Complexity of packing common bases in matroids

One of the most intriguing unsolved questions of matroid optimization is the characterization of the existence of $k$ disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases, such as Woodall's conjecture on packing disjoint dijoins in a directed graph, or Rota's beautiful conjecture on rearrangements of bases. In the present paper we prove that the problem is difficult under the rank oracle model, i.e., we show that there is no algorithm which decides if the common ground set of two matroids can be partitioned into $k$ common bases by using a polynomial number of independence queries. Our complexity result holds even for the very special case when $k=2$. Through a series of reductions, we also show that the abstract problem of packing common bases in two matroids includes the NAE-SAT problem and the Perfect Even Factor problem in directed graphs. These results in turn imply that the problem is not only difficult in the independence oracle model but also includes NP-complete special cases already when $k=2$, one of the matroids is a partition matroid, while the other matroid is linear and is given by an explicit representation.Comment: 14 pages, 9 figure

On packing connectors

AbstractGiven an undirected graphG=(V,E) and a partition {S,T} ofV, anS−Tconnector is a set of edgesF⊆Esuch that every component of the subgraph (V,F) intersects bothSandT. We show thatGhaskedge-disjointS-Tconnectors if and only if |δG(V1)∪…∪δG(Vt)|⩾ktfor every collection {V1,…,Vt} of disjoint nonempty subsets ofSand for every such collection of subsets ofT. This is a common generalization of a theorem of Tutte and Nash-Williams on disjoint spanning trees and a theorem of König on disjoint edge covers in a bipartite graph

A Note on Packing Connectors

Given an undirected graph G = (V; E) and a partition fS; Tg of V , an S-T connector is a set of edges F ` E such that every component of the subgraph (V; F ) intersects both S and T . We show that G has k edge-disjoint S-T connectors if and only if jffi G (V 1 ) [ : : : [ ffi G (V t )j kt for every collection fV 1 ; : : : ; V t g of disjoint nonempty subsets of S and for every such collection of subsets of T . This is a common generalization of a theorem of Tutte and Nash-Williams on disjoint spanning trees and a theorem of König on disjoint edge covers in a bipartite graph