exact algorithms for np-hard problems: a survey abstract. we discuss fast exponential time solutions

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  • Exact Algorithms for NP-Hard Problems: A Survey

    Gerhard J. Woeginger

    Department of Mathematics University of Twente, P.O. Box 217 7500 AE Enschede, The Netherlands

    Abstract. We discuss fast exponential time solutions for NP-complete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NP-complete problems includes the travelling salesman problem, scheduling under precedence constraints, satisfiability, knap- sack, graph coloring, independent sets in graphs, bandwidth of a graph, and many more.

    1 Introduction

    Every NP-complete problem can be solved by exhaustive search. Unfortunately, when the size of the instances grows the running time for exhaustive search soon becomes forbiddingly large, even for instances of fairly small size. For some problems it is possible to design algorithms that are significantly faster than exhaustive search, though still not polynomial time. This survey deals with such fast, super-polynomial time algorithms that solve NP-complete problems to opti- mality. In recent years there has been growing interest in the design and analysis of such super-polynomial time algorithms. This interest has many causes.

    – It is now commonly believed that P�=NP, and that super-polynomial time algorithms are the best we can hope for when we are dealing with an NP- complete problem. There is a handful of isolated results scattered across the literature, but we are far from developing a general theory. In fact, we have not even started a systematic investigation of the worst case behavior of such super-polynomial time algorithms.

    – Some NP-complete problems have better and faster exact algorithms than others. There is a wide variation in the worst case complexities of known exact (super-polynomial time) algorithms. Classical complexity theory can not explain these differences. Do there exist any relationships among the worst case behaviors of various problems? Is progress on the different prob- lems connected? Can we somehow classify NP-complete problems to see how close we are to the best possible algorithms?

    – With the increased speed of modern computers, large instances of NP- complete problems can be solved effectively. For example it is nowadays routine to solve travelling salesman (TSP) instances with up to 2000 cities.

    M. Jünger et al. (Eds.): Combinatorial Optimization (Edmonds Festschrift), LNCS 2570, pp. 185–207, 2003. c© Springer-Verlag Berlin Heidelberg 2003

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  • 186 G.J. Woeginger

    And if the data is nicely structured, then instances with up to 13000 cities can be handled in practice (Applegate, Bixby, Chvátal & Cook [2]). There is a huge gap between the empirical results from testing implementations and the known theoretical results on exact algorithms.

    – Fast algorithms with exponential running times may actually lead to practi- cal algorithms, at least for moderate instance sizes. For small instances, an algorithm with an exponential time complexity of O(1.01n) should usually run much faster than an algorithm with a polynomial time complexity of O(n4).

    In this article we survey known results and approaches to the worst case analysis of exact algorithms for NP-hard problems, and we provide pointers to the liter- ature. Throughout the survey, we will also formulate many exercises and open problems. Open problems refer to unsolved research problems, while exercises pose smaller questions and puzzles that should be fairly easy to solve.

    Organization of this survey. Section 2 collects some technical preliminaries and some basic definitions that will be used in this article. Sections 3–6 introduce and explain the four main techniques for designing fast exact algorithms: Section 3 deals with dynamic programming across the subsets, Section 4 discusses pruning of search trees, Section 5 illustrates the power of preprocessing the data, and Section 6 considers approaches based on local search. Section 7 discusses methods for proving negative results on the worst case behavior of exact algorith

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