NDLI: A Polynomial Algorithm for 3-Compatible Coloring and the Stubborn List Partition Problem (The Stubborn Problem Is Stubborn No More)

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A Polynomial Algorithm for 3-Compatible Coloring and the Stubborn List Partition Problem (The Stubborn Problem Is Stubborn No More)

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Cygan, Marek Wojtaszczyk, Jakub Onufry Pilipczuk, Marcin Pilipczuk, Micha
Copyright Year	2012
Abstract	One of the driving problems in the CSP area is the dichotomy conjecture, formulated in 1993 by Feder and Vardi [Monotone monadic SNP and constraint satisfaction, in Proceedings of the 25th Annual Symposium on Theory of Computing, ACM, New York, 1993, pp. 612--622], stating that for any fixed relational structure $\Gamma$ the constraint satisfaction problem CSP($\Gamma$) is either NP-complete or polynomial time solvable. A large amount of research has gone into checking various specific cases of this conjecture. One such variant which attracted a lot of attention in recent years is the List Matrix Partition problem. In 2004 Cameron et al. [SIAM J. Discrete Math., 21 (2007), pp. 900--929] classified almost all List Matrix Partition variants for matrices of size at most four. The only case which resisted the classification became known as the Stubborn problem. In this paper we show a result which enables us to finish the classification---thus solving a problem which resisted attacks for a few years. Our approach is based on a combinatorial problem known to be at least as hard as the Stubborn problem---the 3-Compatible Coloring problem. In this problem we are given a complete graph with each edge assigned one of three possible colors and we want to assign one of those three colors to each vertex in such a way that no edge has the same color as both of its endpoints. The tractability of the 3-Compatible Coloring problem has been open for several years and the best known algorithm prior to this paper is due to Feder et al. [Two algorithms for general list matrix partitions, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2005, pp. 870--876]---a quasipolynomial algorithm with a $n^{O(\log n / \log \log n)}$ time complexity. In this paper we present a polynomial-time algorithm for the 3-Compatible Coloring problem and consequently we prove a dichotomy for the k-Compatible Coloring problem.
Starting Page	815
Ending Page	828
Page Count	14
File Format	PDF
ISSN	00975397
DOI	10.1137/110826813
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4
Volume Number	41
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2012-08-16
Access Restriction	Subscribed
Subject Keyword	CSP stubborn problem list matrix partition Graph theory Coloring of graphs and hypergraphs 3-compatible coloring
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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