research-article

The Complexity of Computing the Sign of the Tutte Polynomial

Authors:

Leslie Ann Goldberg,

Mark JerrumAuthors Info & Claims

SIAM Journal on Computing, Volume 43, Issue 6

Pages 1921 - 1952

https://doi.org/10.1137/12088330X

Published: 01 January 2014 Publication History

Abstract

We study the complexity of computing the sign of the Tutte polynomial of a graph. As there are only three possible outcomes (positive, negative, and zero), this seems at first sight more like a decision problem than a counting problem. Surprisingly, however, there are large regions of the parameter space for which computing the sign of the Tutte polynomial is actually \#P-hard. As a trivial consequence, approximating the polynomial is also \#P-hard in this case. Thus, approximately evaluating the Tutte polynomial in these regions is as hard as exactly counting the satisfying assignments to a CNF Boolean formula. For most other points in the parameter space, we show that computing the sign of the polynomial is in FP, whereas approximating the polynomial can be done in polynomial time with an NP oracle. As a special case, we completely resolve the complexity of computing the sign of the chromatic polynomial---this is easily computable at q=2 and when $q\leq 32/27$, and is NP-hard to compute for all other values of the parameter q.

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Cited By

Bakali EChalki AGöbel APagourtzis AZachos S(2022)Guest column: A panorama of counting problems the decision version of which is in P3ACM SIGACT News10.1145/3561064.356107253:3(46-68)Online publication date: 1-Sep-2022
https://dl.acm.org/doi/10.1145/3561064.3561072
Bezáková IGalanis AGoldberg LŠtefankovič D(2021)The Complexity of Approximating the Matching Polynomial in the Complex PlaneACM Transactions on Computation Theory10.1145/344864513:2(1-37)Online publication date: 19-Apr-2021
https://dl.acm.org/doi/10.1145/3448645

Index Terms

The Complexity of Computing the Sign of the Tutte Polynomial
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
  2. Mathematical analysis
    1. Numerical analysis
2. Theory of computation
  1. Design and analysis of algorithms

Index terms have been assigned to the content through auto-classification.

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Published In

cover image SIAM Journal on Computing

SIAM Journal on Computing Volume 43, Issue 6

2014

122 pages

ISSN:0097-5397

DOI:10.1137/smjcat.43.6

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© 2014, Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2014

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Cited By

Bakali EChalki AGöbel APagourtzis AZachos S(2022)Guest column: A panorama of counting problems the decision version of which is in P3ACM SIGACT News10.1145/3561064.356107253:3(46-68)Online publication date: 1-Sep-2022
https://dl.acm.org/doi/10.1145/3561064.3561072
Bezáková IGalanis AGoldberg LŠtefankovič D(2021)The Complexity of Approximating the Matching Polynomial in the Complex PlaneACM Transactions on Computation Theory10.1145/344864513:2(1-37)Online publication date: 19-Apr-2021
https://dl.acm.org/doi/10.1145/3448645

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