Deciding non-realizability of oriented matroids by semidefinite programming

Hiroyuki Miyata, Sonoko Moriyama, Hiroshi Imai

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

The concept of oriented matroid is a combinatorial abstraction of many geometric objects such as hyperplane arrangements. The problem to decide whether an oriented matroid has a geometric realization or not is called the realizability problem. This is a fundamental problem in the oriented matroid theory, and many important issues in combinatorial geometry such as stretchability of pseudoline arrangements [4] can be reduced to this problem. The realizability problem is known to be NP-hard [12] and there are many realizability certificates and non-realizability certificates based on sufficient conditions which can be checked efficiently. However, they cannot decide the realizability of all oriented matroids. Therefore new certificates are needed to determine the realizability of those that cannot be decided by existing methods. In this paper, we propose a new certificate for non-realizability of oriented matroids based on semidefinite programming relaxation of Grassmann-Pl̈ucker relations, and apply our method to oriented matroids with 8 elements and rank 4, and 9 elements and rank 3.

Original languageEnglish
Pages (from-to)211-225
Number of pages15
JournalPacific Journal of Optimization
Volume5
Issue number2
Publication statusPublished - 2009 May 1

Keywords

  • Oriented matroids
  • Realizability problem
  • Semidefinite programming

ASJC Scopus subject areas

  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics

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