Abstract
We consider the classic clique (or, equivalently, the independent set) problem in two settings. In the streaming model, edges are given one by one in an adversarial order, and the algorithm aims to output a good approximation under space restrictions. In the communication complexity setting, two players, each holds a graph on n vertices, and they wish to use a limited amount of communication to distinguish between the cases when the union of the two graphs has a low or a high clique number. The settings are related in that the communication complexity gives a lower bound on the space complexity of streaming algorithms.
We give several results that illustrate different tradeoffs between clique separability and the required communication/space complexity under randomization. The main result is a lower bound of \(\Omega(\frac{n^2}{r^2\log^2{n}})\)-space for any r-approximate randomized streaming algorithm for maximum clique. A simple random sampling argument shows that this is tight up to a logarithmic factor. For the case when r = o(logn), we present another lower bound of \(\Omega(\frac{n^2}{r^4})\). In particular, it implies that any constant approximation randomized streaming algorithm requires Ω(n 2) space, even if the algorithm runs in exponential time. Finally, we give a third lower bound that holds for the extremal case of s − 1 vs. \(\mathcal{R}(s)-1\), where \(\mathcal{R}(s)\) is the s-th Ramsey number. This is the extremal setting of clique numbers that can be separated. The proofs involve some novel combinatorial structures and sophisticated combinatorial constructions.
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References
Ahn, K.J., Guha, S.: Graph Sparsification in the Semi-streaming Model. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 328–338. Springer, Heidelberg (2009)
Alon, N., Shapira, A.: A characterization of easily testable induced subgraphs. In: SODA 2004, pp. 942–951. SIAM (2004)
Alon, N., Boppana, R.B.: The monotone circuit complexity of boolean functions. Combinatorica 7(1), 1–22 (1987)
Babai, L., Frankl, P., Simon, J.: Complexity classes in communication complexity theory. In: FOCS 1986, pp. 337–347. IEEE Computer Society (1986)
Bar-Yossef, Z., Kumar, R., Sivakumar, D.: Reductions in streaming algorithms, with an application to counting triangles in graphs. In: SODA 2002, pp. 623–632 (2002)
Becchetti, L., Boldi, P., Castillo, C., Gionis, A.: Efficient semi-streaming algorithms for local triangle counting in massive graphs. In: KDD 2008, pp. 16–24 (2008)
Bollobás, B.: Complete subgraphs are elusive. Journal of Combinatorial Theory, Series B 21(1), 1–7 (1976)
Bordino, I., Donato, D., Gionis, A., Leonardi, S.: Mining Large Networks with Subgraph Counting. In: 8th IEEE International Conference on Data Mining, pp. 737–742. IEEE Computer Society (2008)
Buriol, L.S., Frahling, G., Leonardi, S., Sohler, C.: Estimating Clustering Indexes in Data Streams. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 618–632. Springer, Heidelberg (2007)
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. II. On completeness for W[1]. Theoretical Computer Science 141(1-2), 109–131 (1995)
Erdős, P.: Some remarks on the theory of graphs. Bull. Amer. Math. Soc. 53, 292–294 (1947)
Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compositio. Math. 2, 463–470 (1935)
Feige, U.: Approximating maximum clique by removing subgraphs. SIAM J. Discrete Math. 18(2), 219–225 (2004)
Feigenbaum, J., Kannan, S., McGregor, A., Suri, S., Zhang, J.: On graph problems in a semi-streaming model. Theoretical Computer Science 348(2), 207–216 (2005)
Feigenbaum, J., Kannan, S., McGregor, A., Suri, S., Zhang, J.: Graph distances in the data-stream model. SIAM J. Comput. 38(5), 1709–1727 (2008)
Grimmett, G.R., McDiarmid, C.J.H.: On colouring random graphs. Mathematical Proceedings of the Cambridge Philosophical Society 77, 313–324 (1975)
Halldórsson, B.V., Halldórsson, M.M., Losievskaja, E., Szegedy, M.: Streaming Algorithms for Independent Sets. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 641–652. Springer, Heidelberg (2010)
Håstad, J.: Clique is hard to approximate within n 1 − ε. Acta Mathematica 182, 105–142 (1999)
Indyk, P., Price, E.: K-median clustering, model-based compressive sensing, and sparse recovery for earth mover distance. In: STOC 2011, pp. 627–636 (2011)
Kalyanasundaram, B., Schnitger, G.: The probabilistic communication complexity of set intersection. SIAM J. Discrete Math. 5(4), 545–557 (1992)
Khot, S., Ponnuswami, A.K.: Better Inapproximability Results for maxClique, Chromatic Number and Min-3Lin-Deletion. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 226–237. Springer, Heidelberg (2006)
Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge Univ. Pr. (1997)
Losievskaja, E.: Approximation Algorithms for Independent Set Problems on Hypergraphs. PhD thesis. Reykjavik University (January 2010)
Manjunath, M., Mehlhorn, K., Panagiotou, K., Sun, H.: Approximate Counting of Cycles in Streams. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 677–688. Springer, Heidelberg (2011)
Ruzsa, I., Szemerédi, E.: Triple systems with no six points carrying three triangles. Combinatorics (Keszthely, 1976), Coll. Math. Soc. J. Bolyai 18, 939–945 (1976)
Yao, A.C.C.: Some complexity questions related to distributive computing (preliminary report). In: STOC 1979, pp. 209–213. ACM (1979)
Zelke, M.: Intractability of min- and max-cut in streaming graphs. Inf. Process. Lett. 111(3), 145–150 (2011)
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Halldórsson, M.M., Sun, X., Szegedy, M., Wang, C. (2012). Streaming and Communication Complexity of Clique Approximation. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_38
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DOI: https://doi.org/10.1007/978-3-642-31594-7_38
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