Matroid packing and covering with circuits through an element

In 1981, Seymour proved a conjecture of Welsh that, in a connected matroid M, the sum of the maximum number of disjoint circuits and the minimum number of circuits needed to cover M is at most r*(M) + 1. This paper considers the set Ce(M) of circuits through a fixed element e such that M/e is connected. Let νe(M) be the maximum size of a subset of Ce(M) in which any two distinct members meet only in {e}, and let θe(M) be the minimum size of a subset of Ce(M) that covers M. The main result proves that νe(M) + θe(M) ≤ r* + 2 and that if M has no Fano-minor using e, then νe + θe,(M) ≤ r*(M) + 1. Seymour\u27s result follows without difficulty from this theorem and there are also some interesting applications to graphs. © 2005 Elsevier Inc. All rights reserved

Dependent k-Set Packing on Polynomoids

Specialized hereditary systems, e.g., matroids, are known to have many applications in algorithm design. We define a new notion called d-polynomoid as a hereditary system (E, ? ? 2^E) so that every two maximal sets in ? have less than d elements in common. We study the problem that, given a d-polynomoid (E, ?), asks if the ground set E contains ? disjoint k-subsets that are not in ?, and obtain a complexity trichotomy result for all pairs of k ? 1 and d ? 0. Our algorithmic result yields a sufficient and necessary condition that decides whether each hypergraph in some classes of r-uniform hypergraphs has a perfect matching, which has a number of algorithmic applications