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A general reduction theorem with applications to pathwidth and the complexity of Max 2-CSP. (English) Zbl 1328.68089
Summary: We prove a general reduction theorem which allows us to extend bounds for certain graph parameters on cubic graphs to bounds for general graphs taking into account the individual vertex degrees. As applications, we give an algorithm for Max 2-CSP whose complexity matches the algorithm of A. D. Scott and G. B. Sorkin [Discrete Optim. 4, No. 3–4, 260–287 (2007; Zbl 1153.90505)] in the case of $$d$$ regular graphs, $$d \leq 5$$, but is otherwise faster. It also improves on the previously fastest known algorithm in terms of the average degree, given by A. Golovnev and K. Kutzkov [Theor. Comput. Sci. 526, 18–27 (2014; Zbl 1290.68144)]. Also from the general theorem, we derive a bound for the pathwidth of a general graph which equals that of F. V. Fomin et al. [Algorithmica 54, No. 2, 181–207 (2009; Zbl 1185.68475)] and S. Gaspers [Exponential time algorithms: structures, measures, and bounds. Saarbrücken: VDM Verlag Dr. Mueller e.K. (2010)] for graphs of degree at most 6, but is smaller otherwise, and use this to give an improved exponential-space algorithm for Max 2-CSP. Finally we use the general result to give a faster algorithm for Max 2-CSP on claw-free graphs.

##### MSC:
 68Q25 Analysis of algorithms and problem complexity 05C38 Paths and cycles 05C85 Graph algorithms (graph-theoretic aspects) 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
MAX-2-SAT
Full Text:
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