: 3-Colorability, Path with Forbidden Pairs, Path without Chord, Kernel of Graph, Partition into Triangles, Partition into Hamiltonian Subgraphs, Cubic Subgraph, 3-Dimensional Matching, 3-Exact Cover, Partition into Paths of Length Two, 3-Domatic Number, 2-Partition into Perfect Matchings and Restricted Perfect Matching. Furthermore, we present a uniform method for proving NP-completeness. This method shows that 3-Colorability is a key problem for proving NP-completeness with linear-time transformations. The computational model on which these results are based is a Turing machine with a fixed number of sortings in addition.

Generalized satisfiability problems: minimal elements and phase transitions

We develop a probabilistic model on the generalized satisfiability problems defined by Schaefer (in: Proceedings of the 10th STOC, San Diego, CA, USA, Association for Computing Machinery, New York, 1978, pp. 216-226) for which the arity of the constraints is fixed in order to study the associated phase transition. We establish new results on minimal elements associated with such generalized satisfiability problems. These results are the keys of the exploration we conduct on the location and on the nature of the phase transition for generalized satisfiability. We first prove that the phase transition occurs at the same scale for every reasonable problem and we provide lower and upper bounds for the associated critical ratio. Our framework allows one to get these bounds in a uniform way, in particular, we obtain a lower bound proportional to the number of variables for k-SAT without analyzing any algorithm. Finally, we reveal the seed of coarseness for the phase transition of generalized satisfiability: 2-XOR-SAT.

Combinatorial sharpness criterion and phase transition classification for random CSPs

We investigate the nature of the phase transition (sharp or coarse) for random constraint satisfaction problems. We first give a sharp threshold criterion specified for CSPs, which is derived from Friedgut-Bourgains one. Thus, we get a complete and precise classification of the nature of the threshold for symmetric Boolean CSPs. In particular we show that it is governed by two local properties strongly related to the problems 1-SAT and 2-XOR-SAT.

Belief revision within fragments of propositional logic

Belief revision has been extensively studied in the framework of propositional logic, but just recently revision within fragments of propositional logic has gained attention. Hereby it is not only the belief set and the revision formula which are given within a certain language fragment, but also the result of the revision has to be located in the same fragment. So far, research in this direction has been mainly devoted to the Horn fragment of classical logic. In this work, we present a general approach to define new revision operators derived from known operators (as for instance, Satohs and Dalals revision operators), such that the result of the revision remains in the fragment under consideration. Our approach is not limited to the Horn case but applicable to any fragment of propositional logic where the models of the formulas are closed under a Boolean function. Thus we are able to uniformly treat cases as dual-Horn, Krom and affine formulas, as well.

Approximating the Satisfiability Threshold for Random k -XOR-formulas

In this paper we study random linear systems with

Boolean Constraint Satisfaction Problems: When Does Post's Lattice Help?

k > 3