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199
Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming
 Journal of the ACM
, 1995
"... We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution ..."
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Cited by 936 (14 self)
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We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of ...
Polynomial time approximation schemes for Euclidean TSP and other geometric problems
 In Proceedings of the 37th IEEE Symposium on Foundations of Computer Science (FOCS’96
, 1996
"... Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c � 1 and given any n nodes in � 2, a randomized version of the scheme finds a (1 � 1/c)approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes a ..."
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Cited by 315 (3 self)
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Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c � 1 and given any n nodes in � 2, a randomized version of the scheme finds a (1 � 1/c)approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes are in � d, the running time increases to O(n(log n) (O(�dc))d�1). For every fixed c, d the running time is n � poly(log n), that is nearly linear in n. The algorithm can be derandomized, but this increases the running time by a factor O(n d). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2approximation in polynomial time. We also give similar approximation schemes for some other NPhard Euclidean problems: Minimum Steiner Tree, kTSP, and kMST. (The running times of the algorithm for kTSP and kMST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constantfactor approximation. We also give efficient approximation schemes for Euclidean MinCost Matching, a problem that can be solved exactly in polynomial time. All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as �p for p � 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.
Multiobjective Evolutionary Algorithms: Analyzing the StateoftheArt
, 2000
"... Solving optimization problems with multiple (often conflicting) objectives is, generally, a very difficult goal. Evolutionary algorithms (EAs) were initially extended and applied during the mideighties in an attempt to stochastically solve problems of this generic class. During the past decade, ..."
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Cited by 283 (7 self)
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Solving optimization problems with multiple (often conflicting) objectives is, generally, a very difficult goal. Evolutionary algorithms (EAs) were initially extended and applied during the mideighties in an attempt to stochastically solve problems of this generic class. During the past decade, a variety of multiobjective EA (MOEA) techniques have been proposed and applied to many scientific and engineering applications. Our discussion's intent is to rigorously define multiobjective optimization problems and certain related concepts, present an MOEA classification scheme, and evaluate the variety of contemporary MOEAs. Current MOEA theoretical developments are evaluated; specific topics addressed include fitness functions, Pareto ranking, niching, fitness sharing, mating restriction, and secondary populations. Since the development and application of MOEAs is a dynamic and rapidly growing activity, we focus on key analytical insights based upon critical MOEA evaluation of c...
Nearly Linear Time Approximation Schemes for Euclidean TSP and other Geometric Problems
, 1997
"... We present a randomized polynomial time approximation scheme for Euclidean TSP in ! 2 that is substantially more efficient than our earlier scheme in [2] (and the scheme of Mitchell [21]). For any fixed c ? 1 and any set of n nodes in the plane, the new scheme finds a (1+ 1 c )approximation to ..."
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Cited by 91 (4 self)
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We present a randomized polynomial time approximation scheme for Euclidean TSP in ! 2 that is substantially more efficient than our earlier scheme in [2] (and the scheme of Mitchell [21]). For any fixed c ? 1 and any set of n nodes in the plane, the new scheme finds a (1+ 1 c )approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. (Our earlier scheme ran in n O(c) time.) For points in ! d the algorithm runs in O(n(log n) (O( p dc)) d\Gamma1 ) time. This time is polynomial (actually nearly linear) for every fixed c; d. Designing such a polynomialtime algorithm was an open problem (our earlier algorithm in [2] ran in superpolynomial time for d 3). The algorithm generalizes to the same set of Euclidean problems handled by the previous algorithm, including Steiner Tree, kTSP, kMST, etc, although for kTSP and kMST the running time gets multiplied by k. We also use our ideas to design nearlylinear time approximation schemes for Euclidean vers...
MAXMIN Ant System and Local Search for the Traveling Salesman Problem
 IEEE INTERNATIONAL CONFERENCE ON EVOLUTIONARY COMPUTATION (ICEC'97)
, 1997
"... Ant System is a general purpose algorithm inspired by the study of the behavior of Ant Colonies. It is based on a cooperative search paradigm that is applicable to the solution of combinatorial optimization problems. In this paper we introduce MAX MIN Ant System, an improved version of basic Ant S ..."
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Cited by 87 (17 self)
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Ant System is a general purpose algorithm inspired by the study of the behavior of Ant Colonies. It is based on a cooperative search paradigm that is applicable to the solution of combinatorial optimization problems. In this paper we introduce MAX MIN Ant System, an improved version of basic Ant System, and report our results for its application to symmetric and asymmetric instances of the well known Traveling Salesman Problem. We show how MAX MIN Ant System can be significantly improved extending it with local search heuristics. Our results clearly show that MAX MIN Ant System has the property of effectively guiding the local search heuristics towards promising regions of the search space by generating good initial tours. I. Introduction The Ant System algorithm, originally introduced in [3], [4], is a new cooperative search algorithm inspired by the behavior of real ants. Ants are able to find good solutions to shortest path problems between a food source and their home colony...
A Theoretician's Guide to the Experimental Analysis of Algorithms
, 1996
"... This paper presents an informal discussion of issues that arise when one attempts to analyze algorithms experimentally. It is based on lessons learned by the author over the course of more than a decade of experimentation, survey paper writing, refereeing, and lively discussions with other experimen ..."
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Cited by 76 (0 self)
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This paper presents an informal discussion of issues that arise when one attempts to analyze algorithms experimentally. It is based on lessons learned by the author over the course of more than a decade of experimentation, survey paper writing, refereeing, and lively discussions with other experimentalists. Although written from the perspective of a theoretical computer scientist, it is intended to be of use to researchers from all fields who want to study algorithms experimentally. It has two goals: first, to provide a useful guide to new experimentalists about how such work can best be performed and written up, and second, to challenge current researchers to think about whether their own work might be improved from a scientific point of view. With the latter purpose in mind, the author hopes that at least a few of his recommendations will be considered controversial.
A Genetic Local Search Algorithm for Solving Symmetric and Asymmetric Traveling Salesman Problems
 In Proceedings of the 1996 IEEE International Conference on Evolutionary Computation
, 1996
"... The combination of local search heuristics and genetic algorithms is a promising approach for finding nearoptimum solutions to the traveling salesman problem (TSP). In this paper, an approach is presented in which local search techniques are used to find local optima in a given TSP search space, and ..."
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Cited by 76 (12 self)
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The combination of local search heuristics and genetic algorithms is a promising approach for finding nearoptimum solutions to the traveling salesman problem (TSP). In this paper, an approach is presented in which local search techniques are used to find local optima in a given TSP search space, and genetic algorithms are used to search the space of local optima in order to find the global optimum. New genetic operators for realizing the proposed approach are described, and the quality and efficiency of the solutions obtained for a set of symmetric and asymmetric TSP instances are discussed. The results indicate that it is possible to arrive at high quality solutions in reasonable time. I. Introduction In the Traveling Salesman Problem (TSP) [18], [27], a number of cities with distances between them is given and the task is to find the minimumlength closed tour that visits each city once and returns to its starting point. A symmetric TSP (STSP) is one where the distance between any...
Genetic Local Search for the TSP: New Results
 In Proceedings of the 1997 IEEE International Conference on Evolutionary Computation
, 1997
"... The combination of local search heuristics and genetic algorithms has been shown to be an effective approach for finding nearoptimum solutions to the traveling salesman problem. In this paper, previously proposed genetic local search algorithms for the symmetric and asymmetric traveling salesman pr ..."
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Cited by 73 (13 self)
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The combination of local search heuristics and genetic algorithms has been shown to be an effective approach for finding nearoptimum solutions to the traveling salesman problem. In this paper, previously proposed genetic local search algorithms for the symmetric and asymmetric traveling salesman problem are revisited and potential improvements are identified. Since local search is the central component in which most of the computation time is spent, improving the efficiency of the local search operators is crucial for improving the overall performance of the algorithms. The modifications of the algorithms are described and the new results obtained are presented. The results indicate that the improved algorithms are able to arrive at better solutions in significantly less time. I. Introduction Consider a salesman who wants to start from his home city, visit each of a set of n cities exactly once, and then return home. Since the salesman is interested in finding the shortest possible r...
The TSP Phase Transition
 Artificial Intelligence
, 1996
"... We wish to bring to the attention of the OR community the phenomenon of phase transitions in randomly generated problems. These are of considerable practical use for benchmarking algorithms. They also offer insight into problem hardness and algorithm performance. Whilst phase transition experiments ..."
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Cited by 64 (12 self)
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We wish to bring to the attention of the OR community the phenomenon of phase transitions in randomly generated problems. These are of considerable practical use for benchmarking algorithms. They also offer insight into problem hardness and algorithm performance. Whilst phase transition experiments are frequently performed by AI researchers, such experiments do not appear to be in common use in the OR community. To illustrate the value of such experiments, we examine a typical OR problem, the traveling salesman problem. We report in detail many features of the phase transition in this problem, and show how some of these features are also seen in real problems. Acknowledgements The second author is supported by a HCM Postdoctoral Fellowship. We thank Iain Buchanan for comments on a draft of this paper, and Alan Bundy, and the members of the Mathematical Reasoning Group in Edinburgh for their constructive comments and many CPU cycles donated to these and other experiments from SERC grant GR/H/23610. We also thank the MRG group at Trento and the Department of Computer Science at the University of Strathclyde for additional CPU cycles. Finally, we thank Robert Craig for providing us with his code. 1
Hierons, Search algorithms for regression test case prioritization
 IEEE Transactions on Software Engineering
, 2007
"... Abstract—Regression testing is an expensive, but important, process. Unfortunately, there may be insufficient resources to allow for the reexecution of all test cases during regression testing. In this situation, test case prioritization techniques aim to improve the effectiveness of regression test ..."
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Cited by 59 (10 self)
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Abstract—Regression testing is an expensive, but important, process. Unfortunately, there may be insufficient resources to allow for the reexecution of all test cases during regression testing. In this situation, test case prioritization techniques aim to improve the effectiveness of regression testing by ordering the test cases so that the most beneficial are executed first. Previous work on regression test case prioritization has focused on Greedy Algorithms. However, it is known that these algorithms may produce suboptimal results because they may construct results that denote only local minima within the search space. By contrast, metaheuristic and evolutionary search algorithms aim to avoid such problems. This paper presents results from an empirical study of the application of several greedy, metaheuristic, and evolutionary search algorithms to six programs, ranging from 374 to 11,148 lines of code for three choices of fitness metric. The paper addresses the problems of choice of fitness metric, characterization of landscape modality, and determination of the most suitable search technique to apply. The empirical results replicate previous results concerning Greedy Algorithms. They shed light on the nature of the regression testing search space, indicating that it is multimodal. The results also show that Genetic Algorithms perform well, although Greedy approaches are surprisingly effective, given the multimodal nature of the landscape. Index Terms—Search techniques, test case prioritization, regression testing. Ç 1