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).

翻译：克劳斯证明了，若输入被承诺为一个P-拟阵的非退化扩张，则定向拟阵互补问题（OMCP）可通过规约到唯一汇点定向（USO）中的汇点查找问题来求解。本文研究了退化性对此规约的影响。一方面，对退化性的这种理解使我们能够证明，在P-拟阵USO（即可通过克劳斯规约获得的USO集合）中进行汇点查找所需的顶点评估次数存在线性下界。另一方面，它使我们能够调整克劳斯的规约，使其也能处理退化实例。此外，我们引入了P-拟阵定向拟阵互补问题（P-OMCP）的全搜索版本。给定任何定向拟阵M的任何扩张，通过规约到USO汇点查找的全搜索版本，我们既可以求解OMCP，也可以提供一个多项式时间可验证的证明，表明M不是一个P-拟阵。这将P-OMCP的全搜索版本置于复杂性类Unique End of Potential Line（UEOPL）中。