A computational proof of complexity of some restricted counting problems

Abstract

We explore a computational approach to proving the intractability of certain counting problems. These problems can be described in various ways, and they include concrete problems such as counting the number of vertex covers or independent sets for 3-regular graphs. The high level principle of our approach is algebraic, which provides sufficient conditions for interpolation to succeed. Another algebraic component is holographic reductions. We then analyze in detail polynomial maps on R2 induced by some combinatorial constructions. These maps define sufficiently complicated dynamics of R2 that we can only analyze them computationally. In this paper we use both numerical computation (as intuitive guidance) and symbolic computation (as proof theoretic verification) to derive that a certain collection of combinatorial constructions, in myriad combinations, fulfills the algebraic requirements of proving #P-hardness. The final result is a dichotomy theorem for a class of counting problems. This includes a class of generic holant problems with an arbitrary real valued edge signature over (2,3)-regular undirected graphs. In particular, it includes all partition functions with 0–1 vertex assignments and an arbitrary real valued edge function over all 3-regular undirected graphs.