Abstract
The clique problem consists in determining whether an undirected graph G of order n contains a clique of order ℓ. In this paper we are concerned with the decremental version of clique problem, where the property of containing an ℓ-clique is dynamically checked during deletions of nodes. We provide an improved dynamic algorithm for this problem for every fixed value of ℓ ≥ 3. Our algorithm naturally applies to filtering for the constraint satisfaction problem. In particular, we show how to speed up the filtering based on an important local consistency property: the inverse consistency.
This work has been partially supported by the IST Programme of the EU under contract n. IST-1999-14.186 (ALCOM-FT), by the Italian Ministry of University and Research (Project “ALINWEB: Algorithmics for Internet and the Web”).
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References
Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation 9(3), 251–280 (1990)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. The MIT Press/McGraw-Hill Book Company, Cambridge (2001)
Debruyne, R.: A property of path inverse consistency leading to an optimal PIC algorithm. In: European Conference on Artificial Intelligence, pp. 88–92 (2000)
Demetrescu, C., Italiano, G.F.: Fully dynamic transitive closure: Breaking through the O(n2) barrier. In: IEEE Symposium on Foundations of Computer Science, pp. 381–389 (2000)
Demetrescu, C., Italiano, G.F.: Fully dynamic all pairs shortest paths with real edge weights. In: IEEE Symposium on Foundations of Computer Science, pp. 260–267 (2001)
Demetrescu, C., Italiano, G.F.: A new approach to dynamic all pairs shortest paths. In: ACM Symposium on the Theory of Computing, pp. 159–166 (2003)
Eisenbrand, F., Grandoni, F.: On the complexity of fixed parameter clique and dominating set (2003); To appear in Theoretical Computer Science
Elfe, C.D., Freuder, E.C.: Neighborhood inverse consistency preprocessing. In: National Conference on Artificial Intelligence/Innovative Applications of Artificial Intelligence vol. 1, pp. 202–208 (1996)
Frigioni, D., Marchetti-Spaccamela, A., Nanni, U.: Fully dynamic shortest paths and negative cycles detection on digraphs with arbitrary arc weights. In: European Symposium on Algorithms, pp. 320–331 (1998)
Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fullydynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. Journal of the Association for Computing Machinery 48(4), 723–760 (2001)
Huang, X., Pan, V.: Fast rectangular matrix multiplication and applications. Journal of Complexity 14(2), 257–299 (1998)
Itai, A., Rodeh, M.: Finding a minimum circuit in a graph. SIAM Journal on Computing 7(4), 413–423 (1978)
King, V.: Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In: IEEE Symposium on Foundations of Computer Science, pp. 81–91 (1999)
King, V., Sagert, G.: A fully dynamic algorithm for maintaining the transitive closure. In: ACM Symposium on the Theory of Computing, pp. 492–498 (1999)
Mackworth, A.K.: Consistency in networks of relations. Artificial Intelligence 8, 99–118 (1977)
Montanari, U.: Networks of constraints: Fundamental properties and applications to picture processing. Information Sciences 7, 95–132 (1974)
Nešetřil, J., Poljak, S.: On the complexity of the subgraph problem. Commentationes Mathematicae Universitatis Carolinae 26(2), 415–419 (1985)
Shoshan, A., Zwick, U.: All pairs shortest paths in undirected graphs with integer weights. In: IEEE Symposium on Foundations of Computer Science, pp. 605–615 (1999)
Zwick, U.: All pairs shortest paths in weighted directed graphs - exact and almost exact algorithms. In: IEEE Symposium on Foundations of Computer Science, pp. 310–319 (1998)
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Grandoni, F., Italiano, G.F. (2004). Decremental Clique Problem. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2004. Lecture Notes in Computer Science, vol 3353. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30559-0_12
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DOI: https://doi.org/10.1007/978-3-540-30559-0_12
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