Skip to main content

Some 0/1 polytopes need exponential size extended formulations


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.

This is a preview of subscription content, access via your institution.


  1. 1

    Arora S., Barak B.: Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  2. 2

    Balas E.: Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Algebraic Discret. Methods 6(3), 466–486 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3

    Balas E.: Disjunctive programming: properties of the convex hull of feasible points. Discret. Appl. Math. 89(1–3), 3–44 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4

    Barahona F.: On cuts and matchings in planar graphs. Math. Program. 60, 53–68 (1993). doi:10.1007/BF01580600

    MathSciNet  Article  MATH  Google Scholar 

  5. 5

    Bienstock D.: Approximate formulations for 0–1 knapsack sets. Oper. Res. Lett. 36(3), 317–320 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6

    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

    Article  MATH  Google Scholar 

  7. 7

    Dukes W.M.B.: Bounds on the number of generalized partitions and some applications. Aust. J. Combin. 28, 257–261 (2003)

    MathSciNet  MATH  Google Scholar 

  8. 8

    Edmonds J.: Maximum matching and a polyhedron with 0, 1-vertices. J. Res. Nat. Bur. Standards Sect. B 69, 125–130 (1965)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9

    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)

  10. 10

    Fiorini, S., Rothvoß, T., Tiwary, H.: Extended formulations for polygons. Discret. Comput. Geom., pp. 1–11 (2012). doi:10.1007/s00454-012-9421-9

  11. 11

    Furst M., Saxe J.B., Sipser M.: Parity, circuits, and the polynomial-time hierarchy. Math. Syst. Theory 17(1), 13–27 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12

    Frank A., Tardos É.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13

    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)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14

    Goemans, M.: Smallest compact formulation for the permutahedron. Working paper. (2010)

  15. 15

    Kaibel, V.: Extended Formulations in Combinatorial Optimization, ArXiv e-prints (2011)

  16. 16

    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)

  17. 17

    Kaibel, V., Pashkovich, K., Theis, D.O.: Symmetry matters for the sizes of extended formulations. In: IPCO, pp. 135–148 (2010)

  18. 18

    Kipp Martin R.: Using separation algorithms to generate mixed integer model reformulations. Oper. Res. Lett. 10(3), 119–128 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19

    Pritchard, D.: An lp with integrality gap 1+epsilon for multidimensional knapsack. CoRR, abs/ 1005.3324 (2010)

  20. 20

    Schrijver A.: Combinatorial Optimization. Polyhedra and Efficiency. vol. A,B,C, vol 24 of Algorithms and Combinatorics. Springer, Berlin (2003)

    Google Scholar 

  21. 21

    Shannon C.E.: The synthesis of two-terminal switching circuits. Bell Syst. Tech. J. 28, 59–98 (1949)

    MathSciNet  Article  Google Scholar 

  22. 22

    Vavasis S.A.: On the complexity of nonnegative matrix factorization. SIAM J. Optim. 20(3), 1364–1377 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23

    Yannakakis M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24

    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)

Download references

Author information



Corresponding author

Correspondence to Thomas Rothvoß.

Rights and permissions

Reprints 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).

Download citation

Mathematics Subject Classification

  • 90Cxx