Complexity of packing common bases in matroids

Abstract

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.

Computing Diverse Shortest Paths Efficiently: A Theoretical and Experimental Study

“…Here, a subset B ⊆ E is a common base of M 1 and M 2 if B is a base of both matroids. This problem is at least as hard as the problem of packing k common bases of two matroids, implying that this problem cannot be solved with a polynomial number of independence oracle queries (Bérczi and Schwarcz 2021). This raises the following question: Is there another natural problem that is a special case of finding diverse common bases and solvable in polynomial time?…”

On the complexity of packing rainbow spanning trees

“…We prove hardness of Packing Parity Spanning Trees by reduction from Packing Rainbow Spanning Trees. Interestingly, the direction of the reduction between the two problems is just the opposite of the one appearing in [4] where the hardness of packing disjoint common bases was proved by reduction from packing parity bases.…”

“…In [1], Aharoni and Berger considered a relaxation of the problem in which disjoint common generators are required instead of bases, and showed that there always exist min{γ(M 1 ), γ(M 2 )}/2 pairwise disjoint common generators of M 1 and M 2 . [4] proved that there is no algorithm which decides if the ground set of two matroids can be partitioned into common bases by using a polynomial number of independence queries. Their result implies that determining the exact values of β(M 1 , M 2 ) and γ(M 1 , M 2 ) for two matroids is hard under the rank oracle model.…”

Rainbow bases in matroids

“…Unfortunately, this latter problem has been proven to be algorithmically intractable by Bérczi and Schwarcz [1]. Firstly, they showed that this problem cannot be solved efficiently in the independence oracle model and secondly, they showed that some NP-hard problems appear as special cases of this problem.…”

“…Proof. First observe that as both M and M |X are 2-base matroids, we have r M/X (E(M/X)) = r M (E(M )) − r M (X) = 1 2 |E(M )| − 1 2 |X| = 1 2 |E(M/X)|. Further, for any Z ⊆ E(M )−X, as X is a tight set in M , we have…”