Abstract
We introduce the parameter cutwidth for the Cutting Planes (CP) system of Gomory and Chvátal. We provide linear lower bounds on cutwidth for two simple polytopes. Considering CP as a propositional refutation system, one can see that the cutwidth of a CNF contradiction F is always bound above by the Resolution width of F. We provide an example proving that the converse fails: there is an F which has constant cutwidth, but has Resolution width Ω(n). Following a standard method for converting an FO sentence ψ, without finite models, into a sequence of CNFs, F ψ,n , we provide a classification theorem for CP based on the sum cutwidth plus rank. Specifically, the cutwidth+rank of F ψ,n is bound by a constant c (depending on ψ only) iff ψ has no (infinite) models. This result may be seen as a relative of various gap theorems extant in the literature.
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References
Ben-sasson, E., Wigderson, A.: Short proofs are narrow - resolution made simple. Journal of the ACM, 517–526 (1999)
Bockmayr, A., Eisenbrand, F., Hartmann, M., Schulz, A.S.: On the Chvátal rank of polytopes in the 0/1 cube. Discrete Appl. Math. 98(1-2), 21–27 (1999)
Buresh-Oppenheim, J., Galesi, N., Hoory, S., Magen, A., Pitassi, T.: Rank bounds and integrality gaps for cutting planes procedures. Theory of Computing 2(4), 65–90 (2006)
Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Math. 4, 305–337 (1973)
Cook, W., Coullard, C.R., Turán, G.: On the complexity of cutting-plane proofs. Discrete Appl. Math. 18(1), 25–38 (1987)
Dantchev, S.S.: Rank complexity gap for Lovász-Schrijver and Sherali-Adams proof systems. In: STOC 2007: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pp. 311–317. ACM Press, New York (2007)
Eisenbrand, F., Schulz, S.: Bounds on the chvátal rank of polytopes in the 0/1-cube. Combinatorica 23(2), 245–261 (2003)
Gomory, R.E.: Solving linear programming problems in integers. In: Bellman, R., Hall, M. (eds.) Combinatorial Analysis, Proceedings of Symposia in Applied Mathematics, Providence, RI, vol. 10 (1960)
Gomory, R.E.: An algorithm for integer solutions to linear programs. In: Recent advances in mathematical programming, pp. 269–302. McGraw-Hill, New York (1963)
Riis, S.: A complexity gap for tree resolution. Computational Complexity 10(3), 179–209 (2001)
Riis, S.: On the asymptotic nullstellensatz and polynomial calculus proof complexity. In: Proceedings of the Twenty-Third Annual IEEE Symposium on Logic in Computer Science (LICS 2008), pp. 272–283. IEEE Computer Society Press, Los Alamitos (2008)
Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math. 3(3), 411–430 (1990)
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Dantchev, S., Martin, B. (2009). Cutting Planes and the Parameter Cutwidth. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds) Mathematical Theory and Computational Practice. CiE 2009. Lecture Notes in Computer Science, vol 5635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03073-4_15
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DOI: https://doi.org/10.1007/978-3-642-03073-4_15
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