Abstract
We prove that an ω(log3 n) lower bound for the three-party number-on-the-forehead (NOF) communication complexity of the set-disjointness function implies an n ω(1) size lower bound for tree-like Lovász-Schrijver systems that refute unsatisfiable CNFs. More generally, we prove that an n Ω(1) lower bound for the (k+1)-party NOF communication complexity of set-disjointness implies a \(2^{n^{\Omega(1)}}\) size lower bound for all tree-like proof systems whose formulas are degree k polynomial inequalities.
Paul Beame’s research was supported by NSF grants CCR-0098066 and ITR-0219468. Toniann Pitassi’s research was supported by an Ontario Premiere’s Research Excellence Award, an NSERC grant, and the Institute for Advanced Study where this research was done. Nathan Segerlind’s research was supported by NSF Postdoctoral Fellowship DMS-0303258 and done while at the Institute for Advanced Study.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arora, S., Bollobás, B., Lovász, L.: Proving integrality gaps without knowing the linear program. In: Proceedings 43nd Annual Symposium on Foundations of Computer Science, Vancouver, BC, November 2002, pp. 313–322. IEEE, Los Alamitos (2002)
Babai, L., Frankl, P., Simon, J.: Complexity classes in communication complexity theory. In: 27th Annual Symposium on Foundations of Computer Science, Toronto, Ontario, October 1986, pp. 337–347. IEEE, Los Alamitos (1986)
Beame, P., Pitassi, T., Segerlind, N., Wigderson, A.: A direct sum theorem for corruption and the multiparty NOF communication complexity of set disjointness. In: Proceedings Twentieth Annual IEEE Conference on Computational Complexity, San Jose, CA (June 2005)
Bockmayr, A., Eisenbrand, F., Hartmann, M.E., Schulz, A.S.: On the Chvatal rank of polytopes in the 0/1 cube. Discrete Applied Mathematics 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. In: Proceedings 44th Annual Symposium on Foundations of Computer Science, Boston, MA, October 2003, pp. 318–327. IEEE, Los Alamitos (2003)
Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Mathematics 4, 305–337 (1973)
Chvátal, V., Cook, W., Hartmann, M.: On cutting-plane proofs in combinatorial optimization. Linear Algebra and its Applications 114/115, 455–499 (1989)
Cook, W., Coullard, C.R., Turan, G.: On the complexity of cutting plane proofs. Discrete Applied Mathematics 18, 25–38 (1987)
Dash, S.: On the matrix cuts of Lovász and Schrijver and their use in Integer Programming. PhD thesis, Department of Computer Science, Rice University (March 2001)
Dash, S.: An exponential lower bound on the length of some classes of branch-and-cut proofs. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337, pp. 145–160. Springer, Heidelberg (2002)
Eisenbrand, F., Schulz, A.S.: Bounds on the Chvatal rank of polytopes in the 0/1-cube. Combinatorica 23(2), 245–261 (2003)
Grigoriev, D., Hirsch, E.A., Pasechnik, D.V.: Complexity of semi-algebraic proofs. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 419–430. Springer, Heidelberg (2002)
Impagliazzo, R., Pitassi, T., Urquhart, A.: Upper and lower bounds on tree-like cutting planes proofs. In: 9th Annual IEEE Symposium on Logic in Computer Science, Paris, France, pp. 220–228 (1994)
Kalyanasundaram, B., Schnitger, G.: The probabilistic communication complexity of set intersection. In: Proceedings, Structure in Complexity Theory, Second Annual Conference, Cornell University, Ithaca, NY, June 1987, pp. 41–49. IEEE, Los Alamitos (1987)
Khachian, L.G.: A polynomial time algorithm for linear programming. Doklady Akademii Nauk SSSR, n.s. 244(5), 1093–1096 (1979); English translation in Soviet Math. Dokl. 20, 191–194 (1979)
Lovasz, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optimization 1(2), 166–190 (1991)
Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8(3), 261–277 (1988)
Nisan, N.: The communication complexity of threshold gates. In: Mikl’os, V.S.D., Szonyi, T. (eds.) Combinatorics: Paul Erdös is Eighty, vol. I, pp. 301–315. Bolyai Society (1993)
Pudlák, P.: Lower bounds for resolution and cutting plane proofs and monotone computations. Journal of Symbolic Logic 62(3), 981–998 (1997)
Raz, R., Wigderson, A.: Monotone circuits for matching require linear depth. Journal of the ACM 39(3), 736–744 (1992)
Valiant, L., Vazirani, V.: NP is as easy as detecting unique solutions. Theoretical Computer Science, 85–93 (1986)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Beame, P., Pitassi, T., Segerlind, N. (2005). Lower Bounds for Lovász-Schrijver Systems and Beyond Follow from Multiparty Communication Complexity. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds) Automata, Languages and Programming. ICALP 2005. Lecture Notes in Computer Science, vol 3580. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11523468_95
Download citation
DOI: https://doi.org/10.1007/11523468_95
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-27580-0
Online ISBN: 978-3-540-31691-6
eBook Packages: Computer ScienceComputer Science (R0)