Abstract
In this paper, we introduce lower-truncated transversal polymatroids, and develop efficient algorithms of network-flow type for those polymatroids. The lower-truncated transversal polymatroid contains, as special cases, a variety of useful matroids such as cycle matroids of graphs, matroids in plane skeletal structures, etc. We present simple and powerful theorems which enable us to solve various combinatorial optimization problems for those polymatroids by means of network-flow algorithms. Especially, we can solve greedy-type optimization problems concerning those polymatroids in a remarkably efficient manner. As greedy-type problems, we take up the problem of finding a maximum-weight independent vector, that of finding the principal partition and that of covering and packing, and give efficient solutions for them. Applying general algorithms for lower-truncated transversal polymatroids to cycle matroids of graphs and matroids in plane skeletal structures, we obtain various new results. From the viewpoint of applications, Iower-truncated transversal polymatroids are essentially related to discrete systems with internal degrees of freedom which arise in many fields of engineering, so that the algorithms for those polymatroids developed in this paper give efficient methods to analyze such systems in a unifying manner.
