Generalized Satisfiability Problems Research Articles

TrES: Tropical Encryption Scheme Based on Double Key Exchange

Shor’s quantum algorithm establishes a polynomial time attack on the discrete logarithm problem in any group. Recent announcements of progress in building quantum computers highlight the need for new concepts to create cryptosystems that are resistant to quantum attacks. In this paper, we present а new message encryption scheme. To enhance the security of the scheme, we suggest double key-exchange protocol (KEP). The first stage of the key exchange uses a matrix power function (MPF) in a tropical semiring. These functions are based on the action of a matrix semiring acting on some matrix set. MPFs can be considered as one-way functions because they are based on some generalized satisfiability problems that are potentially NP-complete. The obtained shared secret key at the first stage of the key exchange serves as an input for the second stage. The security of the second phase relies on the difficulty of the semiring action problem. In our protocol, we suggest using left or right action of the tropical semiring (which can be both min-plus and max-plus) on the group of commutative matrices (circulant matrices, in our case). The fact that the key-exchange protocol works in two phases contributes to its security, since an attacker needs to solve two difficult problems in order to break it. The main advantages of the presented protocol are: the increased efficiency and improved security.

Generalized satisfiability problems via operator assignments

Schaefer introduced a framework for generalized satisfiability problems on the Boolean domain and characterized the computational complexity of such problems. We investigate an algebraization of Schaefer's framework in which the Fourier transform is used to represent constraints by multilinear polynomials in a unique way. This representation of constraints gives rise to a relaxation of the notion of satisfiability in which the values to variables are linear operators on some Hilbert space. For constraints given by a system of linear equations over the two-element field, earlier work in the foundations of quantum mechanics has shown that there are systems that have no solutions in the Boolean domain, but have solutions via operator assignments on some finite-dimensional Hilbert space. Our main result is a complete characterization of the classes of Boolean relations for which there is a gap between satisfiability in the Boolean domain and the relaxation of satisfiability via operator assignments.

Parameterized Complexity and Kernelizability of Max Ones and Exact Ones Problems

For a finite set Γ of Boolean relations, M ax O nes SAT(Γ) and E xact O nes SAT(Γ) are generalized satisfiability problems where every constraint relation is from Γ, and the task is to find a satisfying assignment with at least/exactly k variables set to 1, respectively. We study the parameterized complexity of these problems, including the question whether they admit polynomial kernels. For M ax O nes SAT(Γ), we give a classification into five different complexity levels: polynomial-time solvable, admits a polynomial kernel, fixed-parameter tractable, solvable in polynomial time for fixed k , and NP-hard already for k = 1. For E xact O nes SAT(Γ), we refine the classification obtained earlier by taking a closer look at the fixed-parameter tractable cases and classifying the sets Γ for which E xact O nes SAT(Γ) admits a polynomial kernel.

Query-Number Preserving Reductions and Linear Lower Bounds for Testing

In this paper, we study lower bounds on the query complexity of testing algorithms for various problems. Given an oracle that returns information of an input object, a testing algorithm distinguishes the case that the object has a given property P from the case that it has a large distance to having P with probability at least 2/3. The query complexity of an algorithm is measured by the number of accesses to the oracle. We introduce two reductions that preserve the query complexity. One is derived from the gap-preserving local reduction and the other is from the L-reduction. By using the former reduction, we show linear lower bounds on the query complexity for testing basic NP-complete properties, i.e., 3-edge-colorability, directed Hamiltonian path/cycle, undirected Hamiltonian path/cycle, 3-dimensional matching and NP-complete generalized satisfiability problems. Also, using the second reduction, we show a linear lower bound on the query complexity of approximating the size of the maximum 3-dimensional matching.

Threshold properties of random boolean constraint satisfaction problems

We study threshold properties of random constraint satisfaction problems under a probabilistic model due to Molloy [Models for random constraint satisfaction problems, in: Proceedings of the 32nd ACM Symposium on Theory of Computing, 2002]. We give a sufficient condition for the existence of a sharp threshold. In the boolean case, it gives an independent proof for the more difficult half of a classification result conjectured by Creignou and Daudé [Generalized satisfiability problems: minimal elements and phase transitions. Theor. Comput. Sci. 302(1–3) (2003) 417–430], proved in a restricted case by the same authors [Combinatorial sharpness criterion and phase transition classification for random CSPs, Inform. Comput. 190(2) (2004) 220–238], and established by them [Coarse and sharp thresholds for random generalized satisfiability problems, in: M. Drmota, P. Flajolet, D. Gardy, B. Gittenberger (Eds.), Mathematics and Computer Science III: Algorithms, Trees, Combinatorics and Probabilities, Birkhauser, Basel, September 2004, pp. 507–517] while this paper was in the refereeing process.

The complexity of Boolean constraint satisfaction local search problems

The class of generalized satisfiability problems, first introduced by Schaefer in 1978, presents a uniform way of studying the complexity of Boolean constraint satisfaction problems with respect to...

A Dichotomy in the Complexity of Propositional Circumscription

The inference problem for propositional circumscription is known to be highly intractable and, in fact, harder than the inference problem for classical propositional logic. More precisely, in its full generality this problem in /spl Pi//sub 2//sup P/-complete, which means that it has the same inherent computational complexity as the satisfiability problem for quantified Boolean formulas with two alternations (universal-existential) of quantifiers. We use T.J. Schaefer's (1978) framework of generalized satisfiability problems to study the family of all restricted cases of the inference problem for propositional circumscription. Our main result fields a complete classification of the "truly hard"(/spl Pi//sub 2//sup P/-complete) and the "easier" cases of this problem (reducible to the inference problem for classical propositional logic). Specifically, we establish a dichotomy theorem which asserts that each such restricted case is either /spl Pi//sub 2//sup P/-complete or is in co-NP. Moreover, we provide efficiently checkable criteria that tell apart the "truly hard" cases from the "easier" ones.

The complexity of minimal satisfiability problems

A dichotomy theorem for a class of decision problems is a result asserting that certain problems in the class are solvable in polynomial time, while the rest are NP-complete. The first remarkable such dichotomy theorem was proved by Schaefer in 1978. It concerns the class of generalized satisfiability problems Sat (S) , whose input is a CNF (S) -formula, i.e., a formula constructed from elements of a fixed set S of generalized connectives using conjunctions and substitutions by variables. Here, we investigate the complexity of minimal satisfiability problems Min Sat (S) , where S is a fixed set of generalized connectives. The input to such a problem is a CNF (S) -formula and a satisfying truth assignment; the question is to decide whether there is another satisfying truth assignment that is strictly smaller than the given truth assignment with respect to the coordinate-wise partial order on truth assignments. Minimal satisfiability problems were first studied by researchers in artificial intelligence while investigating the computational complexity of propositional circumscription. The question of whether dichotomy theorems can be proved for these problems was raised at that time, but was left open. We settle this question affirmatively by establishing a dichotomy theorem for the class of all Min Sat (S) -problems, where S is a finite set of generalized connectives. We also prove a dichotomy theorem for a variant of Min Sat (S) in which the minimization is restricted to a subset of the variables, whereas the remaining variables may vary arbitrarily (this variant is related to extensions of propositional circumscription and was first studied by Cadoli). Moreover, we show that similar dichotomy theorems hold also when some of the variables are assigned constant values. Finally, we give simple criteria that tell apart the polynomial-time solvable cases of these minimal satisfiability problems from the NP-complete ones.

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 .

On generating all solutions of generalized satisfiability problems

We examine whether all solutions of Generalized Satisfiability problems can be generated efficiently. By refining Schaefer's [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.

Complexity of Generalized Satisfiability Counting Problems

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.

New local search approximation techniques for maximum generalized satisfiability problems

We investigate the relationship beetween the classes MAX-NP and GLO by studying the Maximum Generalized Satisfiability problem, which is in the former class. We present a (2−B)-approximate greedy heuristic for this problem and show that no local search c-approximate algorithm exists, based on an h-bounded neighborhood and on the number of satisfied clauses as objective function. This implies that, with the standard definition of local search, MAX-NP is not contained in GLO.

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 = {Pos n , Neg n , Spider n,p,q , Complete-Bipartite n,p : n, p, q ∈ N}, where Pos n ( x 1, ..., x n ) ≡ ( x 1 ∧ ··· ∧ x n ), Neg n ( x 1, ..., x n ) ≡ (¬ x 1 ∧ ··· ∧ ¬ x n ), Spider n, p, q( x 1, ..., x n , y 1, ..., y p , z 1 ..., z q ) ≡ Λ n i=1 ( x i → y 1) ∧ Λ p i=1 ( y 1≡ y i ∧ Λ q i=1 ( y 1 → z i ), and Complete- Bipartite n, p ( x 1, ..., x n , y 1, ..., y p ) ≡ Λ n i=1 Λ p j=1 ( x i → y j ).

Implications of forbidden structures for extremal algorithmic problems

We develop and analyze P-optimal approximation algorithms for various generalized satisfiability problems and determine the corresponding P-optimal thresholds. The most novel aspect of the paper is that we develop new P-optimal algorithms for restricted generalized satisfiability problems which are defined by forbidden subformulas. All our algorithms run in linear time on a RAM.