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130
Branchandprice: Column generation for solving huge integer programs
 Oper. Res
, 1998
"... We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branchandbound tree. We present classes of models for which th ..."
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Cited by 348 (13 self)
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We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branchandbound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. Wethen discuss computational issues and implementation of column generation, branchandbound algorithms, including special branching rules and e cient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality. 1
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.
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.
Approximation algorithms for TSP with neighborhoods in the plane
 J. ALGORITHMS
, 2001
"... In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. As a generalization of the classical Euclidean TSP, TSPN is also NPhard. In this paper, we present new approximation results for the TSPN, incl ..."
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Cited by 90 (9 self)
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In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. As a generalization of the classical Euclidean TSP, TSPN is also NPhard. In this paper, we present new approximation results for the TSPN, including (1) a constantfactor approximation algorithm for the case of arbitrary connected neighborhoods having comparable diameters; and (2) a PTAS for the important special case of disjoint unit disk neighborhoods (or nearly disjoint, nearlyunit disks). Our methods also yield improved approximation ratios for various special classes of neighborhoods, which have previously been studied. Further, we give a lineartime O(1) approximation algorithm for the case of neighborhoods that are (innite) straight lines.
A BranchandCut Algorithm for the Symmetric Generalized Travelling Salesman Problem
, 1995
"... We consider a variant of the classical symmetric Travelling Salesman Problem in which the nodes are partitioned into clusters and the salesman has to visit at least one node for each cluster. This NPhard problem is known in the literature as the symmetric Generalized Travelling Salesman Problem (GT ..."
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Cited by 75 (4 self)
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We consider a variant of the classical symmetric Travelling Salesman Problem in which the nodes are partitioned into clusters and the salesman has to visit at least one node for each cluster. This NPhard problem is known in the literature as the symmetric Generalized Travelling Salesman Problem (GTSP), and finds practical applications in routing, scheduling and locationrouting. In a companion paper [5] we modeled GTSP as an integer linear program, and studied the facial structure of two polytopes associated with the problem. Here we propose exact and heuristic separation procedures for some classes of facetdefining inequalities, which are used within a branchandcut algorithm for the exact solution of GTSP. Heuristic procedures are also described. Extensive computational results for instances taken from the literature and involving up to 442 nodes are reported.
Chained LinKernighan for large traveling salesman problems
, 2000
"... We discuss several issues that arise in the implementation of Martin, Otto, and Felten's Chained LinKernighan heuristic for largescale traveling salesman problems. Computational results are presented for TSPLIB instances ranging in size from 11,849 cities up to 85,900 cities; for each of t ..."
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Cited by 68 (1 self)
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We discuss several issues that arise in the implementation of Martin, Otto, and Felten's Chained LinKernighan heuristic for largescale traveling salesman problems. Computational results are presented for TSPLIB instances ranging in size from 11,849 cities up to 85,900 cities; for each of these instances, solutions within 1% of the optimal value can be found in under 1 CPU minute on a 300 Mhz Pentium II workstation, and solutions within 0.5% of optimal can be found in under 10 CPU minutes. We also demonstrate the scalability of the heuristic, presenting results for randomly generated Euclidean instances having up to 25,000,000 cities. For the largest of these random instances, a tour within 1% of an estimate of the optimal value can be obtained in under 1 CPU day on a 64bit IBM RS6000 workstation.
Sorting Permutations by Reversals and Eulerian Cycle Decompositions
 SIAM Journal on Discrete Mathematics
, 1997
"... We analyze the strong relationship among three combinatorial problems, namely the problem of sorting a permutation by the minimum number of reversals (MINSBR), the problem of finding the maximum number of edgedisjoint alternating cycles in a breakpoint graph associated with a given permutation (MA ..."
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Cited by 65 (9 self)
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We analyze the strong relationship among three combinatorial problems, namely the problem of sorting a permutation by the minimum number of reversals (MINSBR), the problem of finding the maximum number of edgedisjoint alternating cycles in a breakpoint graph associated with a given permutation (MAXACD), and the problem of partitioning the edge set of a Eulerian graph into the maximum number of cycles (MAXECD). We first illustrate a nice characterization of breakpoint graphs, which leads to a linear time algorithm for their recognition. This characterization is used to prove that MAXECD and MAXACD are equivalent, showing the latter is NPhard. We then describe a transformation from MAXACD to MINSBR, which is therefore shown to be NPhard as well, answering an outstanding question which has been open for some years. Finally, we derive the worstcase performance of a well known lower bound for MINSBR, obtained by solving MAXACD, discussing its implications on approximation algori...
Guided local search and its application to the traveling salesman problem
, 1998
"... The Traveling Salesman Problem (TSP) is one of the most famous problems in combinatorial optimization. In this paper, we are going to examine how the techniques of Guided Local Search (GLS) and Fast Local Search (FLS) can be applied to the problem. Guided Local Search sits on top of local search heu ..."
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Cited by 57 (16 self)
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The Traveling Salesman Problem (TSP) is one of the most famous problems in combinatorial optimization. In this paper, we are going to examine how the techniques of Guided Local Search (GLS) and Fast Local Search (FLS) can be applied to the problem. Guided Local Search sits on top of local search heuristics and has as a main aim to guide these procedures in exploring efficiently and effectively the vast search spaces of combinatorial optimization problems. Guided Local Search can be combined with the neighborhood reduction scheme of Fast Local Search which significantly speeds up the operations of the algorithm. The combination of GLS and FLS with TSP local search heuristics of different efficiency and effectiveness is studied in an effort to determine the dependence of GLS on the underlying local search heuristic used. Comparisons are made with some of the best TSP heuristic algorithms and general optimization techniques which demonstrate the advantages of GLS over alternative heuristic approaches suggested for the problem.
Variable Neighborhood Decomposition Search
, 2001
"... The recent Variable Neighborhood Search (VNS) metaheuristic combines local search with systematic changes of neighborhood in the descent and escape from local optimum phases. When solving large instances of various problems, its efficiency may be enhanced through decomposition. The resulting two lev ..."
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Cited by 54 (10 self)
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The recent Variable Neighborhood Search (VNS) metaheuristic combines local search with systematic changes of neighborhood in the descent and escape from local optimum phases. When solving large instances of various problems, its efficiency may be enhanced through decomposition. The resulting two level VNS, called Variable Neighborhood Decomposition Search (VNDS), is presented and illustrated on the pmedian problem. Results on 1400, 3038 and 5934 node instances from the TSP library show VNDS improves notably upon VNS in less computing time, and gives much better results than Fast Interchange (FI), in the same time that FI takes for a single descent. Moreover, Reduced VNS (RVNS), which does not use a descent phase, gives results similar to those of FI in much less computing time.