Abstract
We study nondeterministic communication complexity and related concepts (fooling sets, fractional covering number) of random functions \(f:X\times Y \rightarrow \{0,1\}\) where each value is chosen to be 1 independently with probability \(p=p(n)\), \(n := {\left|{X}\right|}={\left|{Y}\right|}\).
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Alon, N., Spencer, J.H.: The Probabilistic Method. Wiley, New York (2008)
Beasley, L.B., Klauck, H., Lee, T., Theis, D.O.: Communication complexity, linear optimization, and lower bounds for the nonnegative rank of matrices (dagstuhl seminar 13082). Dagstuhl Rep. 3(2), 127–143 (2013)
Bollobás, B.: Random Graphs. Cambridge Studies in Advanced Mathematics, vol. 73, 2nd edn. Cambridge University Press, Cambridge (2001)
Braun, G., Fiorini, S., Pokutta, S.: Average case polyhedral complexity of the maximum stable set problem. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2014, Barcelona, Spain, 4–6 September 2014, pp. 515–530 (2014). http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.515
Dani, V., Moore, C.: Independent sets in random graphs from the weighted second moment method. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds.) APPROX/RANDOM 2011. LNCS, vol. 6845, pp. 472–482. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22935-0_40
Dawande, M., Keskinocak, P., Swaminathan, J.M., Tayur, S.: On bipartite and multipartite clique problems. J. Algorithms 41(2), 388–403 (2001). http://dx.doi.org/10.1006/jagm.2001.1199
Dawande, M., Keskinocak, P., Tayur, S.: On the biclique problem in bipartite graphs. Carnegie Mellon University (1996). GSIA Working Paper
Dietzfelbinger, M., Hromkovič, J., Schnitger, G.: A comparison of two lower-bound methods for communication complexity. Theoret. Comput. Sci. 168(1), 39–51 (1996). http://dx.doi.org/10.1016/S0304-3975(96)00062-X, 19th International Symposium on Mathematical Foundations of Computer Science, Košice (1994)
Fiorini, S., Kaibel, V., Pashkovich, K., Theis, D.O.: Combinatorial bounds on nonnegative rank and extended formulations. Discrete Math. 313(1), 67–83 (2013)
Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., Wolf, R.: Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds. In: STOC (2012)
Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., Wolf, R.D.: Exponential lower bounds for polytopes in combinatorial optimization. J. ACM (JACM) 62(2), 17 (2015)
Froncek, D., Jerebic, J., Klavzar, S., Kovár, P.: Strong isometric dimension, biclique coverings, and sperner’s theorem. Comb. Probab. Comput. 16(2), 271–275 (2007). http://dx.doi.org/10.1017/S0963548306007711
Goemans, M.X.: Smallest compact formulation for the permutahedron. Math. Program. 153(1), 5–11 (2015)
Hajiabolhassan, H., Moazami, F.: Secure frameproof code through biclique cover. Discrete Math. Theor. Comput. Sci. 14(2), 261–270 (2012). http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/2131/4075
Hajiabolhassan, H., Moazami, F.: Some new bounds for cover-free families through biclique covers. Discrete Math. 312(24), 3626–3635 (2012)
Izhakian, Z., Janson, S., Rhodes, J.: Superboolean rank and the size of the largest triangular submatrix of a random matrix. Proc. Am. Math. Soc. 143(1), 407–418 (2015)
Janson, S., Łuczak, T., Rucinski, A.: Random Graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York (2000)
Kaibel, V.: Extended formulations in combinatorial optimization. Optima - Math. Optim. Soc. Newsl. 85, 2–7 (2011). www.mathopt.org/Optima-Issues/optima85.pdf
Karp, R.M., Sipser, M.: Maximum matchings in sparse random graphs. In: FOCS, pp. 364–375 (1981)
Klauck, H., Lee, T., Theis, D.O., Thomas, R.R.: Limitations of convex programming: lower bounds on extended formulations and factorization ranks (dagstuhl seminar 15082). Dagstuhl Rep. 5(2), 109–127 (2015)
Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)
Lonardi, S., Szpankowski, W., Yang, Q.: Finding biclusters by random projections. In: Sahinalp, S.C., Muthukrishnan, S., Dogrusoz, U. (eds.) CPM 2004. LNCS, vol. 3109, pp. 102–116. Springer, Heidelberg (2004). doi:10.1007/978-3-540-27801-6_8
Lonardi, S., Szpankowski, W., Yang, Q.: Finding biclusters by random projections. Theor. Comput. Sci. 368(3), 217–230 (2006)
Lovás, L., Saks, M.: Communication complexity and combinatorial lattice theory. J. Comput. Syst. Sci. 47, 322–349 (1993)
Mitzenmacher, M., Upfal, E.: Probability and Computing – Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2006)
Park, G., Szpankowski, W.: Analysis of biclusters with applications to gene expression data. In: International Conference on Analysis of Algorithms. DMTCS Proc. AD, vol. 267, p. 274 (2005)
Roughgarden, T.: Communication complexity (for algorithm designers). arXiv preprint arXiv:1509.06257 (2015)
Schrijver, A.: Combinatorial Optimization. Polyhedra and Efficiency. Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)
Sun, X., Nobel, A.B.: On the size and recovery of submatrices of ones in a random binary matrix. J. Mach. Learn. Res 9, 2431–2453 (2008)
Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991). http://dx.doi.org/10.1016/0022-0000(91)90024-Y
Acknowledgments
The authors would like to thank the anonymous referees for their valuable comments.
Dirk Oliver Theis is supported by Estonian Research Council, ETAG (Eesti Teadusagentuur), through PUT Exploratory Grant #620. Mozhgan Pourmoradnasseri is recipient of the Estonian IT Academy Scholarship. This research is supported by the European Regional Fund through the Estonian Center of Excellence in Computer Science, EXCS.
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Pourmoradnasseri, M., Theis, D.O. (2017). Nondeterministic Communication Complexity of Random Boolean Functions (Extended Abstract). In: Gopal, T., Jäger , G., Steila, S. (eds) Theory and Applications of Models of Computation. TAMC 2017. Lecture Notes in Computer Science(), vol 10185. Springer, Cham. https://doi.org/10.1007/978-3-319-55911-7_38
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