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."

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

Meine Dissertation befasst sich mit Boole’schen Constraint Satisfaction Problemen (kurz: CSP). Ein Constraint besteht aus einer Menge von Variablen und einer (Boole’schen) Relation, die die Belegungen bestimmter Tupel von Variablen einschränkt. Ein CSP ist dann die Frage, ob es zu einer gegebenen Menge von Constraints eine Belegung aller Variablen gibt, die alle Constraints gleichzeitig erfüllt. Dieses CSP und einige seiner Derivationen werden unter komplexitätstheoretischen Aspekten betrachtet. Als wichtiges Instrument zur Bestimmung der Komplexität von CSP wird die algebraische Methode verwendet. Diese nutzt die von Emil Post gefundene vollständige Klassifikation aller unter Superposition abgeschlossener Klassen Boole’scher Funktionen (Clones). Der Abschluss einer Menge von Funktionen unter Superposition bedeutet dabei, dass die Menge der Funktionen unter beliebigen Kompositionen abgeschlossen ist. Der nach ihm benannte Post’sche Graph zeigt die vollständige Inklusionsstruktur der Clones. Hiermit konnten in Verbindung mit der Galoistheorie schon viele elegante Beweise geführt werden. Unter anderem wurde so auch das Dichotomieergebnis von Thomas Schaefer erneut bewiesen. Dieses besagt, dass das Boole’sche CSP in Abhängigkeit von den zugelassenen Boole’schen Constraints entweder in P oder NP-vollständig ist.

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.

This dissertation furthers a systematic study of the complexity classification of counting problems. A central goal of this study is to prove complexity classification theorems which state that every problem in some large class is either polynomial-time computable (tractable) or #P-hard. Such classification results are important as they tend to give a unified explanation for the tractability of certain counting problems and a reasonable basis for the conjecture that the remaining problems are inherently intractable. In this dissertation, we focus on the framework of Holant problems on Boolean variables, as well as other frameworks that are expressible as Holant problems, such as counting constraint satisfaction problems and counting Eulerian orientation problems. First, we prove a complexity dichotomy for Holant problems on the Boolean domain with arbitrary sets of real-valued constraint functions. It is proved that for every set F of real-valued constraint functions, Holant(F) is either tractable or #P-hard. The classification has an explicit criterion. This is a culmination of much research on this decade-long study, and it uses many previous results and techniques. On the other hand, to achieve the present result, many new tools were developed, and a novel connection with quantum information theory was built. In particular, two functions exhibiting intriguing and extraordinary closure properties are related to Bell states in quantum information theory. Dealing with these functions plays an important role in the proof. Then, we consider the complexity of Holant problems with respect to planar graphs, where physicists had discovered some remarkable algorithms, such as the FKT algorithm for counting planar perfecting matchings in polynomial time. For a basic case of Holant problems, called six-vertex models, we discover a new tractable class over planar graphs beyond the reach of the FKT algorithm. After carving out this new planar tractable class which had not been discovered for six-vertex models in the past six decades, we prove that everything else is #P-hard, even for the planar case. This leads to a complete complexity classification for planar six-vertex models. This result is the first substantive advance towards a planar Holant classification with asymmetric constraints. We hope this work can help us better understand a fundamental question in theoretical computer science: What does it mean for a computational counting problem to be easy or to be hard?

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.