Abstract
We show that an effective version of Siegel’s theorem on finiteness of integer solutions for a specific algebraic curve and an application of elementary Galois theory are key ingredients in a complexity classification of some Holant problems. These Holant problems, denoted by $${\text {Holant}}(f)$$ Holant ( f ) , are defined by a symmetric ternary function f that is invariant under any permutation of the $$\kappa \ge 3$$ κ ≥ 3 domain elements. We prove that $${\text {Holant}}(f)$$ Holant ( f ) exhibits a complexity dichotomy. The hardness, and thus the dichotomy, holds even when restricted to planar multigraphs. A special case of this result is that counting edge $$\kappa $$ κ -colorings is #P-hard over planar 3-regular multigraphs for all $$\kappa \ge 3$$ κ ≥ 3 . In fact, we prove that counting edge $$\kappa $$ κ -colorings is #P-hard over planar r-regular multigraphs for all $$\kappa \ge r \ge 3$$ κ ≥ r ≥ 3 . The problem is polynomial time computable in all other parameter settings. The proof of the dichotomy theorem for $${\text {Holant}}(f)$$ Holant ( f ) depends on the fact that a specific polynomial p(x, y) has an explicitly listed finite set of integer solutions and the determination of the Galois groups of some specific polynomials. In the process, we also encounter the Tutte polynomial, medial graphs, Eulerian partitions, Puiseux series, and a certain lattice condition on the (logarithm of) the roots of polynomials.
Highlights
1 Introduction What do Siegel’s theorem and Galois theory have to do with complexity theory? In this paper, we show that an effective version of Siegel’s theorem on finiteness of integer solutions for a specific algebraic curve and an application of elementary Galois theory are key ingredients in a chain of steps that lead to a complexity classification of some counting problems
We consider a certain class of counting problems that are expressible as Holant problems with an arbitrary domain of size κ over 3-regular multigraphs and prove a dichotomy theorem for this class of problems
We further prove that the problem using κ colors over r-regular multigraphs is #P-hard for all κ ≥ r ≥ 3, even when restricted to planar multigraphs
Summary
What do Siegel’s theorem and Galois theory have to do with complexity theory? In this paper, we show that an effective version of Siegel’s theorem on finiteness of integer solutions for a specific algebraic curve and an application of elementary Galois theory are key ingredients in a chain of steps that lead to a complexity classification of some counting problems. A special case of this dichotomy theorem is the problem of counting edge colorings over planar 3-regular multigraphs using κ colors. In this case, the corresponding constraint function is the All-Distinct3,κ function, which takes value 1 when all three inputs from [κ] are distinct and 0 otherwise. The techniques we develop to prove Theorem 1.1 naturally extend to a class of Holant problems with domain size κ ≥ 3 over planar 3-regular multigraphs. Functions such as All-Distinct3,κ are symmetric, which means that they are invariant under any permutation of its three inputs.
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