On degeneracy in the P-matroid oriented matroid complementarity problem

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization Pub Date : 2025-06-11 DOI:10.1016/j.disopt.2025.100891

Michaela Borzechowski , Simon Weber

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引用次数: 0

Abstract

Klaus showed that the Oriented Matroid Complementarity Problem (OMCP) can be solved by a reduction to the problem of sink-finding in a unique sink orientation (USO) if the input is promised to be given by a non-degenerate extension of a P-matroid. In this paper, we investigate the effect of degeneracy on this reduction. On the one hand, this understanding of degeneracies allows us to prove a linear lower bound on the number of vertex evaluations required for sink-finding in P-matroid USOs, the set of USOs obtainable through Klaus’ reduction. On the other hand, it allows us to adjust Klaus’ reduction to also work with degenerate instances. Furthermore, we introduce a total search version of the P-Matroid Oriented Matroid Complementarity Problem (P-OMCP). Given any extension of any oriented matroid

M

, by reduction to a total search version of USO sink-finding we can either solve the OMCP, or provide a polynomial-time verifiable certificate that

M

is not a P-matroid. This places the total search version of the P-OMCP in the complexity class Unique End of Potential Line (UEOPL).

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论面向p -拟阵互补问题的退化性

Klaus证明了有向矩阵互补问题（OMCP）可以简化为在唯一集方向（USO）上寻找集的问题，如果输入是由p -矩阵的非退化扩展给出的话。在本文中，我们研究了简并对这种约简的影响。一方面，这种对退化的理解使我们能够证明p -矩阵USOs（通过Klaus约简可得到的USOs集合）中寻找sink所需顶点计算次数的线性下界。另一方面，它允许我们调整克劳斯的减少，也适用于退化的实例。进一步，我们引入了面向p -矩阵互补问题（P-OMCP）的一个全搜索版本。给定任意有向矩阵M的任意扩展，通过还原为USO sink-finding的总搜索版本，我们可以解出OMCP，或者提供一个多项式时间可验证的证明M不是p -矩阵。这将P-OMCP的总搜索版本放在复杂性类唯一潜在行结束（UEOPL）中。

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

Discrete Optimization 管理科学-应用数学

CiteScore

2.10

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.