Rainbow Bases in Matroids

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-05-07 DOI:10.1137/22m1516750

Florian Hörsch, Tomáš Kaiser, Matthias Kriesell

引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1472-1491, June 2024.
Abstract. Recently, it was proved by Bérczi and Schwarcz that the problem of factorizing a matroid into rainbow bases with respect to a given partition of its ground set is algorithmically intractable. On the other hand, many special cases were left open. We first show that the problem remains hard if the matroid is graphic, answering a question of Bérczi and Schwarcz. As another special case, we consider the problem of deciding whether a given digraph can be factorized into subgraphs which are spanning trees in the underlying sense and respect upper bounds on the indegree of every vertex. We prove that this problem is also hard. This answers a question of Frank. In the second part of the article, we deal with the relaxed problem of covering the ground set of a matroid by rainbow bases. Among other results, we show that there is a linear function [math] such that every matroid that can be factorized into [math] bases for some [math] can be covered by [math] rainbow bases if every partition class contains at most 2 elements.

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矩阵中的彩虹基

SIAM 离散数学杂志》，第 38 卷，第 2 期，第 1472-1491 页，2024 年 6 月。摘要。最近，Bérczi 和 Schwarcz 证明了将一个 matroid 分解成彩虹基的问题在算法上是难以解决的。另一方面，还有许多特殊情况没有解决。我们首先证明，如果 matroid 是图形的，这个问题仍然很难解决，从而回答了 Bérczi 和 Schwarcz 提出的一个问题。作为另一个特例，我们考虑了这样一个问题：决定一个给定的数图是否可以被因子化为子图，这些子图在基本意义上是生成树，并且尊重每个顶点的indegree上限。我们证明这个问题也很难解决。这回答了弗兰克的一个问题。在文章的第二部分，我们讨论了用彩虹基覆盖 matroid 地面集的松弛问题。除其他结果外，我们还证明了一个线性函数 [math]，即如果每个分区类最多包含 2 个元素，那么每个可以因式分解为某个 [math] 基的 matroid 都可以被 [math] 彩虹基覆盖。

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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来源期刊

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.