On the Complexity of Some Enumeration Problems for Matroids

Authors:
L. Khachiyan;E. Boros;K. Elbassioni;V. Gurvich;K. Makino
Affiliations:
-;-;-;-;-
Venue:
SIAM Journal on Discrete Mathematics
Year:
2005

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Abstract

Let $M$ be a matroid defined by an independence oracle on ground set $S$, and let $A\subseteq S$. We present an incremental polynomial-time algorithm for enumerating all minimal (maximal) subsets of $S$ which span (do not span) $A$. Special cases of these problems include the generation of bases, circuits, hyperplanes, flats of given rank, circuits through a given element, generalized Steiner trees, and multiway cuts in graphs, as well as some other applications. We also consider some tractable and NP-hard generation problems related to systems of polymatroid inequalities and (generalized) packing and spanning in matroids.