Oriented Euler complexes and signed 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.

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Acknowledgments

We thank Marta Maria Casetti and Julian Merschen for stimulating discussions during our joint research on labeled Gale strings and perfect matchings, which led to the questions answered in this paper. We also thank three anonymous referees for their helpful comments.

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Correspondence to Bernhard von Stengel.

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Végh, L.A., von Stengel, B. Oriented Euler complexes and signed perfect matchings. Math. Program. 150, 153–178 (2015). https://doi.org/10.1007/s10107-014-0770-4

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Keywords

  • Complementary pivoting
  • Euler complex
  • Linear complementarity problem
  • Nash equilibrium
  • Perfect matching
  • Pfaffian orientation
  • PPAD

Mathematics Subject Classification (2010)

  • 90C33