Abstract
We prove that there are 0/1 polytopes \({P \subseteq \mathbb{R}^{n}}\) that do not admit a compact LP formulation. More precisely we show that for every n there is a set \({X \subseteq \{ 0,1\}^n}\) such that conv(X) must have extension complexity at least \({2^{n/2\cdot(1-o(1))}}\) . In other words, every polyhedron Q that can be linearly projected on conv(X) must have exponentially many facets. In fact, the same result also applies if conv(X) is restricted to be a matroid polytope. Conditioning on \({\mathbf{NP}\not\subseteq \mathbf{P_{/poly}}}\) , our result rules out the existence of a compact formulation for any \({\mathbf{NP}}\) -hard optimization problem even if the formulation may contain arbitrary real numbers.
Similar content being viewed by others
References
Arora S., Barak B.: Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge (2009)
Balas E.: Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Algebraic Discret. Methods 6(3), 466–486 (1985)
Balas E.: Disjunctive programming: properties of the convex hull of feasible points. Discret. Appl. Math. 89(1–3), 3–44 (1998)
Barahona F.: On cuts and matchings in planar graphs. Math. Program. 60, 53–68 (1993). doi:10.1007/BF01580600
Bienstock D.: Approximate formulations for 0–1 knapsack sets. Oper. Res. Lett. 36(3), 317–320 (2008)
Conforti M., Cornuéjols G., Zambelli G.: Extended formulations in combinatorial optimization. 4OR Q. J. Oper. Res. 8, 1–48 (2010). doi:10.1007/s10288-010-0122-z
Dukes W.M.B.: Bounds on the number of generalized partitions and some applications. Aust. J. Combin. 28, 257–261 (2003)
Edmonds J.: Maximum matching and a polyhedron with 0, 1-vertices. J. Res. Nat. Bur. Standards Sect. B 69, 125–130 (1965)
Fiorini, S., Massar, S., Pokutta, S., Tiwary, H., de Wolf, R.: Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds. CoRR abs/1111.0837, 2011. (To appear in STOC) (2012)
Fiorini, S., Rothvoß, T., Tiwary, H.: Extended formulations for polygons. Discret. Comput. Geom., pp. 1–11 (2012). doi:10.1007/s00454-012-9421-9
Furst M., Saxe J.B., Sipser M.: Parity, circuits, and the polynomial-time hierarchy. Math. Syst. Theory 17(1), 13–27 (1984)
Frank A., Tardos É.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)
Gerards A.M.H.: Compact systems for t-join and perfect matching polyhedra of graphs with bounded genus. Oper. Res. Lett. 10(7), 377–382 (1991)
Goemans, M.: Smallest compact formulation for the permutahedron. Working paper. http://math.mit.edu/~goemans/PAPERS/permutahedron.pdf (2010)
Kaibel, V.: Extended Formulations in Combinatorial Optimization, ArXiv e-prints (2011)
Khachiyan, L.G.: A polynomial algorithm for linear programming. Soviet Math. Doklady 20, 191–194, (Russian original in Doklady Akademiia Nauk SSSR, 244, 1093–1096) (1979)
Kaibel, V., Pashkovich, K., Theis, D.O.: Symmetry matters for the sizes of extended formulations. In: IPCO, pp. 135–148 (2010)
Kipp Martin R.: Using separation algorithms to generate mixed integer model reformulations. Oper. Res. Lett. 10(3), 119–128 (1991)
Pritchard, D.: An lp with integrality gap 1+epsilon for multidimensional knapsack. CoRR, abs/ 1005.3324 (2010)
Schrijver A.: Combinatorial Optimization. Polyhedra and Efficiency. vol. A,B,C, vol 24 of Algorithms and Combinatorics. Springer, Berlin (2003)
Shannon C.E.: The synthesis of two-terminal switching circuits. Bell Syst. Tech. J. 28, 59–98 (1949)
Vavasis S.A.: On the complexity of nonnegative matrix factorization. SIAM J. Optim. 20(3), 1364–1377 (2009)
Yannakakis M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991)
Ziegler, G.M.: Lectures on 0/1-polytopes. In: Polytopes—combinatorics and computation (Oberwolfach, 1997), vol. 29 of DMV Sem., pp. 1–41. Birkhäuser, Basel (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rothvoß, T. Some 0/1 polytopes need exponential size extended formulations. Math. Program. 142, 255–268 (2013). https://doi.org/10.1007/s10107-012-0574-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-012-0574-3