On the Complexity of the Interlace Polynomial

TL;DR

This paper investigates the computational complexity of the interlace polynomial, proving it is #P-hard to evaluate at most points, with some exceptions, and explores related specializations and approximation hardness.

Contribution

It establishes the #P-hardness of evaluating the interlace polynomial at nearly all points, solving an open problem and analyzing special cases and approximation difficulty.

Findings

Interlace polynomial is #P-hard to evaluate at almost all points.

Evaluation complexity is known for a few specific lines, with some still open.

Independent set polynomial is hard to approximate except at zero.

Abstract

We consider the two-variable interlace polynomial introduced by Arratia, Bollobas and Sorkin (2004). We develop graph transformations which allow us to derive point-to-point reductions for the interlace polynomial. Exploiting these reductions we obtain new results concerning the computational complexity of evaluating the interlace polynomial at a fixed point. Regarding exact evaluation, we prove that the interlace polynomial is #P-hard to evaluate at every point of the plane, except on one line, where it is trivially polynomial time computable, and four lines, where the complexity is still open. This solves a problem posed by Arratia, Bollobas and Sorkin (2004). In particular, three specializations of the two-variable interlace polynomial, the vertex-nullity interlace polynomial, the vertex-rank interlace polynomial and the independent set polynomial, are almost everywhere #P-hard to evaluate, too. For the independent set polynomial, our reductions allow us to prove that it is even hard to approximate at any point except at 0.