Polymatroids and electrical networks
This paper is concerned with the intimate relationship that exists between the theories of Electrical Networks and Polymatroids. We illustrate this fact by discussing a number of (physically significant) variations of the notion of the hybrid rank of an electrical network (the minimum size of a subset H of branches of its graph such that when some of these are shorted and others opened, the branches outside H become either self loops or coloops). We generalize this definition in two different directions. We show that both the primitive definition and its first demineralization are related to the (combinatorial) convolution operation. A natural consequence is the notion of the principal partition of a polymatroid with respect to a modular function. We also show that the second demineralization is related to the Dilworth truncation operation and a natural consequence is the principal lattice of partitions of a submodular function. Among other things we give a new pseudopolynomial algorithm for solving the membership problem over a polymatroid under the assumption that the expansion of the polymatroid in terms of a matroid is given. The complexity of our algorithm is O(E{sub 2}r) calls to the independence oracle of the underlying matroid, where E, r are the cardinality and rank respectively of this matroid. Lastly we present a vector space based definition of the hybrid rank which refines the second generalization. We study optimization problems related to this definition and show the existence of a structure analogous to the principal partition.
- OSTI ID:
- 36329
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
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