Years and Authors of Summarized Original Work
2006; Björklund, Husfeldt
Problem Definition
A k-coloring of a graph G = (V, E) assigns one of k colors to each vertex such that neighboring vertices have different colors. This is sometimes called vertex coloring.
The smallest integer k for which the graph G admits a k-coloring is denoted χ(G) and called the chromatic number. The number of k-colorings of G is denoted P(G; k) and called the chromatic polynomial.
Key Results
The central observation is that χ(G) and P(G; k) can be expressed by an inclusion-exclusion formula whose terms are determined by the number of independent sets of induced subgraphs of G. For \(X \subseteq V\), let s(X) denote the number of nonempty independent vertex subsets disjoint from X, and let s r (X) denote the number of ways to choose r nonempty independent vertex subsets \(S_{1},\ldots ,S_{r}\) (possibly overlapping and with repetitions), all disjoint from X, such that \(\vert S_{1}\vert + \cdots +\vert...
Recommended Reading
Björklund A, Husfeldt T (2008) Exact algorithms for exact satisfiability and number of perfect matchings. Algorithmica 52(2):226–249
Björklund A, Husfeldt T, Kaski P, Koivisto M (2007) Fourier meets Möbius: fast subset convolution. In: Proceedings of the 39th annual ACM symposium on theory of computing (STOC), San Diego, 11–13 June 2007. Association for Computing Machinery, New York, pp 67–74
Björklund A, Husfeldt T, Koivisto M (2009) Set partitioning via inclusion-exclusion. SIAM J Comput 39(2):546–563
Björklund A, Husfeldt T, Kaski P, Koivisto M (2011) Covering and packing in linear space. Inf Process Lett 111(21–22):1033–1036
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Björklund, A., Husfeldt, T. (2015). Exact Graph Coloring Using Inclusion-Exclusion. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-3-642-27848-8_134-2
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DOI: https://doi.org/10.1007/978-3-642-27848-8_134-2
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