Matroid Steiner problems, the Tutte polynomial and network reliability

• Published in: Journal of combinatorial theory. Series B Vol. 47; no. 1; pp. 20 - 31
• Main Authors: Colbourn, Charles J, Pulleyblank, William R
• Format: Journal Article
• Language: English
• Published: Duluth, MN Elsevier Inc 01.08.1989; Academic Press
• Subjects: Applied sciences; Exact sciences and technology; Flows in networks. Combinatorial problems; Operational research and scientific management; Operational research. Management science; Matroid; Decomposition method; Network; System reliability; Generalization; Algorithm; Optimization
• ISSN: 0095-8956; 1096-0902
• DOI: 10.1016/0095-8956(89)90062-2

A matroid Steiner problem is obtained by selecting a suitable subfamily C of the cocircuits, and then defining a Steiner “tree” to be a minimal set having nonempty intersection with all members of C . The family of all sets whose complements contain Steiner trees forms an independence system which we call the Steiner complex. We show that this Steiner complex can be partitioned into as many intervals as there are bases in the underlying matroid. As a special case, we obtain explicit partitions of the independent sets of any matroid. It also shows that both F-complexes associated with network reliability problems and the family of bipartite subgraphs of a graph can be partitioned into as many intervals as there are spanning trees. We also describe a generalization of the Tutte polynomial for matroids to an extended Tutte polynomial for Steiner complexes. This provides an alternative method for evaluating the independence or reliability polynomial. We also discuss applications to network reliability.