On the complexity of packing rainbow spanning trees

Krist'of B'erczi, Gergely Cs'aji, Tam'as Kir'aly
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引用次数: 2

Abstract

One of the most important questions in matroid optimization is to find 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. B\'erczi and Schwarcz showed that the problem is hard in general, therefore identifying the borderline between tractable and intractable instances is of interest. In the present paper, we study the special case when one of the matroids is a partition matroid while the other one is a graphic matroid. This setting is equivalent to the problem of packing rainbow spanning trees, an extension of the problem of packing arborescences in directed graphs which was answered by Edmonds' seminal result on disjoint arborescences. We complement his result by showing that it is NP-complete to decide whether an edge-colored graph contains two disjoint rainbow spanning trees. Our complexity result holds even for the very special case when the graph is the union of two spanning trees and each color class contains exactly two edges. As a corollary, we give a negative answer to a question on the decomposition of oriented $k$-partition-connected digraphs.
关于填充彩虹生成树的复杂性
求两个拟阵的不相交公共基是拟阵优化中的一个重要问题。这个问题的重要性可以通过一长串可以表述为特殊情况的猜想来很好地说明。B\ erczi和Schwarcz表明,这个问题一般来说是困难的,因此确定易处理和难处理实例之间的界限是有意义的。本文研究了一类矩阵是分割矩阵,另一类是图形矩阵的特殊情况。这种设置等价于彩虹生成树的填充问题,是Edmonds关于不相交树形的开创性结果在有向图中填充树形问题的扩展。我们补充了他的结果,证明了判定一个边色图是否包含两个不相交的彩虹生成树是np完全的。我们的复杂度结果甚至适用于非常特殊的情况,即图是两棵生成树的并集,并且每个颜色类恰好包含两条边。作为一个推论,我们给出了一个关于有向k分连通有向图分解问题的否定答案。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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