Supermodularity in Unweighted Graph Optimization I: Branchings and Matchings

TL;DR

This paper introduces a new min-max theorem for supermodular functions that extends classical graph optimization problems like branchings and matchings, revealing computational complexity limitations.

Contribution

It presents a novel min-max theorem for covering supermodular functions with degree-constrained bipartite graphs, extending classical results.

Findings

New min-max theorem for supermodular functions

Extension of Edmonds' disjoint branchings theorem

NP-completeness of the minimum cost extension

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 $k$ disjoint branchings $B_i$ each having a specified number $\mu _i$ 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.