Supermodularity in Unweighted Graph Optimization I: Branchings and Matchings
Abstract
The main result of the paper is motivated by the following two, apparently unrelated graph optimization problems: (A) as an extension of Edmonds' disjoint branchings theorem, characterize digraphs comprising disjoint branchings each having a specified number of arcs, (B) as an extension of Ryser's maximum term rank formula, determine the largest possible matching number of simple bipartite graphs complying with degree-constraints. The solutions to these problems and to their generalizations will be obtained from a new min-max theorem on covering a supermodular function by a simple degree-constrained bipartite graph. A specific feature of the result is that its minimum cost extension is already NP-complete. Therefore classic polyhedral tools themselves definitely cannot be sufficient for solving the problem, even though they make some good service in our approach.
Cite
@article{arxiv.1608.05722,
title = {Supermodularity in Unweighted Graph Optimization I: Branchings and Matchings},
author = {Kristóf Bérczi and András Frank},
journal= {arXiv preprint arXiv:1608.05722},
year = {2017}
}