Exponentiality of the exchange algorithm for finding another room-partitioning

Authors:
Jack Edmonds;Laura Sanití
Affiliations:
Equipe Combinatoire & Optimisation, Universitéé Pierre et Marie Curie Paris, France;Institute of Mathematics, ícole Polytechnique Fèdèrale de Lausanne, CH-1015 Lausanne, Switzerland
Venue:
Discrete Applied Mathematics
Year:
2014

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Abstract

Let T be a triangulated surface given by the list of vertex-triples of its triangles, called rooms. A room-partitioning for T is a subset R of the rooms such that each vertex of T is in exactly one room in R. Given a room-partitioning R for T, the exchange algorithm walks from room to room until it finds a second different room-partitioning R^'. In fact, this algorithm generalizes the Lemke-Howson algorithm for finding a Nash equilibrium for two-person games. In this paper, we show that the running time of the exchange algorithm is not polynomial relative to the number of rooms, by constructing a sequence of (planar) instances, in which the algorithm walks from room to room an exponential number of times. We also show a similar result for the problem of finding a second perfect matching in Eulerian graphs.