Abstract
In 1990 Karchmer and Widgerson considered the following communication problem \(\mathtt {Bit}\): Alice and Bob know a function \(f: \{0, 1\}^n \rightarrow \{0, 1\}\), Alice receives a point \(x \in f^{-1}(1)\), Bob receives \(y \in f^{-1}(0)\), and their goal is to find a position i such that \(x_i \ne y_i\). Karchmer and Wigderson proved that the minimal size of a boolean formula for the function f equals the size of the smallest communication protocol for the \(\mathtt {Bit}\) relation. In this paper we consider a model of dag-like communication complexity (instead of classical one where protocols correspond to trees). We prove an analogue of Karchmer-Wigderson Theorem for this model and boolean circuits. We also consider a relation between this model and communication PLS games proposed by Razborov in 1995 and simplify the proof of Razborov’s analogue of Karchmer-Wigderson Theorem for PLS games.
We also consider a dag-like analogue of real-valued communication protocols and adapt a lower bound technique for monotone real circuits to prove a lower bound for these protocols.
In 1997 Krajíček suggested an interpolation technique that allows to prove lower bounds on the lengths of resolution proofs and Cutting Plane proofs with small coefficients (\(\mathrm {CP}^*\)). Also in 2016 Krajíček adapted this technique to “random resolution”. The base of this technique is an application of Razborov’s theorem. We use real-valued dag-like communication protocols to generalize the ideas of this technique, which helps us to prove a lower bound on the Cutting Plane proof system (\(\mathrm {CP}\)) and adapt it to “random \(\mathrm {CP}\)”.
Our notion of dag-like communication games allows us to use a Raz-McKenzie transformation [5, 17], which yields a lower bound on the real monotone circuit size for the \(\mathtt {CSP}\text {-}\mathtt {SAT}\) problem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alon, N., Boppana, R.B.: The monotone circuit complexity of boolean functions. Combinatorica 7(1), 1–22 (1987). http://dx.doi.org/10.1007/BF02579196
Beame, P., Pitassi, T., Segerlind, N.: Lower bounds for Lovász-Schrijver systems and beyond follow from multiparty communication complexity. SIAM J. Comput. 37(3), 845–869 (2007). http://dx.doi.org/10.1137/060654645
Buss, S.R., Kolodziejczyk, L.A., Thapen, N.: Fragments of approximate counting. J. Symb. Log. 79(2), 496–525 (2014). http://dx.doi.org/10.1017/jsl.2013.37
Dinur, I., Meir, O.: Toward the KRW composition conjecture: Cubic formula lower bounds via communication complexity. In: 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pp. 3:1–3:51 (2016). http://dx.doi.org/10.4230/LIPIcs.CCC.2016.3
Göös, M., Pitassi, T.: Communication lower bounds via critical block sensitivity. In: Symposium on Theory of Computing, STOC 2014, New York, NY, USA, 31 May–03 June, 2014, pp. 847–856 (2014). http://doi.acm.org/10.1145/2591796.2591838
Haken, A., Cook, S.A.: An exponential lower bound for the size of monotone real circuits. J. Comput. Syst. Sci. 58(2), 326–335 (1999)
Hrubeš, P.: A note on semantic cutting planes. Electron. Colloquium Comput. Complex. (ECCC) 20, 128 (2013). http://eccc.hpi-web.de/report/2013/128
Huynh, T., Nordström, J.: On the virtue of succinct proofs: amplifying communication complexity hardness to time-space trade-offs in proof complexity. In: Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, 19–22 May 2012, pp. 233–248 (2012). http://doi.acm.org/10.1145/2213977.2214000
Karchmer, M., Wigderson, A.: Monotone circuits for connectivity require super-logarithmic depth. SIAM J. Discrete Math. 3(2), 255–265 (1990). http://dx.doi.org/10.1137/0403021
Krajíček, J.: Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic. J. Symb. Log. 62(2), 457–486 (1997). http://dx.doi.org/10.2307/2275541
Krajíček, J.: Interpolation by a game. Math. Log. Q. 44, 450–458 (1998). http://dx.doi.org/10.1002/malq.19980440403
Krajíček, J.: An exponential lower bound for a constraint propagation proof system based on ordered binary decision diagrams. J. Symb. Log. 73(1), 227–237 (2008). http://dx.doi.org/10.2178/jsl/1208358751
Krajíček, J.: A feasible interpolation for random resolution. CoRR abs/1604.06560 (2016). http://arxiv.org/abs/1604.06560
Oliveira, I.: Unconditional lower bounds in complexity theory. Ph.D. thesis, Columbia university (2015)
Pudlák, P.: Lower bounds for resolution and cutting plane proofs and monotone computations. J. Symb. Log. 62(3), 981–998 (1997). http://dx.doi.org/10.2307/2275583
Pudlák, P.: On extracting computations from propositional proofs (a survey). In: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2010, 15–18 December 2010, Chennai, India, pp. 30–41 (2010). http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2010.30
Raz, R., McKenzie, P.: Separation of the monotone NC hierarchy. Combinatorica 19(3), 403–435 (1999). http://dx.doi.org/10.1007/s004930050062
Razborov, A.A.: Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic. Izvestiya RAN. Ser. Mat. 59, 201–224 (1995)
Robert, R., Pitassi, T.: Strongly exponential lower bounds for monotone computation. ECCC Report: TR16-188 (2016)
Sokolov, D.: Dag-like communication and its applications. Electronic Colloquium on Computational Complexity (ECCC) (2016). http://eccc.hpi-web.de/report/2016/202
Acknowledgements
This research is supported by Russian Science Foundation (project 16-11-10123).
The author is grateful to Pavel Pudlák and Dmitry Itsykson for fruitful discussions. The author also thanks Edward Hirsch, Dmitry Itsykson and anonymous reviewers for error correction.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Sokolov, D. (2017). Dag-Like Communication and Its Applications. In: Weil, P. (eds) Computer Science – Theory and Applications. CSR 2017. Lecture Notes in Computer Science(), vol 10304. Springer, Cham. https://doi.org/10.1007/978-3-319-58747-9_26
Download citation
DOI: https://doi.org/10.1007/978-3-319-58747-9_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-58746-2
Online ISBN: 978-3-319-58747-9
eBook Packages: Computer ScienceComputer Science (R0)