The Computational Complexity of Tutte Invariants for Planar Graphs
SIAM Journal on Computing
Inapproximability of the Tutte polynomial
Information and Computation
Journal of Combinatorial Theory Series B
Approximating the partition function of the ferromagnetic Potts model
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Exponential time complexity of the permanent and the Tutte polynomial
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
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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≤32/27, and is NP-hard to compute for all other values of the parameter q.