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,…,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≥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(1.3006n). Furthermore we show that dynamic programming can be used to establish an O(3.8730n) algorithm to compute an optimal L(2,1)-labeling.
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Alon, N., Wormald, N.: High degree graphs contain large-star factors (submitted)
Björklund, A., Husfeldt, T.: Inclusion-exclusion algorithms for counting set partitions. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science 2006, pp. 575–582 (2006)
Bodlaender, H.L., Kloks, T., Tan, R.B., van Leeuwen, J.: Approximations for lambda-colorings of graphs. Comput. J. 47, 193–204 (2004)
Chang, G.J., Kuo, D.: The L(2,1)-labeling problem on graphs. SIAM J. Discrete Math. 9, 309–316 (1996)
Fiala, J., Kratochvíl, J.: Complexity of partial covers of graphs. In: Proceedings of ISAAC 2001. LNCS, vol. 2223, pp. 537–549. Springer, Berlin (2001)
Fiala, J., Kratochvíl, J.: Partial covers of graphs. Discuss. Math. Graph Theory 22, 89–99 (2002)
Fiala, J., Kloks, T., Kratochvíl, J.: Fixed-parameter complexity of λ-labelings. Discrete Appl. Math. 113, 59–72 (2001)
Fiala, J., Golovach, P., Kratochvíl, J.: Distance constrained labelings of graphs of bounded treewidth. In: Proceedings of ICALP 2005. LNCS, vol. 3580, pp. 360–372. Springer, Berlin (2005)
Fiala, J., Kratochvíl, J., Pór, A.: On the computational complexity of partial covers of theta graphs. Electron. Notes Discrete Math. 19, 79–85 (2005)
Fomin, F., Grandoni, F., Kratsch, D.: Measure and conquer: Domination—a case study. In: Proceedings of ICALP 2005. LNCS, vol. 3380, pp. 192–203. Springer, Berlin (2005)
Fomin, F., Heggernes, P., Kratsch, D.: Exact algorithms for graph homomorphisms. Theory Comput. Syst. 41, 381–393 (2007)
Gonçalves, D.: On the L(p,1)-labelling of graphs. In: Discrete Mathematics and Theoretical Computer Science Proceedings, vol. AE, pp. 81–86
Griggs, J.R., Yeh, R.K.: Labelling graphs with a condition at distance 2. SIAM J. Discrete Math. 5, 586–595 (1992)
Hasunuma, T., Ishii, T., Ono, H., Uno, Y.: A linear time algorithm for L(2,1)-labeling of trees. arXiv:0810.0906v1 (2008)
Havet, F., Reed, B., Sereni, J.-S.: L(2,1)-labellings of graphs. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms 2008, pp. 621–630 (2008)
Hell, P., Nešetřil, J.: On the complexity of H-colouring. J. Comb. Theory Ser. B 48, 92–110 (1990)
Johnson, D.S., Yannakakis, M., Papadimitriou, C.H.: On generating all maximal independent sets. Inf. Process. Lett. 27, 119–123 (1988)
Koivisto, M.: An O(2n) algorithm for graph coloring and other partitioning problems via inclusion-exclusion. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science 2006, pp. 583–590 (2006)
Král’, D.: Channel assignment problem with variable weights. SIAM J. Discrete Math. 20, 690–704 (2006)
Kratochvíl, J., Kratsch, D., Liedloff, M.: Exact algorithms for L(2,1)-labeling of graphs. In: Proceedings of MFCS 2007. LNCS, vol. 4708, pp. 513–524. Springer, Berlin (2007)
Leese, R.A., Noble, S.D.: Cyclic labellings with constraints at two distances. Electron. J. Comb. 11 (2004) (Research paper 16)
Liu, D., Zhu, X.: Circular distance two labelings and circular chromatic numbers. Ars Comb. 69, 177–183 (2003)
Liu, D., Zhu, X.: Multilevel distance labelings for paths and cycles. SIAM J. Discrete Math. 19, 610–621 (2005)
Roberts, F.S. Private communication to J. Griggs
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The extended abstract of this paper was presented at the conference “Mathematical Foundations of Computer Science (MFCS 2007)”, Český Krumlov, Czech Republic, 26–31 August 2007 [20].
F. Havet was partially supported by the European project FET-AEOLUS.
M. Klazar and J. Kratochvíl was supported by Research grant 1M0545 of the Czech Ministry of Education.
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Havet, F., Klazar, M., Kratochvíl, J. et al. Exact Algorithms for L(2,1)-Labeling of Graphs. Algorithmica 59, 169–194 (2011). https://doi.org/10.1007/s00453-009-9302-7
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DOI: https://doi.org/10.1007/s00453-009-9302-7