Streaming Algorithms for Independent Sets in Sparse Hypergraphs
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We give the first treatment of the classic independent set problem in graphs and hypergraphs in the streaming setting. The objective is to find space-efficient algorithms that output independent sets that are “combinatorially optimal”, that is, with size guarantee in terms of the degree sequence alone. Our main result is a randomized algorithm that achieves this using space in bits that is linear in the number of vertices. We use this to examine assumptions about the streaming model, and advocate the study of output-efficient algorithms that measure space usage relative to the size of the output solution. In that sense, our main algorithm uses space linear in the output size. We also examine algorithms that use little or no space in addition to the bits storing the output. Our algorithms fall also into an online streaming model, where output-changes can go only in one direction. In particular a feasible solution must be maintained at all times, and items that are removed from the solution can never reenter. We obtain tight bounds on deterministic algorithms for independent sets in graphs in that model.
KeywordsStreaming algorithms Independent sets Turan bound Online streaming
We thank Páll Melsted for helpful discussions.
- 1.Ahn, K.J., Guha, S.: Graph sparsification in the semi-streaming model. In: Automata, Languages and Programming, 36th International Colloquium, ICALP 2009, Rhodes, Greece, 5–12 July 2009, Proceedings, pp. 328–338 (2009)Google Scholar
- 2.Alon, N., Arad, U., Azar, Y.: Independent sets in hypergraphs with applications to routing via fixed paths. In: Proceedings of Third International Workshop on Randomization and Approximation Techniques in Computer Science, and Second International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX’99, pp. 16–27 (1999)Google Scholar
- 4.Caro, Y.: New Results on the Independence Number. Technical report, Tel-Aviv University (1979)Google Scholar
- 8.Emek, Y., Halldórsson, M.M., Rosén, A.: Space-constrained interval selection. In: Automata, Languages, and Programming—39th International Colloquium, ICALP 2012, Warwick, UK, 9–13 July 2012, Proceedings, Part I, pp. 302–313 (2012)Google Scholar
- 9.Emek, Y., Rosén, A.: Semi-streaming set cover - (extended abstract). In: Automata, Languages, and Programming—41st International Colloquium, ICALP 2014, Copenhagen, Denmark, 8–11 July 2014, Proceedings, Part I, pp. 453–464 (2014)Google Scholar
- 13.Halldórsson, B.V., Halldórsson, M.M., Losievskaja, E., Szegedy, M.: Streaming algorithms for independent sets. In: Automata, Languages and Programming—37th International Colloquium, ICALP, Bordeaux, France, Proceedings, pp. 641–652. Springer (2010)Google Scholar
- 15.Halldórsson, M.M., Sun, X., Szegedy, M., Wang, C.: Streaming and communication complexity of clique approximation. In: Automata, Languages, and Programming—39th International Colloquium, ICALP 2012, Warwick, UK, 9–13 July 2012, Proceedings, Part I, pp. 449–460 (2012)Google Scholar
- 19.Wei, V.K.: A lower bound on the stability number of a simple graph. Technical Memorandum No. 81-11217-9, Bell Laboratories (1981)Google Scholar