Abstract
An intersection graph of n vertices assumes that each vertex is equipped with a subset of a global label set. Two vertices share an edge when their label sets intersect. Random Intersection Graphs (RIGs) (as defined in [18,32]) consider label sets formed by the following experiment: each vertex, independently and uniformly, examines all the labels (m in total) one by one. Each examination is independent and the vertex succeeds to put the label in her set with probability p. Such graphs nicely capture interactions in networks due to sharing of resources among nodes. We study here the problem of efficiently coloring (and of finding upper bounds to the chromatic number) of RIGs. We concentrate in a range of parameters not examined in the literature, namely: (a) m = n α for α less than 1 (in this range, RIGs differ substantially from the Erdös-Renyi random graphs) and (b) the selection probability p is quite high (e.g. at least \(\frac{\ln^2{n}}{m}\) in our algorithm) and disallows direct greedy colouring methods.
We manage to get the following results:
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For the case mp ≤ βln n, for any constant β < 1 − α, we prove that np colours are enough to colour most of the vertices of the graph with high probability (whp). This means that even for quite dense graphs, using the same number of colours as those needed to properly colour the clique induced by any label suffices to colour almost all of the vertices of the graph. Note also that this range of values of m, p is quite wider than the one studied in [4].
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We propose and analyze an algorithm CliqueColour for finding a proper colouring of a random instance of \({\cal G}_{n, m, p}\), for any mp ≥ ln 2 n. The algorithm uses information of the label sets assigned to the vertices of G n, m, p and runs in \(O\left(\frac{n^2mp^2}{\ln{n}} \right)\) time, which is polynomial in n and m. We also show by a reduction to the uniform random intersection graphs model that the number of colours required by the algorithm are of the correct order of magnitude with the actual chromatic number of G n, m, p .
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We finally compare the problem of finding a proper colouring for G n, m, p to that of colouring hypergraphs so that no edge is monochromatic. We show how one can find in polynomial time a k-colouring of the vertices of G n, m, p , for any integer k, such that no clique induced by only one label in G n, m, p is monochromatic.
Our techniques are novel and try to exploit as much as possible the hidden structure of random intersection graphs in this interesting range.
This work was partially supported by the ICT Programme of the European Union under contract number ICT-2008-215270 (FRONTS). Also supported by Research Training Group GK-693 of the Paderborn Institute for Scientific Computation (PaSCo).
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References
Alon, N., Spencer, J.: “The Probabilistic Method”. John Wiley & Sons, Inc., Chichester (2000)
Alon, N., Kahale, N.: A spectral technique for colouring random 3-colourable graphs. In: The Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pp. 346–355 (1994)
Beck, J.: An Algorithmic Approach to the Lovász Local Lemma. Random Structures & Algorithms 2, 343–365 (1991)
Behrisch, M., Taraz, A., Ueckerdt, M.: Colouring random intersection graphs and complex networks. SIAM J. Discrete Math. 23(1), 288–299 (2009)
Bollobás, B.: “The chromatic number of random graphs”. Combinatorica 8(1), 49–55 (1988)
Busch, C., Magdon-Ismail, M., Sivrikaya, F., Yener, B.: Contention-free MAC Protocols for Asynchronous Wireless Sensor Networks. Distributed Computing 21(1), 23–42 (2008)
Caragiannis, I., Kaklamanis, C., Kranakis, E., Krizanc, D., Wiese, A.: Communication in Wireless Networks with Directional Antennas. In: The Proceedings of the 20th Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2008), pp. 344–351 (2008)
Deijfen, M., Kets, W.: Random Intersection Graphs With Tunable Degree Distribution and Clustering. CentER Discussion Paper Series No. 2007-08, http://ssrn.com/abstract=962359
Diaz, J., Lotker, Z., Serna, M.J.: The Distant-2 Chromatic Number of Random Proximity and Random Geometric Graphs. Inf. Process. Lett. 106(4), 144–148 (2008)
Dolev, S., Gilbert, S., Guerraoui, R., Newport, C.C.: Secure Communication over Radio Channels. In: The ACM Symposium on Principles of Distributed Computing (PODC), pp. 105–114 (2008)
Dolev, S., Lahiani, L., Yung, M.: secret swarm unit reactive k −Secret sharing. In: Srinathan, K., Rangan, C.P., Yung, M. (eds.) INDOCRYPT 2007. LNCS, vol. 4859, pp. 123–137. Springer, Heidelberg (2007)
Efthymiou, C., Spirakis, P.G.: On the existence of hamiltonian cycles in random intersection graphs. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 690–701. Springer, Heidelberg (2005)
Erdös, P., Selfridge, J.: On a Combinatorial Game. J. Combinatorial Th (A) 14, 298–301 (1973)
Fill, J.A., Sheinerman, E.R., Singer-Cohen, K.B.: Random Intersection Graphs when m = ω(n): An Equivalence Theorem Relating the Evolution of the G(n, m, p) and G(n, p) models, http://citeseer.nj.nec.com/fill98random.html
Frieze, A., Mubayi, D.: Colouring Simple Hypergraphs, arXiv:0901.3699v1 (2008)
Frieze, A.: On the Independence Number of Random Graphs. Disc. Math. 81, 171–175 (1990)
Godehardt, E., Jaworski, J.: Two models of Random Intersection Graphs for Classification. In: Opitz, O., Schwaiger, M. (eds.) Studies in Classification, Data Analysis and Knowledge Organization, pp. 67–82. Springer, Heidelberg (2002)
Karoński, M., Scheinerman, E.R., Singer-Cohen, K.B.: On Random Intersection Graphs: The Subgraph Problem. Combinatorics, Probability and Computing Journal 8, 131–159 (1999)
Kothapalli, K., Scheideler, C., Schindelhauer, C., Onus, M.: Distributed Colouring in \(O(\sqrt{\log{n}})\) bits. In: The 20th IEEE International Parallel & Distributed Processing Symposium (IPDPS) (2006)
Leone, P., Moraru, L., Powell, O., Rolim, J.D.P.: Localization algorithm for wireless ad-hoc sensor networks with traffic overhead minimization by emission inhibition. In: Nikoletseas, S.E., Rolim, J.D.P. (eds.) ALGOSENSORS 2006. LNCS, vol. 4240, pp. 119–129. Springer, Heidelberg (2006)
Łuczak, T.: The chromatic number of random graphs. Combinatorica 11(1), 45–54 (2005)
Molloy, M., Reed, B.: Graph Colouring and the Probabilistic Method. Springer, Heidelberg (2002)
Molloy, M., Reed, B.: Colouring Graphs whose Chromatic Number is almost their Maximum Degree. In: The Proceedings of Latin American Theoretical Informatics, pp. 216–225 (1998)
Molloy, M., Reed, B.: Further Algorithmic Aspects of the Local Lemma. In: The Proceedings of the 30th ACM Symposium on Theory of Computing, pp. 524–529 (1998)
Nikoletseas, S., Raptopoulos, C., Spirakis, P.: Colouring Non-Sparse Random Intersection Graphs, http://wwwhni.uni-paderborn.de/alg/mitarbeiter/raptopox/
Nikoletseas, S.E., Raptopoulos, C., Spirakis, P.G.: The existence and efficient construction of large independent sets in general random intersection graphs. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1029–1040. Springer, Heidelberg (2004)
Nikoletseas, S., Raptopoulos, C., Spirakis, P.: Large Independent Sets in General Random Intersection Graphs. Theoretical Computer Science (TCS) Journal, Special Issue on Global Computing (2008)
Nikoletseas, S., Raptopoulos, C., Spirakis, P.: On the Independence Number and Hamiltonicity of Uniform Random Intersection Graphs. In: accepted in the 23rd IEEE International Parallel and Distributed Processing Symposium (IPDPS) (2009)
Raptopoulos, C., Spirakis, P.G.: Simple and efficient greedy algorithms for hamilton cycles in random intersection graphs. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 493–504. Springer, Heidelberg (2005)
Ross, S.M.: “Stochastic Processes”, 2nd edn. John Wiley & Sons, Inc., Chichester (2000)
Schindelhauer, C., Voss, K.: Oblivious Parallel Probabilistic Channel Utilization without Control Channels. In: The 20th IEEE International Parallel & Distributed Processing Symposium (IPDPS) (2006)
Singer-Cohen, K.B.: Random Intersection Graphs, PhD thesis. John Hopkins University (1995)
Stark, D.: The Vertex Degree Distribution of Random Intersection Graphs. Random Structures & Algorithms 24(3), 249–258 (2004)
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Nikoletseas, S., Raptopoulos, C., Spirakis, P.G. (2009). Colouring Non-sparse Random Intersection Graphs. In: Královič, R., Niwiński, D. (eds) Mathematical Foundations of Computer Science 2009. MFCS 2009. Lecture Notes in Computer Science, vol 5734. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03816-7_51
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