Polynomial-time approximation algorithms for the Ising model
SIAM Journal on Computing
Complexity: knots, colourings and counting
Complexity: knots, colourings and counting
Automating Po´lya theory: the computational complexity of the cycle index polynomial
Information and Computation
Surveys in combinatorics, 1995
On Unapproximable Versions of NP-Complete Problems
SIAM Journal on Computing
The Computational Complexity of Tutte Invariants for Planar Graphs
SIAM Journal on Computing
The Complexity of Ferromagnetic Ising with Local Fields
Combinatorics, Probability and Computing
Inapproximability of the Tutte polynomial
Information and Computation
Computing the Tutte Polynomial in Vertex-Exponential Time
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Approximating the partition function of the ferromagnetic Potts model
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
The expressibility of functions on the boolean domain, with applications to counting CSPs
Journal of the ACM (JACM)
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We consider the problem of approximating certain combinatorial polynomials. First, we consider the problem of approximating the Tutte polynomial of a binary matroid with parameters q=2 and @c. (Relative to the classical (x,y) parameterisation, q=(x-1)(y-1) and @c=y-1.) A graph is a special case of a binary matroid, so earlier work by the authors shows inapproximability (subject to certain complexity assumptions) for q2, apart from the trivial case @c=0. The situation for q=2 is different. Previous results for graphs imply inapproximability in the region -2=0, the approximation problem is hard for the complexity class #RH@P"1 under approximation-preserving (AP) reducibility. The latter result indicates a gap in approximation complexity at q=2: whereas an FPRAS is known in the graphical case, there can be none in the binary matroid case, unless there is an FPRAS for all of #RH@P"1. The result also implies that it is computationally difficult to approximate the weight enumerator of a binary linear code, apart from at the special weights at which the problem is exactly solvable in polynomial time. As a consequence, we show that approximating the cycle index polynomial of a permutation group is hard for #RH@P"1 under AP-reducibility, partially resolving a question that we first posed in 1992.