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 and therefore classical polyhedral tools do not help.
Bibtex entry:
@techreport{egres-16-09,
AUTHOR | = | {B{\'e}rczi, Krist{\'o}f and Frank, Andr{\'a}s}, |
TITLE | = | {Supermodularity in unweighted graph optimization I: Branchings and matchings}, |
NOTE | = | {{\tt www.cs.elte.hu/egres}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2016}, |
NUMBER | = | {TR-2016-09} |