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Exact Algorithms for L(2,1)-Labeling of Graphs

Abstract : The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph $G=(V,E)$ into an interval of integers $\{0, \dots ,k\}$ is an $L(2,1)$-labeling of $G$ of span $k$ if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbor are mapped onto distinct integers. It is known that for any fixed $k\ge 4$, deciding the existence of such a labeling is an NP-complete problem. We present exact exponential time algorithms that are faster than the naive $O^*((k+1)^n)$ algorithm that would try all possible mappings. The improvement is best seen in the first NP-complete case of $k=4$, where the running time of our algorithm is $O(1.3006^n)$. Furthermore we show that dynamic programming can be used to establish an $O(3.8730^n)$ algorithm to compute an optimal $L(2,1)$-labeling.
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Contributor : Mathieu Liedloff <>
Submitted on : Tuesday, March 2, 2010 - 4:44:32 PM
Last modification on : Monday, October 12, 2020 - 10:30:20 AM

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Frédéric Havet, Martin Klazar, Jan Kratochvil, Dieter Kratsch, Mathieu Liedloff. Exact Algorithms for L(2,1)-Labeling of Graphs. Algorithmica, Springer Verlag, 2011, 59 (2), pp.169-194. ⟨10.1007/s00453-009-9302-7⟩. ⟨hal-00460873⟩



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