Abstract
We consider the problem of inferring the most likely social network given connectivity constraints imposed by observations of outbreaks within the network. Given a set of vertices (or agents) V and constraints (or observations) S i ⊆ V we seek to find a minimum log-likelihood cost (or maximum likelihood) set of edges (or connections) E such that each S i induces a connected subgraph of (V,E). For the offline version of the problem, we prove an Ω(log(n)) hardness of approximation result for uniform cost networks and give an algorithm that almost matches this bound, even for arbitrary costs. Then we consider the online problem, where the constraints are satisfied as they arrive. We give an O(nlog(n))-competitive algorithm for the arbitrary cost online problem, which has an Ω(n)-competitive lower bound. We look at the uniform cost case as well and give an O(n 2/3log2/3(n))-competitive algorithm against an oblivious adversary, as well as an \(\Omega(\sqrt{n})\)-competitive lower bound against an adaptive adversary. We examine cases when the underlying network graph is known to be a star or a path, and prove matching upper and lower bounds of Θ(log(n)) on the competitive ratio for them.
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References
Akutsu, T., Tamura, T., Horimoto, K.: Completing networks using observed data. In: ALT, pp. 126–140 (2009)
Alon, N., Asodi, V.: Learning a hidden subgraph. SIAM J. Discrete Math. 18(4), 697–712 (2005)
Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: The online set cover problem. In: Proceedings of the 35th annual ACM symposium on Theory of computing, pp. 100–105 (2003)
Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: The online set cover problem. In: STOC, pp. 100–105 (2003)
Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: A general approach to online network optimization problems. ACM Transactions on Algorithms 2(4), 640–660 (2006)
Alon, N., Beigel, R., Kasif, S., Rudich, S., Sudakov, B.: Learning a hidden matching. SIAM J. Comput. 33(2), 487–501 (2004)
Angluin, D., Aspnes, J., Reyzin, L.: Optimally learning social networks with activations and suppressions. In: 19th International Conference on Algorithmic Learning Theory, pp. 272–286 (2008)
Angluin, D., Chen, J.: Learning a hidden graph using O(logn) queries per edge. J. Comput. Syst. Sci. 74(4), 546–556 (2008)
Beigel, R., Alon, N., Kasif, S., Apaydin, M.S., Fortnow, L.: An optimal procedure for gap closing in whole genome shotgun sequencing. In: RECOMB, pp. 22–30 (2001)
Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)
Buchbinder, N.: Designing Competitive Online Algorithms Via A Primal-Dual Approach. PhD thesis, Technion – Israel Institute of Technology, Haifa, Israel (2008)
Erdös, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–61 (1960)
Feige, U.: A threshold of ln for approximating set cover. J. ACM 45(4), 634–652 (1998)
Grebinski, V., Kucherov, G.: Reconstructing a Hamiltonian cycle by querying the graph: Application to DNA physical mapping. Discrete Applied Mathematics 88(1-3), 147–165 (1998)
Gupta, A., Krishnaswamy, R., Ravi, R.: Online and stochastic survivable network design. In: STOC, pp. 685–694 (2009)
Korach, E., Stern, M.: The clustering matroid and the optimal clustering tree. Mathematical Programming 98(1-3), 345–414 (2003)
Korach, E., Stern, M.: The complete optimal stars-clustering-tree problem. Discrete Applied Mathematics 156(4), 444–450 (2008)
Reyzin, L., Srivastava, N.: Learning and verifying graphs using queries with a focus on edge counting. In: Hutter, M., Servedio, R.A., Takimoto, E. (eds.) ALT 2007. LNCS (LNAI), vol. 4754, pp. 285–297. Springer, Heidelberg (2007)
Wolsey, L.A.: An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 2(4), 385–393 (1982)
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Angluin, D., Aspnes, J., Reyzin, L. (2010). Inferring Social Networks from Outbreaks. In: Hutter, M., Stephan, F., Vovk, V., Zeugmann, T. (eds) Algorithmic Learning Theory. ALT 2010. Lecture Notes in Computer Science(), vol 6331. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16108-7_12
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DOI: https://doi.org/10.1007/978-3-642-16108-7_12
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