Separation Algorithms for Classes of STSP Inequalities Arising from a New STSP Relaxation

The problem ofseparating a class of inequalities that are valid for the symmetric traveling salesman problem (STSP) consists of devising an efficient method that, given a fractional pointx*. for a relaxation of the STSP, either finds an inequality in the given class of STSP inequalities that is violated byx*. or determines that there are no such violated inequalities. We can define important classes of STSP inequalities by performing Naddef's and Rinaldi's (1993)node-lifting operation on STSP inequalities defined on small graphs.We present efficient methods forexactly separating large classes of STSP inequalities that arise from an STSP relaxation related to node lifting. In particular, we show how to find efficiently the most-violated inequality in the class of STSP inequalities having abackbone set with a constant numberk of vertices, and thus separate in polynomial time the class of all STSP inequalities whose backbone set has at mostk vertices.

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

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

[3]  Andrea Lodi,et al.  Polynomial-Time Separation of Simple Comb Inequalities , 2002, IPCO.

[4]  Giovanni Rinaldi,et al.  The graphical relaxation: A new framework for the symmetric traveling salesman polytope , 1993, Math. Program..

[5]  Gérard Cornuéjols,et al.  The traveling salesman problem on a graph and some related integer polyhedra , 1985, Math. Program..

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

[7]  William H. Cunningham,et al.  Small Travelling Salesman Polytopes , 1991, Math. Oper. Res..

[8]  T. C. Hu,et al.  Multi-Terminal Network Flows , 1961 .

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

[10]  G. Rinaldi,et al.  Chapter 4 The traveling salesman problem , 1995 .

[11]  M. Yannakakis.  Expressing combinatorial optimization problems by linear programs , 1991, Symposium on the Theory of Computing.

[12]  Matteo Fischetti,et al.  On the separation of maximally violated mod-k cuts , 1999, Math. Program..

[13]  Michel X. Goemans,et al.  Worst-case comparison of valid inequalities for the TSP , 1995, Math. Program..

[14]  Manfred W. Padberg,et al.  A branch-and-cut approach to a traveling salesman problem with side constraints , 1989 .

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

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

[17]  Robert D. Carr.  Some Results on Node Lifting of TSP Inequalities , 2000, J. Comb. Optim..

[18]  Robert D. Carr,et al.  Towards a 4/3 approximation for the asymmetric traveling salesman problem , 1999, SODA '00.

[19]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[20]  Éva Tardos,et al.  Separating Maximally Violated Comb Inequalities in Planar Graphs , 1999, Math. Oper. Res..

[21]  Gerhard Reinelt,et al.  Traveling salesman problem , 2012 .

[22]  Robert Carr.  Separating over Classes of TSP Inequalities Defined by 0 Node-Lifting in Polynominal Time , 1996, IPCO.

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

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