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27
New Constructive Aspects of the Lovász Local Lemma
"... The Lovász Local Lemma (LLL) is a powerful tool that gives sufficient conditions for avoiding all of a given set of “bad ” events, with positive probability. A series of results have provided algorithms to efficiently construct structures whose existence is nonconstructively guaranteed by the LLL, ..."
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The Lovász Local Lemma (LLL) is a powerful tool that gives sufficient conditions for avoiding all of a given set of “bad ” events, with positive probability. A series of results have provided algorithms to efficiently construct structures whose existence is nonconstructively guaranteed by the LLL, culminating in the recent breakthrough of Moser & Tardos. We show that the output distribution of the MoserTardos algorithm wellapproximates the conditional LLLdistribution – the distribution obtained by conditioning on all bad events being avoided. We show how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and nonconstructive results. We also show that when an LLL application provides a small amount of slack, the number of resamplings of the
Complexity of twolevel logic minimization
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
"... Abstract—The complexity of twolevel logic minimization is a topic of interest to both computeraided design (CAD) specialists and computer science theoreticians. In the logic synthesis community, twolevel logic minimization forms the foundation for more complex optimization procedures that have si ..."
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Abstract—The complexity of twolevel logic minimization is a topic of interest to both computeraided design (CAD) specialists and computer science theoreticians. In the logic synthesis community, twolevel logic minimization forms the foundation for more complex optimization procedures that have significant realworld impact. At the same time, the computational complexity of twolevel logic minimization has posed challenges since the beginning of the field in the 1960s; indeed, some central questions have been resolved only within the last few years, and others remain open. This recent activity has classified some logic optimization problems of high practical relevance, such as finding the minimal sumofproducts (SOP) form and maximal term expansion and reduction. This paper surveys progress in the field with selfcontained expositions of fundamental early results, an account of the recent advances, and some new classifications. It includes an introduction to the relevant concepts and terminology from computational complexity, as well a discussion of the major remaining open problems in the complexity of logic minimization. Index Terms—Computational complexity, logic design, logic minimization, twolevel logic. I.
The Power of Nondeterminism in SelfAssembly
"... We investigate the role of nondeterminism in Winfree’s abstract tile assembly model, which was conceived to model artificial molecular selfassembling systems constructed from DNA. By nondeterminism we do not mean a magical ability such as that possessed by a nondeterministic algorithm to search an ..."
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We investigate the role of nondeterminism in Winfree’s abstract tile assembly model, which was conceived to model artificial molecular selfassembling systems constructed from DNA. By nondeterminism we do not mean a magical ability such as that possessed by a nondeterministic algorithm to search an exponentialsize space in polynomial time. Rather, we study realistically implementable systems that retain a different sense of determinism in that they are guaranteed to produce a unique shape but are nondeterministic in that they do not guarantee which tile types will be placed where within the shape. We show a “molecular computability ” result: there is an infinite shape S that is uniquely assembled by a tile system but not by any deterministic tile system. We show a “molecular complexity ” result: there is a finite shape S that is uniquely assembled by a tile system with c tile types, but every deterministic tile system that uniquely assembles S has more than c tile types. In fact we extend the technique to derive a stronger (classical complexity theoretic) result, showing that the problem of finding the minimum number of tile types that uniquely assemble a given finite shape is Σ P 2complete. In contrast, the problem of finding the minimum number of deterministic tile types that uniquely assemble a shape is NPcomplete [5].
The complexity of boolean formula minimization
 Journal of Computer and Systems Sciences
"... The Minimum Equivalent Expression problem is a natural optimization problem in the second level of the PolynomialTime Hierarchy. It has long been conjectured to be Σ P 2complete and indeed appears as an open problem in Garey and Johnson [GJ79]. The depth2 variant was only shown to be Σ P 2comple ..."
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Cited by 13 (3 self)
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The Minimum Equivalent Expression problem is a natural optimization problem in the second level of the PolynomialTime Hierarchy. It has long been conjectured to be Σ P 2complete and indeed appears as an open problem in Garey and Johnson [GJ79]. The depth2 variant was only shown to be Σ P 2complete in 1998 [Uma98, Uma01], and even resolving the complexity of the depth3 version has been mentioned as a challenging open problem. We prove that the depthk version is Σ P 2complete under Turing reductions for all k ≥ 3. We also settle the complexity of the original, unbounded depth Minimum Equivalent Expression problem, by showing that it too is Σ P 2complete under Turing reductions. Supported by NSF CCF0830787, and BSF 2004329.
The complexity of nonrepetitive edge coloring of graphs. http://arxiv.org/abs/0709.4497 (accessed February 15th
, 2008
"... A squarefree word is a sequence w of symbols such that there are no strings x, y, and z for which w = xyyz. A nonrepetitive coloring of a graph is an edge coloring in which the sequence of colors along any open path is squarefree. The Thue number π(G) of a graph G is the least n for which the graph ..."
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A squarefree word is a sequence w of symbols such that there are no strings x, y, and z for which w = xyyz. A nonrepetitive coloring of a graph is an edge coloring in which the sequence of colors along any open path is squarefree. The Thue number π(G) of a graph G is the least n for which the graph can be nonrepetitively colored in n colors. A number of recent papers have shown both exact and approximation results for Thue numbers of various classes of graphs. We show that determining whether a graph G has φ(G) ≤ k is Σ p 2complete. When we restrict to paths of length at most n, the problem becomes NPcomplete for fixed n. For n = 2, this is the edge coloring problem; thus the boundedpath version can be thought of as a generalization of edge coloring. 1
Complexity of clique coloring and related problems
, 2004
"... A kcliquecoloring of a graph G is an assignment of k colors to the vertices of G such that every maximal (i.e., not extendable) clique of G contains two vertices with different colors. We show that deciding whether a graph has a kcliquecoloring is Σ p 2 complete for every k ≥ 2. The complexity ..."
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A kcliquecoloring of a graph G is an assignment of k colors to the vertices of G such that every maximal (i.e., not extendable) clique of G contains two vertices with different colors. We show that deciding whether a graph has a kcliquecoloring is Σ p 2 complete for every k ≥ 2. The complexity of two related problems are also considered. A graph is kcliquechoosable, if for every klistassignment on the vertices, there is a clique coloring where each vertex receives a color from its list. This problem turns out to be Π p 3complete for every k ≥ 2. A graph G is hereditary kcliquecolorable if every induced subgraph of G is kcliquecolorable. We prove that deciding hereditarycomplete for every k ≥ 3. kcliquecolorability is also Π p 3 1
Quantified maximum satisfiability: A coreguided approach
 In International Conference Theory and Applications of Satisfiability Testing
, 2013
"... Abstract. In recent years, there have been significant improvements in algorithms both for Quantified Boolean Formulas (QBF) and for Maximum Satisfiability (MaxSAT). This paper studies the problem of solving quantified formulas subject to a cost function, and considers the problem in a quantified ..."
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Abstract. In recent years, there have been significant improvements in algorithms both for Quantified Boolean Formulas (QBF) and for Maximum Satisfiability (MaxSAT). This paper studies the problem of solving quantified formulas subject to a cost function, and considers the problem in a quantified MaxSAT setting. Two approaches are investigated. One is based on relaxing the soft clauses and performing a linear search on the cost function. The other approach, which is the main contribution of the paper, is inspired by recent work on MaxSAT, and exploits the iterative identification of unsatisfiable cores. The paper investigates the application of these approaches to the concrete problem of computing smallest minimal unsatisfiable subformulas (SMUS), a decision version of which is a wellknown problem in the second level of the polynomial hierarchy. Experimental results, obtained on representative problem instances, indicate that the coreguided approach for the SMUS problem outperforms the use of linear search over the values of the cost function. More significantly, the coreguided approach also outperforms the stateoftheart SMUS extractor Digger. 1
On the Complexity of Entailment in Existential Conjunctive First Order Logic with Atomic Negation
, 2011
"... We consider the entailment problem in the fragment of firstorder logic (FOL) composed of existentially closed conjunctions of literals (without functions), denoted FOL(∃, ∧, ¬a). This problem can be recast as several fundamental problems in artificial intelligence and databases, namely query contai ..."
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We consider the entailment problem in the fragment of firstorder logic (FOL) composed of existentially closed conjunctions of literals (without functions), denoted FOL(∃, ∧, ¬a). This problem can be recast as several fundamental problems in artificial intelligence and databases, namely query containment for conjunctive queries with negation, clause entailment for clauses without functions and query answering with incomplete information for Boolean conjunctive queries with negation over a fact base. Entailment in FOL(∃, ∧, ¬a) is Π P 2complete, whereas it is only NPcomplete when the formulas contain no negation. We investigate the role of specific literals in this complexity increase. These literals have the property of being “exchangeable”, with this notion taking the structure of the formulas into account. To focus on the structure of formulas, we shall see them as labeled graphs. Graph homomorphism, which provides a sound and complete proof procedure for positive formulas, is at the core of this study. Let ENTAILMENTk be the following family of problems: given two formulas g and h in FOL(∃, ∧, ¬a), such that g has at most k pairs of exchangeable literals, is g entailed by h? The main
On the hardness of satisfiability with bounded occurrences in the polynomialtime hierarchy
 THEORY OF COMPUTING
, 2007
"... In 1991, Papadimitriou and Yannakakis gave a reduction implying the NPhardness of approximating the problem 3SAT with bounded occurrences. Their reduction is based on expander graphs. We present an analogue of this result for the second level of the polynomialtime hierarchy based on superconcen ..."
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In 1991, Papadimitriou and Yannakakis gave a reduction implying the NPhardness of approximating the problem 3SAT with bounded occurrences. Their reduction is based on expander graphs. We present an analogue of this result for the second level of the polynomialtime hierarchy based on superconcentrator graphs. This resolves an open question of Ko and Lin (1995) and should be useful in deriving inapproximability results in the polynomialtime hierarchy. More precisely, we show that given an instance of ∀∃3SAT in which every variable occurs at most B times (for some absolute constant B), it is Π2hard to distinguish between the following two cases: YES instances, in which for any assignment to the universal variables there exists an assignment to the existential variables that satisfies all the clauses, and NO instances in which there exists an assignment to the universal variables such that any