A Complete Classification Of The Approximability Of Maximization Problems Derived From Boolean Constraint Satisfaction

Abstract: "In this paper we study the approximability of boolean constraint satisfaction problems. A problem in this class consists of some collection of 'constraints' (i.e., functions f: [0,1][superscript k] -> [0,1]); an instance of a problem is a set of constraints applied to specified subsets of n boolean variables. Schaefer earlier studied the question of whether one could find in polynomial time a setting of the variables satisfying all constraints; he showed that every such problem is either in P or is NP-complete. We consider optimization variants of these problems in which one either tries to maximize the number of satisfied constraints (as in MAX 3SAT or MAX CUT) or tries to find an assignment satisfying all constraints which maximizes the number of variables set to 1 (as in MAX CUT or MAX CLIQUE). We completely classify the approximability of all such problems. In the first case, we show that any such optimization problem is either in P or is MAX SNP-hard. In the second case, we show that such problems fall precisely into one of five classes: solvable in polynomial-time, approximable to within constant factors in polynomial time (but no better), approximable to within polynomial factors in polynomial time (but no better), not approximable to within any factor but decidable in polynomial time, and not decidable in polynomial time (unless P = NP). This result proves formally for this class of problems two results which to this point have only been empirical observations; namely, that NP-hard problems in MAX SNP alwyas turn out to be MAX SNP-hard, and that there seem to be no natural maximization problems approximable to within polylogarithmic factors but no better."

Many fundamental combinatorial problems, arising in such diverse fields as artificial intelligence, logic, graph theory, and linear algebra, can be formulated as Boolean constraint satisfaction problems (CSP). This book is devoted to the study of the complexity of such problems. The authors' goal is to develop a framework for classifying the complexity of Boolean CSP in a uniform way. In doing so, they bring out common themes underlying many concepts and results in both algorithms and complexity theory. The results and techniques presented here show that Boolean CSP provide an excellent framework for discovering and formally validating "global" inferences about the nature of computation.

Presents a novel form of a compendium that classifies an infinite number of problems by using a rule-based approach.

This book documents the state of the art in combinatorial optimization, presenting approximate solutions of virtually all relevant classes of NP-hard optimization problems. The wealth of problems, algorithms, results, and techniques make it an indispensible source of reference for professionals. The text smoothly integrates numerous illustrations, examples, and exercises.

Combinatorial optimization is a multidisciplinary scientific area, lying in the interface of three major scientific domains: mathematics, theoretical computer science and management. The three volumes of the Combinatorial Optimization series aim to cover a wide range of topics in this area. These topics also deal with fundamental notions and approaches as with several classical applications of combinatorial optimization. Concepts of Combinatorial Optimization, is divided into three parts: - On the complexity of combinatorial optimization problems, presenting basics about worst-case and randomized complexity; - Classical solution methods, presenting the two most-known methods for solving hard combinatorial optimization problems, that are Branch-and-Bound and Dynamic Programming; - Elements from mathematical programming, presenting fundamentals from mathematical programming based methods that are in the heart of Operations Research since the origins of this field.

This book constitutes the refereed proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2000, held in Saarbrücken, Germany in September 2000. The 22 revised full papers presented together with four invited contributions were carefully reviewed and selected from 68 submissions. The topics dealt with include design and analysis of approximation algorithms, inapproximibility results, on-line problems, randomization techniques, average-case analysis, approximation classes, scheduling problems, routing and flow problems, coloring and partitioning, cuts and connectivity, packing and covering, geometric problems, network design, and various applications.

This book constitutes the refereed proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science, MFCS 2000, held in Bratislava/Slovakia in August/September 2000. The 57 revised full papers presented together with eight invited papers were carefully reviewed and selected from a total of 147 submissions. The book gives an excellent overview on current research in theoretical informatics. All relevant foundational issues, from mathematical logics as well as from discrete mathematics are covered. Anybody interested in theoretical computer science or the theory of computing will benefit from this book.

This book constitutes the refereed proceedings of the 14th International Conference on Theory and Applications of Satisfiability Testing, SAT 2011, held in Ann Arbor, MI, USA in June 2011. The 25 revised full papers presented together with abstracts of 2 invited talks and 10 poster papers were carefully reviewed and selected from 57 submissions. The papers are organized in topical sections on complexity analysis, binary decision diagrams, theoretical analysis, extraction of minimal unsatisfiable subsets, SAT algorithms, quantified Boolean formulae, model enumeration and local search, and empirical evaluation.

This book constitutes the refereed proceedings of the 24th Seminar on Current Trends in Theory and Practice of Informatics, SOFSEM'97, held in Milovy, Czech Republic, in November 1997. SOFSEM is special in being a mix of a winter school, an international conference, and an advanced workshop meeting the demand for ongoing education in the area of computer science. The volume presents 22 invited contributions by leading experts together with 24 revised contributed papers selected from 63 submissions. The invited presentations are organized in topical sections on foundations, distributed and parallel computing, software engineering and methodology, and databases and information systems.

This book constitutes the strictly refereed proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science, STACS 98, held in Paris, France, in February 1998. The volume presents three invited surveys together with 52 revised full papers selected from a total of 155 submissions. The papers are organized in topical sections on algorithms and data structures, logic, complexity, and automata and formal languages.