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Combinatorics, Complexity, and ChanceA Tribute to Dominic Welsh$
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Geoffrey Grimmett and Colin McDiarmid

Print publication date: 2007

Print ISBN-13: 9780198571278

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198571278.001.0001

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APPROXIMATING THE TUTTE POLYNOMIAL

APPROXIMATING THE TUTTE POLYNOMIAL

Chapter:
(p.144) 9 APPROXIMATING THE TUTTE POLYNOMIAL
Source:
Combinatorics, Complexity, and Chance
Author(s):

Mark Jerrum

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198571278.003.0009

This chapter examines some algorithmic problems associated with matroids. It focuses on determining a ‘fully polynomial randomized approximation scheme’ or ‘FPRAS’. First, the problem of counting matroid bases is considered. Feder and Mihail's polynomial time algorithm for approximating the number of bases in ‘balanced’ matroids is then described, and an improved analysis from Jerrum and Son is presented. The more general problem of evaluating the Tutte polynomial is discussed. This is a two-variable polynomial T(M; x, y) which is associated with a matroid M that encodes much information about M. What is known about the complexity of approximating the Tutte polynomial is reviewed, and the boundary is extended.

Keywords:   matroids, fully polynomial randomized approximation scheme, Tutte polynomial

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