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197
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 ..."
Abstract

Cited by 1000 (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.
No Free Lunch Theorems for Search
, 1995
"... We show that all algorithms that search for an extremum of a cost function perform exactly the same, when averaged over all possible cost functions. In particular, if algorithm A outperforms algorithm B on some cost functions, then loosely speaking there must exist exactly as many other functions wh ..."
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Cited by 292 (2 self)
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We show that all algorithms that search for an extremum of a cost function perform exactly the same, when averaged over all possible cost functions. In particular, if algorithm A outperforms algorithm B on some cost functions, then loosely speaking there must exist exactly as many other functions where B outperforms A. Starting from this we analyze a number of the other a priori characteristics of the search problem, like its geometry and its informationtheoretic aspects. This analysis allows us to derive mathematical benchmarks for assessing a particular search algorithm 's performance. We also investigate minimax aspects of the search problem, the validity of using characteristics of a partial search over a cost function to predict future behavior of the search algorithm on that cost function, and timevarying cost functions. We conclude with some discussion of the justifiability of biologicallyinspired search methods.
Ant colonies for the travelling salesman problem
, 1997
"... We describe an artificial ant colony capable of solving the travelling salesman problem (TSP). Ants of the artificial colony are able to generate successively shorter feasible tours by using information accumulated in the form of a pheromone trail deposited on the edges of the TSP graph. Computer si ..."
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Cited by 276 (5 self)
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We describe an artificial ant colony capable of solving the travelling salesman problem (TSP). Ants of the artificial colony are able to generate successively shorter feasible tours by using information accumulated in the form of a pheromone trail deposited on the edges of the TSP graph. Computer simulations demonstrate that the artificial ant colony is capable of generating good solutions to both symmetric and asymmetric instances of the TSP. The method is an example, like simulated annealing, neural networks and evolutionary computation, of the successful use of a natural metaphor to design an optimization algorithm.
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 226 (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.
Very LargeScale Neighborhood Search for the Quadratic Assignment Problem
 DISCRETE APPLIED MATHEMATICS
, 2002
"... The Quadratic Assignment Problem (QAP) consists of assigning n facilities to n locations so as to minimize the total weighted cost of interactions between facilities. The QAP arises in many diverse settings, is known to be NPhard, and can be solved to optimality only for fairly small size instances ..."
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Cited by 148 (13 self)
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The Quadratic Assignment Problem (QAP) consists of assigning n facilities to n locations so as to minimize the total weighted cost of interactions between facilities. The QAP arises in many diverse settings, is known to be NPhard, and can be solved to optimality only for fairly small size instances (typically, n < 25). Neighborhood search algorithms are the most popular heuristic algorithms to solve larger size instances of the QAP. The most extensively used neighborhood structure for the QAP is the 2exchange neighborhood. This neighborhood is obtained by swapping the locations of two facilities and thus has size O(n²). Previous efforts to explore larger size neighborhoods (such as 3exchange or 4exchange neighborhoods) were not very successful, as it took too long to evaluate the larger set of neighbors. In this paper, we propose very largescale neighborhood (VLSN) search algorithms where the size of the neighborhood is very large and we propose a novel search procedure to heuristically enumerate good neighbors. Our search procedure relies on the concept of improvement graph which allows us to evaluate neighbors much faster than the existing methods. We present extensive computational results of our algorithms on standard benchmark instances. These investigations reveal that very largescale neighborhood search algorithms give consistently better solutions compared the popular 2exchange neighborhood algorithms considering both the solution time and solution accuracy.
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 137 (15 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...
AntQ: A Reinforcement Learning approach to the traveling salesman problem
, 1995
"... In this paper we introduce AntQ, a family of algorithms which present many similarities with Qlearning (Watkins, 1989), and which we apply to the solution of symmetric and asymmetric instances of the traveling salesman problem (TSP). AntQ algorithms were inspired by work on the ant system ( ..."
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Cited by 127 (15 self)
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In this paper we introduce AntQ, a family of algorithms which present many similarities with Qlearning (Watkins, 1989), and which we apply to the solution of symmetric and asymmetric instances of the traveling salesman problem (TSP). AntQ algorithms were inspired by work on the ant system (AS), a distributed algorithm for combinatorial optimization based on the metaphor of ant colonies which was recently proposed in (Dorigo, 1992; Dorigo, Maniezzo and Colorni, 1996). We show that AS is a particular instance of the AntQ family, and that there are instances of this family which perform better than AS. We experimentally investigate the functioning of AntQ and we show that the results obtained by AntQ on symmetric TSP's are competitive with those obtained by other heuristic approaches based on neural networks or local search. Finally, we apply AntQ to some difficult asymmetric TSP's obtaining very good results: AntQ was able to find solutions of a quality which...
MAXMIN Ant System
, 1999
"... Ant System, the first Ant Colony Optimization algorithm, showed to be a viable method for attacking hard combinatorial optimization problems. Yet, its performance, when compared to more finetuned algorithms, was rather poor for large instances of traditional benchmark problems like the Traveling Sa ..."
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Cited by 122 (3 self)
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Ant System, the first Ant Colony Optimization algorithm, showed to be a viable method for attacking hard combinatorial optimization problems. Yet, its performance, when compared to more finetuned algorithms, was rather poor for large instances of traditional benchmark problems like the Traveling Salesman Problem. To show that Ant Colony Optimization algorithms could be good alternatives to existing algorithms for hard combinatorial optimization problems, recent research in this ares has mainly focused on the development of algorithmic variants which achieve better performance than AS. In this article, we present¨�©� � –¨��� � Ant System, an Ant Colony Optimization algorithm derived from Ant System.¨�©� � –¨��� � Ant System differs from Ant System in several important aspects, whose usefulness we demonstrate by means of an experimental study. Additionally, we relate one of the characteristics specific to¨� ¨ AS — that of using a greedier search than Ant System — to results from the search space analysis of the combinatorial optimization problems attacked in this paper. Our computational results on the Traveling Salesman Problem and the Quadratic Assignment Problem show that ¨�©� � –¨��� � Ant System is currently among the best performing algorithms for these problems.
Solving Symmetric and Asymmetric TSPs by Ant Colonies
, 1996
"... In this paper we present ACS, a distributed algorithm for the solution of combinatorial optimization problems which was inspired by the observation of real colonies of ants. We apply ACS to both symmetric and asymmetric traveling salesman problems. Results show that ACS is able to find good sol ..."
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Cited by 102 (20 self)
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In this paper we present ACS, a distributed algorithm for the solution of combinatorial optimization problems which was inspired by the observation of real colonies of ants. We apply ACS to both symmetric and asymmetric traveling salesman problems. Results show that ACS is able to find good solutions to these problems. I. Introduction In this paper we present Ant Colony System (ACS), a novel distributed approach to combinatorial optimization based on the observation of real ant colonies behavior. ACS finds its ground in one of the authors previous work on the socalled Ant System (AS) [1],[2],[5],[7] and in AntQ [8] an extension of AS with Qlearning [12], a reinforcement learning technique. In particular, ACS is a revisited version of AntQ where a different way to update the experience accumulated by the artificial ants has been introduced [6]. All the mentioned systems belong to the Artificial Ant Colonies (AAC) family of algorithms that has been applied to various combinat...
Adaptive simulated annealing (ASA): Lessons learned
 Control and Cybernetics
, 1996
"... Adaptive simulated annealing (ASA) is a global optimization algorithm based on an associated proof that the parameter space can be sampled much more efficiently than by using other previous simulated annealing algorithms. The author's ASA code has been publicly available for over two years. ..."
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Cited by 93 (13 self)
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Adaptive simulated annealing (ASA) is a global optimization algorithm based on an associated proof that the parameter space can be sampled much more efficiently than by using other previous simulated annealing algorithms. The author's ASA code has been publicly available for over two years. During this time the author has volunteered to help people via email, and the feedback obtained has been used to further develop the code.