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Abstract: "In this paper we introduce a new approach to the logical definability of NP optimization problems by focusing on the expressibility of feasible solutions. We show that in this framework first-order sentences capture exactly all polynomially bounded optimization problems. We also show that, assuming P [not equal to] NP, it is an undecidable problem to tell if a given first-order sentence defines an approximable optimization problem. We then isolate a syntactically defined class of NP minimization problems that contains the MIN SET COVER problem and has the property that every problem in it has a logarithmic approximation algorithm. We conclude by giving a machine-independent characterization of the NP [equals?] co-NP problem in terms of logical expressibility of the MAX CLIQUE problem."
Abstract: "Extending a well known property of polynomially bounded NP optimization problems, we observe that, by attaching weights to tuples over the domain of the input, all NP optimization problems admit a certain logical characterization. We show that any NP optimization problem can be stated as a problem in which the constraint conditions can be expressed by a II2 first-order formula and this is the best possible result. We further analyze the weighted analogue of all syntactically defined classes of optimization problems that are known to have good approximation properties: MAX NP, MAX SNP, MAX SNP([pi]), MIN F[pi]1 and MIN F[pi]2(1). All these classes continue to have the same approximation properties in the case of positive weights. Using reductions from multiprover interactive systems, we show that if NP [not subset of] DTIME[2[superscript log[superscript O(1)] n]], the approximation properties of the above classes devaluates considerably when negative weights are also allowed (with the exception of MIN F+II1, where only a weaker deterioration could be proven). It follows that the general weighted versions of MAX 2SAT, Set Cover, Priority Ordering and of some other closely related natural problems are not approximable in quasipolynomial time within a factor of 2[superscript log [superscript [mu]] n] for some [mu]> 0 (which depends on the problem), unless NP [subset of] DTIME[2[superscript log [superscript O(1)] n]]. Under the same hypothesis, we show that the maximization variant of Set Partition is also not superpolylog approximable. A stronger result is proven for the minimization variant of Set Partition: if P [not equal to] NP, then MIN Set Partition cannot be approximated in polynomial time within a factor of n[supersript O(1)]."
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.
This book constitutes the thoroughly refereed post-conference proceedings of the 6th International Symposium on Parameterized and Exact Computation, IPEC 2011, in Saarbrücken, Germany, in September 2011. The 21 revised full papers presented were carefully reviewed and selected from 40 submissions. The topics addressed cover research in all aspects of parameterized and exact computation and complexity, including but not limited to new techniques for the design and analysis of parameterized and exact algorithms, fixed-parameter tractability results, parameterized complexity theory, relationship between parameterized complexity and traditional complexity classifications, applications of parameterized and exact computation, and implementation issues of parameterized and exact algorithms.
This book constitutes the proceedings of the 25th Seminar on Current Trends in Theory and Practice of Informatics, SOFSEM'98, held in Jasna, Slovakia, in November 1998. The volume presents 19 invited survey articles by internationally well-known authorities together with 18 revised full research papers carefully reviewed and selected for inclusion in the book. The areas covered include history of models of computation, algorithms, formal methods, practical aspects of software engineering, database systems, parallel and distributed systems, electronic commerce, and electronic documents and digital libraries.
This is a supplementary volume to the major three-volume Handbook of Combinatorial Optimization set. It can also be regarded as a stand-alone volume presenting chapters dealing with various aspects of the subject in a self-contained way.
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.
The fusion between graph theory and combinatorial optimization has led to theoretically profound and practically useful algorithms, yet there is no book that currently covers both areas together. Handbook of Graph Theory, Combinatorial Optimization, and Algorithms is the first to present a unified, comprehensive treatment of both graph theory and c
The LNCS series reports state-of-the-art results in computer science research, development, and education, at a high level and in both printed and electronic form. Enjoying tight cooperation with the R&D community, with numerous individuals, as well as with prestigious organizations and societies, LNCS has grown into the most comprehensive computer science research forum available. The scope of LNCS, including its subseries LNAI and LNBI, spans the whole range of computer science and information technology including interdisciplinary topics in a variety of application fields. In parallel to the printed book, each new volume is published electronically in LNCS Online.
This book constitutes the refereed proceedings of the Third International Workshop on Parameterized and Exact Computation, IWPEC 2008, held in Victoria, Canada, in May 2008 - co-located with the 40th ACM Symposium on Theory of Computing, STOC 2008. The 17 revised full papers presented together with 3 invited lectures were carefully reviewed and selected from 32 submissions. The topics addressed cover research in all aspects of parameterized and exact computation and complexity, including but not limited to new techniques for the design and analysis of parameterized and exact algorithms, parameterized complexity theory, relationship between parameterized complexity and traditional complexity classifications, applications of parameterized computation, implementation and experiments, high-performance computing and fixed-parameter tractability.