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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 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.
Voronoi diagrams  a survey of a fundamental geometric data structure
 ACM COMPUTING SURVEYS
, 1991
"... This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. ..."
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Cited by 753 (5 self)
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This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.
On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts  Towards Memetic Algorithms
, 1989
"... Short abstract, isn't it? P.A.C.S. numbers 05.20, 02.50, 87.10 1 Introduction Large Numbers "...the optimal tour displayed (see Figure 6) is the possible unique tour having one arc fixed from among 10 655 tours that are possible among 318 points and have one arc fixed. Assuming that ..."
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Cited by 241 (10 self)
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Short abstract, isn't it? P.A.C.S. numbers 05.20, 02.50, 87.10 1 Introduction Large Numbers "...the optimal tour displayed (see Figure 6) is the possible unique tour having one arc fixed from among 10 655 tours that are possible among 318 points and have one arc fixed. Assuming that one could possibly enumerate 10 9 tours per second on a computer it would thus take roughly 10 639 years of computing to establish the optimality of this tour by exhaustive enumeration." This quote shows the real difficulty of a combinatorial optimization problem. The huge number of configurations is the primary difficulty when dealing with one of these problems. The quote belongs to M.W Padberg and M. Grotschel, Chap. 9., "Polyhedral computations", from the book The Traveling Salesman Problem: A Guided tour of Combinatorial Optimization [124]. It is interesting to compare the number of configurations of realworld problems in combinatorial optimization with those large numbers arising in Cosmol...
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 194 (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
An effective implementation of the linkernighan traveling salesman heuristic
 European Journal of Operational Research
, 2000
"... This report describes an implementation of the LinKernighan heuristic, one of the most successful methods for generating optimal or nearoptimal solutions for the symmetric traveling salesman problem. Computational tests show that the implementation is highly effective. It has found optimal solution ..."
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Cited by 188 (1 self)
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This report describes an implementation of the LinKernighan heuristic, one of the most successful methods for generating optimal or nearoptimal solutions for the symmetric traveling salesman problem. Computational tests show that the implementation is highly effective. It has found optimal solutions for all solved problem instances we have been able to obtain, including a 7397city problem (the largest nontrivial problem instance solved to optimality today). Furthermore, the algorithm has improved the best known solutions for a series of largescale problems with unknown optima, among these an 85900city problem. 1.
Multicast Tree Generation in Networks with Asymmetric Links
 IEEE/ACM Transactions on Networking
, 1996
"... We formulate the problem of multicast tree generation as one of computing a directed Steiner tree of minimal cost. In this context, we present a polynomialtime algorithm that provides for tradeoff selection, using a single parameter , between the treecost (Steiner cost) and the runtime efficiency ..."
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Cited by 82 (0 self)
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We formulate the problem of multicast tree generation as one of computing a directed Steiner tree of minimal cost. In this context, we present a polynomialtime algorithm that provides for tradeoff selection, using a single parameter , between the treecost (Steiner cost) and the runtime efficiency. Further, the same algorithm may be used for delay optimization or treecost minimization simply by configuring the value of appropriately. We present theoretical and experimental analysis characterizing the problem and the performance of our algorithm. Theoretically, we (1) show that it is highly unlikely that there exists a polynomialtime algorithm with a performance guarantee of constant times optimum cost, (2) introduce metrics for measuring the asymmetry of graphs, and (3) show that the worstcase cost of the tree produced by our algorithm is at most twice the optimum cost times the asymmetry, for two of these asymmetry metrics. For graphs with bounded asymmetry, this gives constant ...
Approximation Algorithms For The Geometric Covering Salesman Problem
 Discrete Applied Mathematics
, 1995
"... We introduce a geometric version of the Covering Salesman Problem: Each of the n salesman's clients specifies a neighborhood in which they are willing to meet the salesman. Identifying a tour of minimum length that visits all neighborhoods is an NPhard problem, since it is a generalization of ..."
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Cited by 78 (3 self)
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We introduce a geometric version of the Covering Salesman Problem: Each of the n salesman's clients specifies a neighborhood in which they are willing to meet the salesman. Identifying a tour of minimum length that visits all neighborhoods is an NPhard problem, since it is a generalization of the Traveling Salesman Problem. We present simple heuristic procedures for constructing tours, for a variety of neighborhood types, whose length is guaranteed to be within a constant factor of the length of an optimal tour. The neighborhoods we consider include, parallel unit segments, translates of a polygonal region, and circles. y Partially supported by NSF Grants DMS 8903304 and ECSE8857642. 1 Introduction A salesman wants to meet a set of potential buyers. Each buyer specifies a compact set in the plane, his neighborhood, within which he is willing to meet. For example, the neighborhoods may be disks centered at the buyers locations, and each disk's radius specifies the distance that a ...
Experimental Analysis of Heuristics for the STSP
 Local Search in Combinatorial Optimization
, 2001
"... In this and the following chapter, we consider what approaches one should take when one is confronted with a realworld application of the TSP. What algorithms should be used under which circumstances? We ..."
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Cited by 66 (1 self)
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In this and the following chapter, we consider what approaches one should take when one is confronted with a realworld application of the TSP. What algorithms should be used under which circumstances? We
Online Steiner Trees in the Euclidean Plane
 Discrete and Computational Geometry
, 1993
"... Suppose we are given a sequence of n points in the Euclidean plane, and our objective is to construct, online, a connected graph that connects all of them, trying to minimize the total sum of lengths of its edges. The points appear one at a time, and at each step the online algorithm must construc ..."
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Cited by 47 (3 self)
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Suppose we are given a sequence of n points in the Euclidean plane, and our objective is to construct, online, a connected graph that connects all of them, trying to minimize the total sum of lengths of its edges. The points appear one at a time, and at each step the online algorithm must construct a connected graph that contains all current points by connecting the new point to the previously constructed graph. This can be done by joining the new point (not necessarily by a straight line) to any point of the previous graph, (not necessarily one of the given points). The performance of our algorithm is measured by its competitive ratio: the supremum, over all sequences of points, of the ratio between the total length of the graph constructed by our algorithm and the total length of the best Steiner tree that connects all the points. There are known online algorithms whose competitive ratio is O(log n) even for all metric spaces, but the only lower bound known is of [IW] for some con...
MINIMUMWEIGHT TWOCONNECTED SPANNING NETWORKS
, 1990
"... We consider the problem of constructing a minimumweight, twoconnected network spanning all the points in a set V. We assume a symmetric, nonnegative distance function d ( ' ) defined on V x V which satisfies the triangle inequality. We obtain a structural characterization of optimal solution ..."
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Cited by 47 (3 self)
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We consider the problem of constructing a minimumweight, twoconnected network spanning all the points in a set V. We assume a symmetric, nonnegative distance function d ( ' ) defined on V x V which satisfies the triangle inequality. We obtain a structural characterization of optimal solutions. Specifically, there exists an optimal twoconnected solution whose vertices all have degree 2 or 3, and such that the removal of any edge or pair of edges leaves a bridge in the resulting connected components. These are the strongest possible conditions on the structure of an optimal solution since we also show thar any twoconnected graph satisfying these conditions is the unique optimal solution for a particular choice of 'canonical' distances satisfying the triangle inequality. we use these properties to show that the weight of an optimal traveling salesman cycle i, uiort f times the weight of an optimal twoconnected solution; examples are provided which approach this bound arbiirarily closely. In addition, we obtain similar results for the variation of thii problem where the network need only span a prespecified subset of the points.