NDLI: The Complexity of Counting Edge Colorings and a Dichotomy for Some Higher Domain Holant Problems

Content Provider	IEEE Xplore Digital Library
Author	Jin-Yi Cai Heng Guo Williams, T.
Copyright Year	2014
Description	Author affiliation: Comput. Sci. Dept., Univ. of Wisconsin-Madison, Madison, WI, USA (Jin-Yi Cai; Heng Guo; Williams, T.)
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 Holant(f), are defined by a symmetric ternary function f that is invariant under any permutation of the κ ≥ 3 domain elements. We prove that Holant(f) exhibits a complexity dichotomy. The hardness, and thus the dichotomy, holds even when restricted to planar graphs. A special case of this result is that counting edge κ-colorings is #P-hard over planar 3-regular multigraphs for all κ ≥ 3. In fact, we prove that counting edge κ-colorings is #P-hard over planar r-regular multigraphs for all κ ≥ 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.
Sponsorship	IEEE Comput. Soc. Tech. Comm. Math. Found. Comput.
Starting Page	601
Ending Page	610
File Size	352554
Page Count	10
File Format	PDF
ISBN	9781479965175
ISSN	02725428
DOI	10.1109/FOCS.2014.70
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2014-10-18
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Polynomials Complexity theory Eigenvalues and eigenfunctions Lattices Color Interpolation Transmission line matrix methods edge coloring counting problems dichotomy theorem Holant problems
Content Type	Text
Resource Type	Article

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4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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The Complexity of Counting Edge Colorings and a Dichotomy for Some Higher Domain Holant Problems

The Complexity of Counting Edge Colorings and a Dichotomy for Some Higher Domain Holant Problems

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