In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NP-hard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. It is known that every Boolean (that is, two-valued) Max CSP problem with a finite set of allowed constraint types is either solvable exactly in polynomial time or else APX-complete (and hence can have no polynomial time approximation scheme unless P=NP. It has been an open problem for several years whether this result can be extended to non-Boolean Max CSP, which is much more difficult to analyze than the Boolean case. In this paper, we make the first step in this direction by establishing this result for Max CSP over a three-element domain. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known algebraic combinatorial property of supermodularity. We also show that every hard three-valued Max CSP problem contains, in a certain specified sense, one of the two basic hard Max CSP problems which are the Maximum k-colourable subgraph problems for k=2,3.
The Max-CSP framework has been well-studied in the Boolean case. Many fundamental results have been obtained, concerning both complexity classifications and approximation properties (see, e.g., @cite_18 @cite_21 @cite_8 @cite_3 @cite_2 @cite_26 ). In the non-Boolean case, a number of results have been obtained that concern exact (superpolynomial) algorithms or approximation properties (see, e.g., @cite_5 @cite_1 @cite_0 @cite_10 ). The main research problem we will look at in this paper is the following.
@cite_18 @cite_26 @cite_8 @cite_21 @cite_1 @cite_3 @cite_0 @cite_2 @cite_5 @cite_10
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We prove optimal, up to an arbitrary e > 0, inapproximability results for Max-E k-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover. Preface 1. Introduction 2. Complexity Classes 3. Boolean Constraint Satisfaction Problems 4. Characterizations of Constraint Functions 5. Implementation of Functions and Reductions 6. Classification Theorems for Decision, Counting and Quantified Problems 7. Classification Theorems for Optimization Problems 8. Input-Restricted Constrained Satisfaction Problems 9. The Complexity of the Meta-Problems 10. Concluding Remarks Bibliography Index. A boolean constraint satisfaction problem consists of some finite set of constraints (i.e., functions from 0 1-vectors to 0, 1 ) and an instance of such a problem is a set of constraints applied to specified subsets of n boolean variables. The goal is to find an assignment to the variables which satisfy all constraint applications. The computational complexity of optimization problems in connection with such problems has been studied extensively but the results have relied on the assumption that the weights are non-negative. The goal of this article is to study variants of these optimization problems where arbitrary weights are allowed. For the four problems that we consider, we give necessary and sufficient conditions for when the problems can be solved in polynomial time. In addition, we show that the problems are NP-equivalent in all other cases. (C) 2000 Elsevier Science B.V. All rights reserved. By the breakthrough work of Hastad [J ACM 48(4) (2001), 798–859], several constraint satisfaction problems are now known to have the following approximation resistance property: Satisfying more clauses than what picking a random assignment would achieve is NP-hard. This is the case for example for Max E3-Sat, Max E3-Lin, and Max E4-Set Splitting. A notable exception to this extreme hardness is constraint satisfaction over two variables (2-CSP); as a corollary of the celebrated Goemans-Williamson algorithm [J ACM 42(6) (1995), 1115–1145], we know that every Boolean 2-CSP has a nontrivial approximation algorithm whose performance ratio is better than that obtained by picking a random assignment to the variables. An intriguing question then is whether this is also the case for 2-CSPs over larger, non-Boolean domains. This question is still open, and is equivalent to whether the generalization of Max 2-SAT to domains of size d, can be approximated to a factor better than (1 − 1 d2). In an attempt to make progress towards this question, in this paper we prove, first, that a slight restriction of this problem, namely, a generalization of linear inequations with two variables per constraint, is not approximation resistant, and, second, that the Not-All-Equal Sat problem over domain size d with three variables per constraint, is approximation resistant, for every d ≥ 3. In the Boolean case, Not-All-Equal Sat with three variables per constraint is equivalent to Max 2-SAT and thus has a nontrivial approximation algorithm; for larger domain sizes, Max 2-SAT can be reduced to Not-All-Equal Sat with three variables per constraint. Our approximation algorithm implies that a wide class of 2-CSPs called regular 2-CSPs can all be approximated beyond their random assignment threshold. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004 We study optimization problems that may be expressed as "Boolean constraint satisfaction problems." An instance of a Boolean constraint satisfaction problem is given by m constraints applied to n Boolean variables. Different computational problems arise from constraint satisfaction problems depending on the nature of the "underlying" constraints as well as on the goal of the optimization task. Here we consider four possible goals: Max CSP (Min CSP) is the class of problems where the goal is to find an assignment maximizing the number of satisfied constraints (minimizing the number of unsatisfied constraints). Max Ones (Min Ones) is the class of optimization problems where the goal is to find an assignment satisfying all constraints with maximum (minimum) number of variables set to 1. Each class consists of infinitely many problems and a problem within a class is specified by a finite collection of finite Boolean functions that describe the possible constraints that may be used. Tight bounds on the approximability of every problem in Max CSP were obtained by Creignou [ J. Comput. System Sci., 51 (1995), pp. 511--522]. In this work we determine tight bounds on the "approximability" (i.e., the ratio to within which each problem may be approximated in polynomial time) of every problem in Max Ones, Min CSP, and Min Ones. Combined with the result of Creignou, this completely classifies all optimization problems derived from Boolean constraint satisfaction. Our results capture a diverse collection of optimization problems such as MAX 3-SAT, Max Cut, Max Clique, Min Cut, Nearest Codeword, etc. Our results unify recent results on the (in-)approximability of these optimization problems and yield a compact presentation of most known results. Moreover, these results provide a formal basis to many statements on the behavior of natural optimization problems that have so far been observed only empirically. We consider the problem MAX CSP over multi-valued domains with variables ranging over sets of size si ≤ s and constraints involving kj ≤ k variables. We study two algorithms with approximation ratios A and B. respectively, so we obtain a solution with approximation ratio max (A, B).The first algorithm is based on the linear programming algorithm of Serna, Trevisan, and Xhafa [Proc. 15th Annual Symp. on Theoret. Aspects of Comput. Sci., 1998, pp. 488-498] and gives ratio A which is bounded below by s1-k. For k = 2, our bound in terms of the individual set sizes is the minimum over all constraints involving two variables of (1 2√s1+ 1 2√s2)2, where s1 and s2 are the set sizes for the two variables.We then give a simple combinatorial algorithm which has approximation ratio B, with B > A e. The bound is greater than s1-k e in general, and greater than s1-k(1 - (s - 1) 2(k - 1)) for s ≤ k - 1, thus close to the s1-k linear programming bound for large k. For k = 2, the bound is 4 9 if s = 2, 1 2(s - 1) if s ≥ 3, and in general greater than the minimum of 1 4S1 + 1 4s2 over constraints with set sizes s1 and s2, thus within a factor of two of the linear programming bound.For the case of k = 2 and s = 2 we prove an integrality gap of 4 9 (1 + O(n-1 2)). This shows that our analysis is tight for any method that uses the linear programming upper bound. We present parallel approximation algorithms for maximization problems expressible by integer linear programs of a restricted syntactic form introduced by [BKT96]. One of our motivations was to show whether the approximation results in the framework of holds in the parallel setting. Our results are a confirmation of this, and thus we have a new common framework for both computational settings. Also, we prove almost tight non-approximability results, thus solving a main open question of We obtain the results through the constraint satisfaction problem over multi-valued domains, for which we show non-approximability results and develop parallel approximation algorithms. Our parallel approximation algorithms are based on linear programming and random rounding; they are better than previously known sequential algorithms. The non-approximability results are based on new recent progress in the fields of Probabilistically Checkable Proofs and Multi-Prover One-Round Proof Systems [Raz95, Has97, AS97, RS97].