Fractional packing in ideal clutters

Authors:
Yuji Matsuoka
Affiliations:
University of Tokyo, Tokyo, Japan, and Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
Venue:
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2007

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Abstract

In this paper we present a polyhedral framework for fractional packing in ideal clutters. Consider an ideal clutter with a nonnegative capacity function on its vertices. It follows from ideality that for any nonnegative capacity the total multiplicity of an optimal fractional packing is equal to the minimum capacity of a vertex cover. Our framework finds an optimal packing using at most n edges with positive multiplicities, performing at most n times minimizations for the clutter and at most n2 times minimizations for its blocker, where n denotes the cardinality of the vertex set. Applied to the clutter of dijoins (directed cut covers), the framework provides the first combinatorial polynomial-time algorithm for fractional packing of dijoins.