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Representations of Monotone Boolean Functions by Linear Programs

Authors Mateus de Oliveira Oliveira, Pavel Pudlák

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Mateus de Oliveira Oliveira
Pavel Pudlák

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Mateus de Oliveira Oliveira and Pavel Pudlák. Representations of Monotone Boolean Functions by Linear Programs. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.CCC.2017.3

Abstract

We introduce the notion of monotone linear-programming circuits (MLP circuits),  a model of computation for partial Boolean functions. Using this model, we prove the following results.

1. MLP circuits are superpolynomially stronger than monotone Boolean circuits.

2. MLP circuits are exponentially stronger than monotone span programs.

3. MLP circuits can be used to provide monotone feasibility interpolation theorems for Lovasz-Schrijver  proof systems, and for mixed Lovasz-Schrijver  proof systems.

4. The Lovasz-Schrijver  proof system cannot be polynomially simulated by the cutting planes proof system. This is the first result showing a separation between these two proof systems.

Finally, we discuss connections between the problem of proving lower bounds on the size of MLPs and the problem of proving lower bounds on extended formulations of polytopes.

Subject Classification

Keywords
  • Monotone Linear Programming Circuits
  • Lovász-Schrijver Proof System
  • Cutting-Planes Proof System
  • Feasible Interpolation
  • Lower Bounds

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References

  1. Michael Alekhnovich and Alexander A. Razborov. Satisfiability, branch-width and Tseitin tautologies. In Proc. of the 43rd Symposium on Foundations of Computer Science, pages 593-603, 2002. Google Scholar
  2. László Babai, Anna Gál, and Avi Wigderson. Superpolynomial lower bounds for monotone span programs. Combinatorica, 19(3):301-319, 1999. Google Scholar
  3. Paul Beame, Russell Impagliazzo, Jan Krajíček, Toniann Pitassi, and Pavel Pudlák. Lower bounds on Hilbert’s Nullstellensatz and propositional proofs. Proceedings of the London Mathematical Society, 3(1):1-26, 1996. Google Scholar
  4. Gábor Braun, Samuel Fiorini, Sebastian Pokutta, and David Steurer. Approximation limits of linear programs (beyond hierarchies). Mathematics of Operations Research, 40(3):756-772, 2015. Google Scholar
  5. Mark Braverman and Ankur Moitra. An information complexity approach to extended formulations. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing, pages 161-170. ACM, 2013. Google Scholar
  6. Samuel R. Buss and Toniann Pitassi. Good degree bounds on Nullstellensatz refutations of the induction principle. Journal of Computer and System Sciences, 57(2):162-171, 1998. Google Scholar
  7. Stephen A. Cook, Toniann Pitassi, Robert Robere, and Benjamin Rossman. Exponential lower bounds for monotone span programs. ECCC, TR16-64, 2016. Google Scholar
  8. Sanjeeb Dash. Exponential lower bounds on the lengths of some classes of branch-and-cut proofs. Mathematics of Operations Research, 30(3):678-700, 2005. Google Scholar
  9. Samuel Fiorini, Serge Massar, Sebastian Pokutta, Hans Raj Tiwary, and Ronald De Wolf. Exponential lower bounds for polytopes in combinatorial optimization. Journal of the ACM (JACM), 62(2):17, 2015. Google Scholar
  10. Xudong Fu. Lower bounds on sizes of cutting planes proofs for modular coloring principles. Proof Complexity and Feasible Arithmetics, pages 135-148, 1998. Google Scholar
  11. Anna Gál and Pavel Pudlák. A note on monotone complexity and the rank of matrices. Information Processing Letters, 87(6):321-326, 2003. Google Scholar
  12. Michel X. Goemans. Smallest compact formulation for the permutahedron. Mathematical Programming, 153(1):5-11, 2015. Google Scholar
  13. Dima Grigoriev. Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity. Theoretical Computer Science, 259(1):613-622, 2001. Google Scholar
  14. Armin Haken. The intractability of resolution. Theoretical Computer Science, 39:297-308, 1985. Google Scholar
  15. Armin Haken and Stephen A. Cook. An exponential lower bound for the size of monotone real circuits. Journal of Computer and System Sciences, 58(2):326-335, 1999. Google Scholar
  16. Russell Impagliazzo, Pavel Pudlák, and Jiri Sgall. Lower bounds for the polynomial calculus and the gröbner basis algorithm. Computational Complexity, 8(2):127-144, 1999. Google Scholar
  17. M. Karchmer and A. Wigderson. On span programs. In Proceedings of the 8th Annual Structure in Complexity Theory Conference, pages 102-111. IEEE CS, 1993. Google Scholar
  18. Jan Krajíček. Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic. The Journal of Symbolic Logic, 62(02):457-486, 1997. Google Scholar
  19. Jan Krajíček. Interpolation and approximate semantic derivations. Mathematical Logic Quarterly, 48(4):602-606, 2002. Google Scholar
  20. László Lovász and Alexander Schrijver. Cones of matrices and set-functions and 0-1 optimization. SIAM Journal on Optimization, 1(2):166-190, 1991. Google Scholar
  21. Toniann Pitassi and Nathan Segerlind. Exponential lower bounds and integrality gaps for tree-like lovasz-schrijver procedures. SIAM Journal on Computing, 41(1):128-159, 2012. Google Scholar
  22. Pavel Pudlák. Lower bounds for resolution and cutting plane proofs and monotone computations. The Journal of Symbolic Logic, 62(03):981-998, 1997. Google Scholar
  23. Pavel Pudlak and Jiri Sgall. Algebraic models of computation and interpolation for algebraic proof systems. In Proc. of Feasible Arithmetic and Proof Complexity, DIMACS Series in Discrete Math. and Theoretical Comp. Sci., volume 39, pages 279-295, 1998. Google Scholar
  24. Ran Raz and Avi Wigderson. Monotone circuits for matching require linear depth. Journal of the ACM (JACM), 39(3):736-744, 1992. Google Scholar
  25. Alexander A. Razborov. Lower bounds on monotone complexity of the logical permanent. Mathematical Notes, 37(6):485-493, 1985. Google Scholar
  26. Alexander A. Razborov. Lower bounds for monotone complexity of boolean functions. American Mathematical Society Translations, 147:75-84, 1990. Google Scholar
  27. Alexander A. Razborov. Proof complexity and beyond. ACM SIGACT News, 47(2):66-86, 2016. Google Scholar
  28. Thomas Rothvoß. The matching polytope has exponential extension complexity. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 263-272. ACM, 2014. Google Scholar
  29. Alexander Schrijver. Combinatorial Optimization: Polyhedra and Efficiency, volume 24 of Algorithms and Combinatorics. Springer, 2003. Google Scholar
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