Oriented Euler Complexes and Signed Perfect Matchings

TL;DR

This paper introduces oriented pivoting systems and extends orientation concepts to Euler complexes, providing a unified proof for sign properties of pivoting paths and a near-linear algorithm for finding opposite sign perfect matchings.

Contribution

It develops a unified framework for complementary pivoting, extending orientations to Euler complexes, and offers an efficient algorithm for finding opposite sign perfect matchings.

Findings

Endpoints of pivoting paths have opposite signs.

Orientation extends to Euler complexes and oiks.

Near-linear time algorithm for opposite sign perfect matchings.

Abstract

This paper presents "oriented pivoting systems" as an abstract framework for complementary pivoting. It gives a unified simple proof that the endpoints of complementary pivoting paths have opposite sign. A special case are the Nash equilibria of a bimatrix game at the ends of Lemke-Howson paths, which have opposite index. For Euler complexes or "oiks", an orientation is defined which extends the known concept of oriented abstract simplicial manifolds. Ordered "room partitions" for a family of oriented oiks come in pairs of opposite sign. For an oriented oik of even dimension, this sign property holds also for unordered room partitions. In the case of a two-dimensional oik, these are perfect matchings of an Euler graph, with the sign as defined for Pfaffian orientations of graphs. A near-linear time algorithm is given for the following problem: given a graph with an Eulerian orientation with a perfect matching, find another perfect matching of opposite sign. In contrast, the complementary pivoting algorithm for this problem may be exponential.