Exact algorithms for counting 3-colorings of graphs

Graph Coloring Problem, as one of the best known NP-complete problems, has been extensively studied by researchers in a wide range of fields, leading to many applications and theories in mathematics and computer science. In this paper, we focus on the design of exact algorithms for counting 3-colori...

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Veröffentlicht in:Discrete Applied Mathematics Jg. 322; S. 74 - 93
Hauptverfasser: Zhu, Enqiang, Wu, Pu, Shao, Zehui
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier B.V 15.12.2022
Schlagworte:
3-Coloring
Branch and reduce
Dynamic programming
Exact exponential-time algorithms
Measure and conquer
Measure and conquer
Branch and reduce
3-Coloring
Dynamic programming
Exact exponential-time algorithms
ISSN:0166-218X, 1872-6771
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Zusammenfassung:Graph Coloring Problem, as one of the best known NP-complete problems, has been extensively studied by researchers in a wide range of fields, leading to many applications and theories in mathematics and computer science. In this paper, we focus on the design of exact algorithms for counting 3-colorings of a graph (denoted by #3-Coloring). Our approach is based on branch and reduce paradigm. We use the measure and conquer method to analyze the algorithms, in which we design two sets of measures (weights of vertices) intended for two distinct situations. In particular, we use the tree-width based technique to handle a special case by leveraging dynamic programming. As a result, we obtain an O(1.588n)-time algorithm for the #3-Coloring problem, which improves the previous O(1.6262n)-time algorithm by Fomin et al. (2007).
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2022.08.002