The Dulmage--Mendelsohn decomposition (or the DM-decomposition) gives a unique partition of the vertex set of a bipartite graph reflecting the structure of all the maximum matchings therein. A bipartite graph is said to be DM-irreducible if its DM-decomposition consists of a single component. In this paper, we focus on the problem of making a given bipartite graph DM-irreducible by adding edges. When the input bipartite graph is balanced (i.e., both sides have the same number of vertices) and has a perfect matching, this problem is equivalent to making a directed graph strongly connected by adding edges, for which the minimum number of additional edges was characterized by Eswaran and Tarjan (1976). We give a general solution to this problem, which is divided into three parts. We first show that our problem can be formulated as a special case of a general framework of covering supermodular functions, which was introduced by Frank and Jord 'an (1995) to investigate the directed connectivity augmentation problem. Secondly, when the input graph is not balanced, the problem is solved via matroid intersection. This result can be extended to the minimum cost version in which the addition of an edge gives rise to an individual cost. Thirdly, for balanced input graphs, we devise a combinatorial algorithm that finds a minimum number of additional edges to attain the DM-irreducibility, while the minimum cost version of this problem is NP-hard. These results also lead to min-max characterizations of the minimum number, which generalize the result of Eswaran and Tarjan.
The subject of sparse matrices has its roots in such diverse fields as management science, power systems analysis, surveying, circuit theory, and structural analysis. Mathematical models in all these areas give rise to very large systems of linear equations which could not be solved were it not for the fact that the matrices contain relatively few non-zero entries. Only comparatively recently, in the last fifteen years or so, has it become apparent that the equations can be solved even when the pattern is irregular, and it is primarily the solution of such problems that is considered in this book. The subject is intensely practical and this book is written with practicalities ever in mind. Whenever two methods are applicable, their rival merits are considered, and conclusions are based on practical experience where theoretical comparison is not possible. Non-numeric computing techniques have been included, as well as frequent illustrations, in an attempt to bridge the usually wide gap between the printed page and the working computer code. Despite this practical bias, it is recognized that many aspects of the subject are of interest in their own right, and the book aims to be suitable also for a student course, probably at MSc level. Exercises have been included to illustrate and strengthen understanding of the material, as well as to extend it. Efficient use of sparsity is a key to solving large problems in many fields. This book will supply both insight and answers for those attempting to solve these problems.