Matroidal Structure Research Articles

The Edmonds-Gallai Decomposition for the $k$-Piece Packing Problem

Generalizing Kaneko's long path packing problem, Hartvigsen, Hell and Szabó consider a new type of undirected graph packing problem, called the $k$-piece packing problem. A $k$-piece is a simple, connected graph with highest degree exactly $k$ so in the case $k=1$ we get the classical matching problem. They give a polynomial algorithm, a Tutte-type characterization and a Berge-type minimax formula for the $k$-piece packing problem. However, they leave open the question of an Edmonds-Gallai type decomposition. This paper fills this gap by describing such a decomposition. We also prove that the vertex sets coverable by $k$-piece packings have a certain matroidal structure.

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

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.

Constrained matroidal bottleneck problems

Algorithms for the constrained bottleneck matroid base problem and the constrained bottleneck independent set problem are presented. The algorithms have improved complexities upon the best known general algorithm. Sufficient conditions are obtained when the proposed algorithms have linear complexity and are therefore the best possible ones. Several specific classes of matroidal problems are considered, including problems on partition, graphic and uniform matroids, special variants of the assignment and scheduling problems. The proposed approach is based on shrinking iteratively the dimensions of auxiliary problems. It is demonstrated that the approach cannot be extended beyond the class of matroidal problems, and the assumption of matroidal structure is not only sufficient but also necessary for the applicability of the approach.

The Delivery Man Problem and Cumulative Matroids

Given a complete directed graph G = (V, A), the delivery man problem (DMP) consists of determining a Hamiltonian circuit minimizing the sum of distances (along the circuit) from a given vertex v1, to every vertex of V, including v1 itself. There exists a number of applications of the DMP in the fields of distribution and machine scheduling. The DMP is NP-hard. The objective of this paper is to develop new theoretical results and an exact algorithm for the problem. A new, integer linear programming formulation is provided, and results on the matroidal structure of a class of combinatorial problems are developed. These are used to derive lower bounds for the DMP. These bounds are embedded into an enumerative algorithm. The largest problems solved to optimality with the proposed algorithm involve 60 vertices. This compares favorably with previously published methods.