Complexity of packing common bases in matroids

• 成果类型: Article
• 署名作者: Berczi, Kristof; Schwarcz, Tamas
• 署名单位: Eotvos Lorand University; Eotvos Lorand University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-020-01497-y
• 发表日期: 2021
• 页码: 1-18
• 关键词: transversals; conjecture

摘要

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.