6,049 research outputs found

    On the random satisfiable process

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    In this work we suggest a new model for generating random satisfiable k-CNF formulas. To generate such formulas -- randomly permute all 2^k\binom{n}{k} possible clauses over the variables x_1, ..., x_n, and starting from the empty formula, go over the clauses one by one, including each new clause as you go along if after its addition the formula remains satisfiable. We study the evolution of this process, namely the distribution over formulas obtained after scanning through the first m clauses (in the random permutation's order). Random processes with conditioning on a certain property being respected are widely studied in the context of graph properties. This study was pioneered by Ruci\'nski and Wormald in 1992 for graphs with a fixed degree sequence, and also by Erd\H{o}s, Suen, and Winkler in 1995 for triangle-free and bipartite graphs. Since then many other graph properties were studied such as planarity and H-freeness. Thus our model is a natural extension of this approach to the satisfiability setting. Our main contribution is as follows. For m \geq cn, c=c(k) a sufficiently large constant, we are able to characterize the structure of the solution space of a typical formula in this distribution. Specifically, we show that typically all satisfying assignments are essentially clustered in one cluster, and all but e^{-\Omega(m/n)} n of the variables take the same value in all satisfying assignments. We also describe a polynomial time algorithm that finds with high probability a satisfying assignment for such formulas

    Random k-SAT and the Power of Two Choices

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    We study an Achlioptas-process version of the random k-SAT process: a bounded number of k-clauses are drawn uniformly at random at each step, and exactly one added to the growing formula according to a particular rule. We prove the existence of a rule that shifts the satisfiability threshold. This extends a well-studied area of probabilistic combinatorics (Achlioptas processes) to random CSP's. In particular, while a rule to delay the 2-SAT threshold was known previously, this is the first proof of a rule to shift the threshold of k-SAT for k >= 3. We then propose a gap decision problem based upon this semi-random model. The aim of the problem is to investigate the hardness of the random k-SAT decision problem, as opposed to the problem of finding an assignment or certificate of unsatisfiability. Finally, we discuss connections to the study of Achlioptas random graph processes.Comment: 13 page

    Simplifying Random Satisfiability Problem by Removing Frustrating Interactions

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    How can we remove some interactions in a constraint satisfaction problem (CSP) such that it still remains satisfiable? In this paper we study a modified survey propagation algorithm that enables us to address this question for a prototypical CSP, i.e. random K-satisfiability problem. The average number of removed interactions is controlled by a tuning parameter in the algorithm. If the original problem is satisfiable then we are able to construct satisfiable subproblems ranging from the original one to a minimal one with minimum possible number of interactions. The minimal satisfiable subproblems will provide directly the solutions of the original problem.Comment: 21 pages, 16 figure

    Lower Bounds on Query Complexity for Testing Bounded-Degree CSPs

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    In this paper, we consider lower bounds on the query complexity for testing CSPs in the bounded-degree model. First, for any ``symmetric'' predicate P:0,1k→0,1P:{0,1}^{k} \to {0,1} except \equ where k≥3k\geq 3, we show that every (randomized) algorithm that distinguishes satisfiable instances of CSP(P) from instances (∣P−1(0)∣/2k−ϵ)(|P^{-1}(0)|/2^k-\epsilon)-far from satisfiability requires Ω(n1/2+δ)\Omega(n^{1/2+\delta}) queries where nn is the number of variables and δ>0\delta>0 is a constant that depends on PP and ϵ\epsilon. This breaks a natural lower bound Ω(n1/2)\Omega(n^{1/2}), which is obtained by the birthday paradox. We also show that every one-sided error tester requires Ω(n)\Omega(n) queries for such PP. These results are hereditary in the sense that the same results hold for any predicate QQ such that P−1(1)⊆Q−1(1)P^{-1}(1) \subseteq Q^{-1}(1). For EQU, we give a one-sided error tester whose query complexity is O~(n1/2)\tilde{O}(n^{1/2}). Also, for 2-XOR (or, equivalently E2LIN2), we show an Ω(n1/2+δ)\Omega(n^{1/2+\delta}) lower bound for distinguishing instances between ϵ\epsilon-close to and (1/2−ϵ)(1/2-\epsilon)-far from satisfiability. Next, for the general k-CSP over the binary domain, we show that every algorithm that distinguishes satisfiable instances from instances (1−2k/2k−ϵ)(1-2k/2^k-\epsilon)-far from satisfiability requires Ω(n)\Omega(n) queries. The matching NP-hardness is not known, even assuming the Unique Games Conjecture or the dd-to-11 Conjecture. As a corollary, for Maximum Independent Set on graphs with nn vertices and a degree bound dd, we show that every approximation algorithm within a factor d/\poly\log d and an additive error of ϵn\epsilon n requires Ω(n)\Omega(n) queries. Previously, only super-constant lower bounds were known

    Simplest random K-satisfiability problem

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    We study a simple and exactly solvable model for the generation of random satisfiability problems. These consist of γN\gamma N random boolean constraints which are to be satisfied simultaneously by NN logical variables. In statistical-mechanics language, the considered model can be seen as a diluted p-spin model at zero temperature. While such problems become extraordinarily hard to solve by local search methods in a large region of the parameter space, still at least one solution may be superimposed by construction. The statistical properties of the model can be studied exactly by the replica method and each single instance can be analyzed in polynomial time by a simple global solution method. The geometrical/topological structures responsible for dynamic and static phase transitions as well as for the onset of computational complexity in local search method are thoroughly analyzed. Numerical analysis on very large samples allows for a precise characterization of the critical scaling behaviour.Comment: 14 pages, 5 figures, to appear in Phys. Rev. E (Feb 2001). v2: minor errors and references correcte

    Phase Transition in Matched Formulas and a Heuristic for Biclique Satisfiability

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    A matched formula is a CNF formula whose incidence graph admits a matching which matches a distinct variable to every clause. We study phase transition in a context of matched formulas and their generalization of biclique satisfiable formulas. We have performed experiments to find a phase transition of property "being matched" with respect to the ratio m/nm/n where mm is the number of clauses and nn is the number of variables of the input formula φ\varphi. We compare the results of experiments to a theoretical lower bound which was shown by Franco and Gelder (2003). Any matched formula is satisfiable, moreover, it remains satisfiable even if we change polarities of any literal occurrences. Szeider (2005) generalized matched formulas into two classes having the same property -- var-satisfiable and biclique satisfiable formulas. A formula is biclique satisfiable if its incidence graph admits covering by pairwise disjoint bounded bicliques. Recognizing if a formula is biclique satisfiable is NP-complete. In this paper we describe a heuristic algorithm for recognizing whether a formula is biclique satisfiable and we evaluate it by experiments on random formulas. We also describe an encoding of the problem of checking whether a formula is biclique satisfiable into SAT and we use it to evaluate the performance of our heuristicComment: Conference version submitted to SOFSEM 2018 (https://beda.dcs.fmph.uniba.sk/sofsem2019/) 18 pages(17 without refernces), 3 figures, 8 tables, an algorithm pseudocod
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