Standard pairs for lawrence-type matrices and their applications to several lawrence-type integer programs

Abstract

Computational algebraic approaches using Gröbner bases or standard pairs of toric ideals to integer programming problems have been studied in recent years. In combinatorial optimization problems, matrices of Lawrence type arise in many situations as capacitated integer programming problems and some multidimensional transportation problems. While the reduced Gröbner basis of the toric ideal for the Lawrence lifting Λ(A) of A relates a set of circuits of the vector matroid defined by a matrix A, a relation of the vector matroid with a standard pair decomposition for Λ(A) is not well-known. In this paper, we show a bijection between the set of bases of a matrix A and the set of standard pairs Λ(A) which relates with the dual feasible bases for a linear programming problem with a coefficient matrix Λ(A). This relation gives a matroidal structure of standard pairs in a sense that an adjacency of standard pairs corresponds to that of vertices in a base polyhedron of a vector matroid. As applications to Lawrence type integer programming problems, we analyze the number of dual feasible bases for minimum cost flow problems and for multidimensional transportation problems of type 2 × ··· × 2 × M × N.