On the separation of split cuts and related inequalities

Abstract. The split cuts of Cook, Kannan and Schrijver are general-purpose valid inequalities for integer programming which include a variety of other well-known cuts as special cases. To detect violated split cuts, one has to solve the associated separation problem. The complexity of split cut separation was recently cited as an open problem by Cornuéjols & Li CL01. In this paper we settle this question by proving strong 𝒩𝒫-completeness of separation for split cuts. As a by-product we also show 𝒩𝒫-completeness of separation for several other classes of inequalities, including the MIR-inequalities of Nemhauser and Wolsey and some new inequalities which we call balanced split cuts and binary split cuts. We also strengthen 𝒩𝒫-completeness results of Caprara & Fischetti CF96 (for -cuts) and Eisenbrand E99 (for Chvátal-Gomory cuts). To compensate for this bleak picture, we also give a positive result for the Symmetric Travelling Salesman Problem. We show how to separate in polynomial time over a class of split cuts which includes all comb inequalities with a fixed handle.

[1]  R. Gomory.  AN ALGORITHM FOR THE MIXED INTEGER PROBLEM , 1960 .

[2]  Bert Gerards,et al.  Matrices with the edmonds—Johnson property , 1986, Comb..

[3]  Matteo Fischetti,et al.  {0, 1/2}-Chvátal-Gomory cuts , 1996, Math. Program..

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

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

[6]  Friedrich Eisenbrand,et al.  NOTE – On the Membership Problem for the Elementary Closure of a Polyhedron , 1999, Comb..

[7]  Adam N. Letchford.  On Disjunctive Cuts for Combinatorial Optimization , 2001, J. Comb. Optim..

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

[9]  Gérard Cornuéjols,et al.  Elementary closures for integer programs , 2001, Oper. Res. Lett..

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

[11]  Eugene L. Lawler,et al.  Traveling Salesman Problem , 2016 .

[12]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[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]  Laurence A. Wolsey,et al.  A recursive procedure to generate all cuts for 0–1 mixed integer programs , 1990, Math. Program..

[15]  Egon Balas,et al.  Intersection Cuts - A New Type of Cutting Planes for Integer Programming , 1971, Oper. Res..

[16]  Mihalis Yannakakis,et al.  Expressing combinatorial optimization problems by linear programs , 1991, STOC '88.

[17]  William J. Cook,et al.  Chvátal closures for mixed integer programming problems , 1990, Math. Program..

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

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

[20]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

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

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

[24]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..

[25]  Egon Balas,et al.  programming: Properties of the convex hull of feasible points * , 1998 .

[26]  Giovanni Rinaldi,et al.  A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Traveling Salesman Problems , 1991, SIAM Rev..

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

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

[29]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[30]  Manfred W. Padberg,et al.  On the facial structure of set packing polyhedra , 1973, Math. Program..