Separating Maximally Violated Comb Inequalities in Planar Graphs

The Traveling Salesman Problem TSP is a benchmark problem in combinatorial optimization. It was one of the very first problems used for developing and testing approaches to solving large integer programs, including cutting plane algorithms and branch-and-cut algorithms. Much of the research in this area has been focused on finding new classes of facets for the TSP polytope, and much less attention has been paid to algorithms for separating from these classes of facets. In this paper, we consider the problem of finding violated comb inequalities. If there are no violated subtour constraints in a fractional solution of the TSP, a comb inequality may not be violated by more than 1. Given a fractional solution in the subtour elimination polytope whose graph is planar, we either find a violated comb inequality or determine that there are no comb inequalities violated by 1. Our algorithm runs in On + MCn time, where MCn is the time to compute a cactus representation of all minimum cuts of a weighted planar graph on n vertices.

[1]  Giovanni Rinaldi,et al.  Facet identification for the symmetric traveling salesman polytope , 1990, Math. Program..

[2]  M. R. Rao,et al.  Odd Minimum Cut-Sets and b-Matchings , 1982, Math. Oper. Res..

[3]  Martin Grötschel,et al.  On the symmetric travelling salesman problem II: Lifting theorems and facets , 1979, Math. Program..

[4]  David R. Karger,et al.  Minimum cuts in near-linear time , 1996, STOC '96.

[5]  Michel X. Goemans,et al.  2-Change for k-connected networks , 1991, Oper. Res. Lett..

[6]  James B. Orlin,et al.  A faster algorithm for finding the minimum cut in a graph , 1992, SODA '92.

[7]  Martin Grötschel,et al.  Solution of large-scale symmetric travelling salesman problems , 1991, Math. Program..

[8]  Adam N. Letchford Separating a Superclass of Comb Inequalities in Planar Graphs , 2000, Math. Oper. Res..

[9]  Eugene L. Lawler,et al.  The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization , 1985 .

[10]  E. BixbyR. The Minimum Number of Edges and Vertices in a Graph with Edge Connectivity n and m n-Bonds , 1975 .

[11]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988, Wiley interscience series in discrete mathematics and optimization.

[12]  András A. Benczúr,et al.  Augmenting undirected connectivity in RNC and in randomized Õ(n3) time , 1994, STOC '94.

[13]  Robert Carr Separating Clique Trees and Bipartition Inequalities Having a Fixed Number of Handles and Teeth in Polynomial Time , 1997, Math. Oper. Res..

[14]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[15]  Toshihide Ibaraki,et al.  Computing Edge-Connectivity in Multigraphs and Capacitated Graphs , 1992, SIAM J. Discret. Math..

[16]  Gerhard Reinelt,et al.  Traveling Salesman Problem , 2012 .

[17]  M. Padberg,et al.  On the symmetric travelling salesman problem II , 1979 .

[18]  Martin Grötschel,et al.  On the symmetric travelling salesman problem I: Inequalities , 1979, Math. Program..

[19]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[20]  E. A. Timofeev,et al.  Efficient algorithm for finding all minimal edge cuts of a nonoriented graph , 1986 .

[21]  Toshihide Ibaraki,et al.  A fast algorithm for cactus representations of minimum cuts , 2000 .

[22]  Matteo Fischetti,et al.  On the Separation of Maximally Violated mod-k Cuts , 1999, IPCO.

[23]  Éva Tardos,et al.  Separating Maximally Violated Comb Inequalities in Planar Graphs , 1996, IPCO.

[24]  M. Padberg,et al.  Addendum: Optimization of a 532-city symmetric traveling salesman problem by branch and cut , 1990 .

[25]  Martin Grötschel,et al.  On the symmetric travelling salesman problem: Solution of a 120-city problem , 1980 .

[26]  András A. Benczúr,et al.  Cut structures and randomized algorithms in edge-connectivity problems , 1997 .

[27]  H. Crowder,et al.  Solving Large-Scale Symmetric Travelling Salesman Problems to Optimality , 1980 .

[28]  Harold N. Gabow,et al.  Applications of a poset representation to edge connectivity and graph rigidity , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[29]  Lisa Fleischer Building Chain and Cactus Representations of All Minimum Cuts from Hao-Orlin in the Same Asymptotic Run Time , 1999, J. Algorithms.

[30]  Richard M. Karp,et al.  The Traveling-Salesman Problem and Minimum Spanning Trees , 1970, Oper. Res..

[31]  Vijay V. Vazirani,et al.  Representing and Enumerating Edge Connectivity Cuts in RNC , 1991, WADS.

[32]  Vasek Chvátal,et al.  Edmonds polytopes and weakly hamiltonian graphs , 1973, Math. Program..

[33]  David R. Karger,et al.  A new approach to the minimum cut problem , 1996, JACM.

[34]  David Applegate,et al.  Finding Cuts in the TSP (A preliminary report) , 1995 .

[35]  Kellogg S. Booth,et al.  Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms , 1976, J. Comput. Syst. Sci..

[36]  Hiroshi Nagamochi,et al.  Canonical cactus representation for minimum cuts , 1994 .

[37]  Robert D. Carr Separating Clique Tree and Bipartition Inequalities in Polynominal Time , 1995, IPCO.