Inapproximability of the Tutte polynomial

TL;DR

This paper investigates the computational difficulty of approximating the Tutte polynomial for various graph parameters, establishing intractability results and connecting them to counting problems like perfect matchings.

Contribution

It provides new intractability results for approximating the Tutte polynomial at specific points, extending understanding of its computational complexity.

Findings

No FPRAS exists for (x,y) in certain half-planes under RP ≠ NP

Approximate counting of perfect matchings is as hard as certain Tutte polynomial evaluations

No FPRAS for counting nowhere-zero λ-flows for λ>2

Abstract

The Tutte polynomial of a graph G is a two-variable polynomial T(G;x,y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: take as input a graph G, and output a value which is a good approximation to T(G;x,y). Jaeger, Vertigan and Welsh have completely mapped the complexity of exactly computing the Tutte polynomial. They have shown that this is #P-hard, except along the hyperbola (x-1)(y-1)=1 and at four special points. We are interested in determining for which points (x,y) there is a "fully polynomial randomised approximation scheme" (FPRAS) for T(G;x,y). Under the assumption RP is not equal to NP, we prove that there is no FPRAS at (x,y) if (x,y) is in one of the half-planes x<-1 or y<-1 (excluding the easy-to-compute cases mentioned above). Two exceptions to this result are the half-line x<-1, y=1 (which is still open) and the portion of the hyperbola (x-1)(y-1)=2 corresponding to y<-1 which we show to be equivalent in difficulty to approximately counting perfect matchings. We give further intractability results for (x,y) in the vicinity of the origin. A corollary of our results is that, under the assumption RP is not equal to NP, there is no FPRAS at the point (x,y)=(0,1--lambda) when \lambda>2 is a positive integer. Thus there is no FPRAS for counting nowhere-zero \lambda flows for \lambda>2. This is an interesting consequence of our work since the corresponding decision problem is in P for example for \lambda=6.