Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Degeneracy in the P-Matroid Oriented Matroid Complementarity Problem

Michaela Borzechowski, Simon Weber

TL;DR

This work analyzes degeneracy in the P-OMCP and its Klaus-style reduction to USO, showing how degeneracy disrupts edge orientation and proposing a total-search framework (Total-P-OMCP and Total-USO) to handle degenerate inputs within UEOPL. It establishes a linear lower bound of $n$ queries for sink-finding on uniform P-matroid USOs by constructing adversarial localizations and perturbations, and demonstrates how degeneracies can be remedied to extend Klaus' reduction to degenerate instances via PPUs and lexicographic extensions. The paper also connects P-OMCP to the P-LCP, extends the reduction to total search settings, and situates Total-P-OMCP in UEOPL, highlighting the broader impact on TFNP subclasses and the ongoing challenge of identifying complete problems in UEOPL.

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).

On Degeneracy in the P-Matroid Oriented Matroid Complementarity Problem

TL;DR

This work analyzes degeneracy in the P-OMCP and its Klaus-style reduction to USO, showing how degeneracy disrupts edge orientation and proposing a total-search framework (Total-P-OMCP and Total-USO) to handle degenerate inputs within UEOPL. It establishes a linear lower bound of queries for sink-finding on uniform P-matroid USOs by constructing adversarial localizations and perturbations, and demonstrates how degeneracies can be remedied to extend Klaus' reduction to degenerate instances via PPUs and lexicographic extensions. The paper also connects P-OMCP to the P-LCP, extends the reduction to total search settings, and situates Total-P-OMCP in UEOPL, highlighting the broader impact on TFNP subclasses and the ongoing challenge of identifying complete problems in UEOPL.

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).
Paper Structure (13 sections, 16 theorems, 11 equations, 6 figures)

This paper contains 13 sections, 16 theorems, 11 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Paper Overview
  4. Preliminaries
  5. Oriented Matroids
  6. P-OMCP
  7. Unique Sink Orientation (USO)
  8. Klaus' Reduction
  9. The Effect of Degeneracy on the Resulting USOs
  10. Constructions Based on Degeneracy and Perturbations
  11. P-LCP
  12. The Search Problem Complexity of P-OMCP
  13. Schurr and Szabó's Lower Bound

Key Result

Lemma 3

[lemma]lem:orthogonal Let $X\in \mathcal{C}{$C$}\xspace$ be a circuit and $Y\in\mathcal{C}{$C$}\xspace^*$ be a cocircuit of some oriented matroid ${\mathcal{M}}{$M$}\xspace$. Then, $X$ and $Y$ are orthogonal.

Figures (6)

  • Figure 1: Realization of $\widehat{\mathcal{M}}$.
  • Figure 2: Realization of $\widehat{\mathcal{M}}{$M$}\xspace'$.
  • Figure 3: The USO created from $\widehat{\mathcal{M}}$ by Klaus' reduction.
  • Figure 4: The PPU created by the reduction from the degenerate P-matroid extension $\widehat{\mathcal{M}}{$M$}\xspace'$ is first completely unoriented. After orienting every unoriented edge downwards (red) as in \ref{['lem:completable']}, it becomes a USO.
  • Figure 5: The form of the PPU given by a lexicographic extension of a uniform P-matroid.
  • ...and 1 more figures

Theorems & Definitions (40)

  • Definition 1: Circuit axioms
  • Definition 2
  • Lemma 3: bjoerner1999orientedmatroids
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7: P-matroid
  • Example 8
  • Example 9
  • Lemma 10: Szabó-Welzl Condition szabo2001usos
  • ...and 30 more