The complexity of counting edge colorings and a dichotomy for some higher domain Holant problems
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Publication:313398
DOI10.1186/S40687-016-0067-8zbMATH Open1344.05060arXiv1404.4020OpenAlexW2412274288WikidataQ59474470 ScholiaQ59474470MaRDI QIDQ313398
Author name not available (Why is that?)
Publication date: 9 September 2016
Published in: (Search for Journal in Brave)
Abstract: We show that an effective version of Siegel's Theorem on finiteness of integer solutions and an application of elementary Galois theory are key ingredients in a complexity classification of some Holant problems. These Holant problems, denoted by Holant(f), are defined by a symmetric ternary function f that is invariant under any permutation of the k >= 3 domain elements. We prove that Holant(f) exhibits a complexity dichotomy. This dichotomy holds even when restricted to planar graphs. A special case of this result is that counting edge k-colorings is #P-hard over planar 3-regular graphs for k >= 3. In fact, we prove that counting edge k-colorings is #P-hard over planar r-regular graphs for all k >= r >= 3. The problem is polynomial-time computable in all other parameter settings. The proof of the dichotomy theorem for 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.
Full work available at URL: https://arxiv.org/abs/1404.4020
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