Abstract
An extended formulation of a polyhedron \(P\) is a linear description of a polyhedron \(Q\) together with a linear map \(\pi \) such that \(\pi (Q)=P\). These objects are of fundamental importance in polyhedral combinatorics and optimization theory, and the subject of a number of studies. Yannakakis’ factorization theorem (Yannakakis in J Comput Syst Sci 43(3):441–466, 1991) provides a surprising connection between extended formulations and communication complexity, showing that the smallest size of an extended formulation of \(P\) equals the nonnegative rank of its slack matrix \(S\). Moreover, Yannakakis also shows that the nonnegative rank of \(S\) is at most \(2^c\), where \(c\) is the complexity of any deterministic protocol computing \(S\). In this paper, we show that the latter result can be strengthened when we allow protocols to be randomized. In particular, we prove that the base-\(2\) logarithm of the nonnegative rank of any nonnegative matrix equals the minimum complexity of a randomized communication protocol computing the matrix in expectation. Using Yannakakis’ factorization theorem, this implies that the base-\(2\) logarithm of the smallest size of an extended formulation of a polytope \(P\) equals the minimum complexity of a randomized communication protocol computing the slack matrix of \(P\) in expectation. We show that allowing randomization in the protocol can be crucial for obtaining small extended formulations. Specifically, we prove that for the spanning tree and perfect matching polytopes, small variance in the protocol forces large size in the extended formulation.
Similar content being viewed by others
Notes
Throughout this paper, we use \(\lg \) for binary logarithm.
The extended formulation for \(Q\) given above potentially has a large number of equalities, but recall we only consider the number of inequalities in the size of the extended formulation. The reasons for this are twofold: first, one can ignore most of the equalities after picking a small number of linearly independent equalities; and second, our concern in this paper is mainly the existence of certain extensions.
References
Chattopadhyay, A., Pitassi, T.: The story of set disjointness. SIGACT News 41(3), 59–85 (2010)
Chvátal, V.: On certain polytopes associated with graphs. J. Comb. Theory B 18, 138–154 (1975)
Cohen, J.E., Rothblum, U.G.: Nonnegative ranks, decompositions, and factorizations of nonnegative matrices. Linear Algebra Appl. 190, 149–168 (1993)
Conforti, M., Cornuéjols, G., Zambelli, G.: Extended formulations in combinatorial optimization. 4OR 8(1), 1–48 (2010)
Edmonds, J.: Maximum matching and a polyhedron with 0, 1 vertices. J. Res. Nat. Bur. Stand. 69B, 125–130 (1965)
Edmonds, J.: Matroids and the greedy algorithm. Math. Program. 1, 127–136 (1971)
Faenza, Y., Oriolo, G., Stauffer, G.: Separating stable sets in claw-free graphs via Padberg-Rao and compact linear programs. In: Rabani, Y. (ed.) Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2012), pp. 1298–1308. SIAM, Japan (2012)
Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., de Wolf, R.: Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds. In: Proceedings of the 44th ACM Symposium on Theory of Computing (STOC 2012), pp. 95–106 (2012)
Fiorini, S., Kaibel, V., Pashkovich, K., Theis, D.O.: Combinatorial bounds on nonnegative rank and extended formulations. Discret. Math. 313(1), 67–83 (2013)
Galluccio, A., Gentile, C., Ventura, P.: The stable set polytope of claw-free graphs with large stability number. Electron. Notes Discrete Math. 36, 1025–1032 (2010)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization, volume 2 of Algorithms and Combinatorics., 2nd edn. Springer, Berlin (1993)
Høyer, P., de Wolf, R.: Improved quantum communication complexity bounds for disjointness and equality. In Proceedings of STACS, pp. 299–310 (2002)
Kaibel, V.: Extended formulations in combinatorial optimization. Optima 85, 2–7 (2011)
Kaibel, V., Pashkovich, K., Oliver, D.: Theis. Symmetry matters for the sizes of extended formulations. In: Proceedings of IPCO, pp. 135–148 (2010)
Kalyanasundaram, B., Schnitger, G.: The probabilistic communication complexity of set intersection. SIAM J. Discr. Math. 5(4), 545–557 (1992)
Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)
Krause, M.: Geometric arguments yield better bounds for threshold circuits and distributed computing. Theor. Comput. Sci. 156(1&2), 99–117 (1996)
Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math. 13(4), 383–390 (1975)
Richard, K.M.: Using separation algorithms to generate mixed integer model reformulations. Oper. Res. Lett. 10(3), 119–128 (1991)
Razborov, A.A.: On the distributional complexity of disjointness. Theor. Comput. Sci. 106(2), 385–390 (1992)
Rothvoß, T.: Some 0/1 polytopes need exponential size extended formulations. arXiv:1105.0036 (2011)
Schrijver, A.: Combinatorial optimization. Polyhedra and efficiency. Vol. A and B, Volume 24 of Algorithms and Combinatorics. Springer, Berlin (2003)
Thomason, A.: The extremal function for complete minors. Journal of Combinatorial Theory. Series B, Volume 81, Number 2. Academic Press, Inc., NY (2001)
Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991)
Williams, J.C.: A linear-size zero–one programming model for the minimum spanning tree problem in planar graphs. Networks 39(1), 53–60 (2002)
Wolsey, L.A.: Using extended formulations in practice. Optima 85, 7–9 (2011)
Zhang, S.: Quantum Strategic Game Theory. In Proceedings of the 3rd Innovations in, Theoretical Computer Science, pp. 39–59 (2012)
Ziegler, G.M.: Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer, Berlin (1995)
Acknowledgments
The authors thank Sebastian Pokutta and Ronald de Wolf for their useful feedback. The research of Faenza was supported by the German Research Foundation (DFG) within the Priority Programme 1307 Algorithm Engineering. The research of Grappe was supported by the Progetto di Eccellenza 2008–2009 of the Fondazione Cassa di Risparmio di Padova e Rovigo. The research of Fiorini was partially supported by the Actions de Recherche Concertées (ARC) fund of the French community of Belgium. The research of Tiwary was supported by the Fonds National de la Recherche Scientifique (F.R.S.–FNRS). The authors would also like to thank the anonymous referees for their helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
A previous and reduced version of this paper appeared in the Proceedings of ISCO 2012.
H. R. Tiwary: Postdoctoral Researcher of the Fonds National de la Recherche Scientifique (F.R.S.–FNRS).
Rights and permissions
About this article
Cite this article
Faenza, Y., Fiorini, S., Grappe, R. et al. Extended formulations, nonnegative factorizations, and randomized communication protocols. Math. Program. 153, 75–94 (2015). https://doi.org/10.1007/s10107-014-0755-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-014-0755-3