The complexity of computing the tutte polynomial on transversal matroids

Abstract

The complexity of computing the Tutte polynomialT(M,x,y) is determined for transversal matroidM and algebraic numbersx andy. It is shown that for fixedx andy the problem of computingT(M,x,y) forM a transversal matroid is #P-complete unless the numbersx andy satisfy (x−1)(y−1)=1, in which case it is polynomial-time computable. In particular, the problem of counting bases in a transversal matroid, and of counting various types of “matchable” sets of nodes in a bipartite graph, is #P-complete.

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Colbourn, C.J., Provan, J.S. & Vertigan, D. The complexity of computing the tutte polynomial on transversal matroids. Combinatorica 15, 1–10 (1995). https://doi.org/10.1007/BF01294456

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Mathematics Subject Classification (1991)

  • 05 D 15
  • 68 Q 25
  • 68 R 05