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Ant Colony System: A cooperative learning approach to the traveling salesman problem
 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION
, 1997
"... This paper introduces the ant colony system (ACS), a distributed algorithm that is applied to the traveling salesman problem (TSP). In the ACS, a set of cooperating agents called ants cooperate to find good solutions to TSP’s. Ants cooperate using an indirect form of communication mediated by a pher ..."
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Cited by 1029 (53 self)
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This paper introduces the ant colony system (ACS), a distributed algorithm that is applied to the traveling salesman problem (TSP). In the ACS, a set of cooperating agents called ants cooperate to find good solutions to TSP’s. Ants cooperate using an indirect form of communication mediated by a pheromone they deposit on the edges of the TSP graph while building solutions. We study the ACS by running experiments to understand its operation. The results show that the ACS outperforms other natureinspired algorithms such as simulated annealing and evolutionary computation, and we conclude comparing ACS3opt, a version of the ACS augmented with a local search procedure, to some of the best performing algorithms for symmetric and asymmetric TSP’s.
Greedy Randomized Adaptive Search Procedures
, 2002
"... GRASP is a multistart metaheuristic for combinatorial problems, in which each iteration consists basically of two phases: construction and local search. The construction phase builds a feasible solution, whose neighborhood is investigated until a local minimum is found during the local search phas ..."
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Cited by 647 (82 self)
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GRASP is a multistart metaheuristic for combinatorial problems, in which each iteration consists basically of two phases: construction and local search. The construction phase builds a feasible solution, whose neighborhood is investigated until a local minimum is found during the local search phase. The best overall solution is kept as the result. In this chapter, we first describe the basic components of GRASP. Successful implementation techniques and parameter tuning strategies are discussed and illustrated by numerical results obtained for different applications. Enhanced or alternative solution construction mechanisms and techniques to speed up the search are also described: Reactive GRASP, cost perturbations, bias functions, memory and learning, local search on partially constructed solutions, hashing, and filtering. We also discuss in detail implementation strategies of memorybased intensification and postoptimization techniques using pathrelinking. Hybridizations with other metaheuristics, parallelization strategies, and applications are also reviewed.
Ant algorithms for discrete optimization
 ARTIFICIAL LIFE
, 1999
"... This article presents an overview of recent work on ant algorithms, that is, algorithms for discrete optimization that took inspiration from the observation of ant colonies’ foraging behavior, and introduces the ant colony optimization (ACO) metaheuristic. In the first part of the article the basic ..."
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Cited by 489 (42 self)
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This article presents an overview of recent work on ant algorithms, that is, algorithms for discrete optimization that took inspiration from the observation of ant colonies’ foraging behavior, and introduces the ant colony optimization (ACO) metaheuristic. In the first part of the article the basic biological findings on real ants are reviewed and their artificial counterparts as well as the ACO metaheuristic are defined. In the second part of the article a number of applications of ACO algorithms to combinatorial optimization and routing in communications networks are described. We conclude with a discussion of related work and of some of the most important aspects of the ACO metaheuristic.
Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
 Journal of the ACM
, 1998
"... Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c Ͼ 1 and given any n nodes in 2 , a randomized version of the scheme finds a (1 ϩ 1/c)approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes ..."
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Cited by 397 (2 self)
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Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c Ͼ 1 and given any n nodes in 2 , a randomized version of the scheme finds a (1 ϩ 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) ). For every fixed c, d the running time is n ⅐ 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 ᐉ p for p Ն 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.
The ant colony optimization metaheuristic
 in New Ideas in Optimization
, 1999
"... Ant algorithms are multiagent systems in which the behavior of each single agent, called artificial ant or ant for short in the following, is inspired by the behavior of real ants. Ant algorithms are one of the most successful examples of swarm intelligent systems [3], and have been applied to many ..."
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Cited by 389 (23 self)
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Ant algorithms are multiagent systems in which the behavior of each single agent, called artificial ant or ant for short in the following, is inspired by the behavior of real ants. Ant algorithms are one of the most successful examples of swarm intelligent systems [3], and have been applied to many types of problems, ranging from the classical traveling salesman
A Graduated Assignment Algorithm for Graph Matching
, 1996
"... A graduated assignment algorithm for graph matching is presented which is fast and accurate even in the presence of high noise. By combining graduated nonconvexity, twoway (assignment) constraints, and sparsity, large improvements in accuracy and speed are achieved. Its low order computational comp ..."
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Cited by 373 (15 self)
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A graduated assignment algorithm for graph matching is presented which is fast and accurate even in the presence of high noise. By combining graduated nonconvexity, twoway (assignment) constraints, and sparsity, large improvements in accuracy and speed are achieved. Its low order computational complexity [O(lm), where l and m are the number of links in the two graphs] and robustness in the presence of noise offer advantages over traditional combinatorial approaches. The algorithm, not restricted to any special class of graph, is applied to subgraph isomorphism, weighted graph matching, and attributed relational graph matching. To illustrate the performance of the algorithm, attributed relational graphs derived from objects are matched. Then, results from twentyfive thousand experiments conducted on 100 node random graphs of varying types (graphs with only zeroone links, weighted graphs, and graphs with node attributes and multiple link types) are reported. No comparable results have...
On the Solution of Traveling Salesman Problems
 DOC. MATH. J. DMV
, 1998
"... Following the theoretical studies of J.B. Robinson and H.W. Kuhn in the late 1940s and the early 1950s, G.B. Dantzig, R. Fulkerson, and S.M. Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TS ..."
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Cited by 225 (7 self)
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Following the theoretical studies of J.B. Robinson and H.W. Kuhn in the late 1940s and the early 1950s, G.B. Dantzig, R. Fulkerson, and S.M. Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TSP instances with more than several hundred cities; over the years, it has evolved further through the work of M. Grötschel , S. Hong , M. Jünger , P. Miliotis , D. Naddef , M. Padberg ... some of its refinements that led to the solution of a 13,509city instance.
Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to ..."
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Cited by 187 (15 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
Guillotine subdivisions approximate polygonal subdivisions: Part II  A simple polynomialtime approximation scheme for geometric kMST, TSP, and related problems
, 1996
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A Racing Algorithm for Configuring Metaheuristics
, 2002
"... This paper describes a racing procedure for finding, in a limited amount of time, a configuration of a metaheuristic that performs as good as possible on a given instance class of a combinatorial optimization problem. Taking inspiration from methods proposed in the machine learning literature ..."
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Cited by 164 (34 self)
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This paper describes a racing procedure for finding, in a limited amount of time, a configuration of a metaheuristic that performs as good as possible on a given instance class of a combinatorial optimization problem. Taking inspiration from methods proposed in the machine learning literature for model selection through crossvalidation, we propose a procedure that empirically evaluates a set of candidate configurations by discarding bad ones as soon as statistically sufficient evidence is gathered against them. We empirically evaluate our procedure using as an example the configuration of an ant colony optimization algorithm applied to the traveling salesman problem.