The Complexity of Computing the Sign of the Tutte Polynomial

TL;DR

This paper investigates the computational complexity of determining the sign of the Tutte polynomial for graphs, revealing regions of #P-hardness and polynomial-time computability, and fully resolving the complexity for the chromatic polynomial.

Contribution

It characterizes the complexity landscape of computing the sign of the Tutte polynomial across different parameter regions, including a complete resolution for the chromatic polynomial.

Findings

Computing the sign is #P-hard in large regions of the parameter space.

In most other regions, computing the sign is in FP, and approximation is polynomial with an NP oracle.

The complexity of the chromatic polynomial sign is fully classified, being easy at certain points and NP-hard elsewhere.

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 is less than or equal to 32/27, and is NP-hard to compute for all other values of the parameter q.