# Complexity of packing common bases in matroids

## TL;DR

This paper proves that deciding whether two matroids can be partitioned into k common bases is computationally hard, even for simple cases, and relates this problem to well-known NP-complete problems.

## Contribution

It establishes the computational complexity of packing common bases in matroids, showing hardness results and connections to NP-complete problems.

## Key findings

- No polynomial-time algorithm exists for the problem under the rank oracle model.
- The problem includes NP-complete cases even with simple matroid types.
- It relates the problem to NAE-SAT and Perfect Even Factor problems.

## 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.