Shared vs Private Randomness in Distributed Interactive Proofs
In distributed interactive proofs, the nodes of a graph G interact with a powerful but untrustable prover who tries to convince them, in a small number of rounds and through short messages, that G satisfies some property. This series of interactions is followed by a phase of distributed verification, which may be either deterministic or randomized, where nodes exchange messages with their neighbors.
The nature of this last verification round defines the two types of interactive protocols. We say that the protocol is of Arthur-Merlin type if the verification round is deterministic. We say that the protocol is of Merlin-Arthur type if, in the verification round, the nodes are allowed to use a fresh set of random bits.
In the original model introduced by Kol, Oshman, and Saxena [PODC 2018], the randomness was private in the sense that each node had only access to an individual source of random coins. Crescenzi, Fraigniaud, and Paz [DISC 2019] initiated the study of the impact of shared randomness (the situation where the coin tosses are visible to all nodes) in the distributed interactive model.
In this work, we continue that research line by showing that the impact of the two forms of randomness is very different depending on whether we are considering Arthur-Merlin protocols or Merlin-Arthur protocols. While private randomness gives more power to the first type of protocols, shared randomness provides more power to the second. Our results also connect shared randomness in distributed interactive proofs with distributed verification, and new lower bounds are obtained.
Distributed interactive proofs
Distributed verification
Shared randomness
Private randomness
Theory of computation~Distributed computing models
Theory of computation~Interactive proof systems
Theory of computation~Distributed algorithms
51:1-51:13
Regular Paper
Partially supported by CONICYT via PIA / Apoyo a Centros Científicos y Tecnológicos de Excelencia AFB 170001 (P.M. and I.R.), FONDECYT 1170021 (D.R. and I.R.), FONDECYT 11190482 (P.M.) and PAI + Convocatoria Nacional Subvención a la Incorporación en la Academia Año 2017 + PAI77170068 (P.M.).
A full version of the paper is available at https://arxiv.org/abs/2006.16191.
Pedro
Montealegre
Pedro Montealegre
Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile
Diego
Ramírez-Romero
Diego Ramírez-Romero
Departamento de Ingeniería Matemática, Universidad de Chile, Santiago, Chile
Ivan
Rapaport
Ivan Rapaport
DIM-CMM (UMI 2807 CNRS), Universidad de Chile, Santiago, Chile
10.4230/LIPIcs.ISAAC.2020.51
László Babai and Peter G Kimmel. Randomized simultaneous messages: Solution of a problem of Yao in communication complexity. In Proceedings of Computational Complexity. Twelfth Annual IEEE Conference, pages 239-246. IEEE, 1997.
Mor Baruch, Pierre Fraigniaud, and Boaz Patt-Shamir. Randomized proof-labeling schemes. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, pages 315-324, 2015.
Florent Becker, Pedro Montealegre, Ivan Rapaport, and Ioan Todinca. The simultaneous number-in-hand communication model for networks: Private coins, public coins and determinism. In International Colloquium on Structural Information and Communication Complexity, pages 83-95. Springer, 2014.
Keren Censor-Hillel, Ami Paz, and Mor Perry. Approximate proof-labeling schemes. Theoretical Computer Science, 2018.
Pierluigi Crescenzi, Pierre Fraigniaud, and Ami Paz. Trade-offs in distributed interactive proofs. In 33rd International Symposium on Distributed Computing (DISC 2019). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2019.
Orr Fischer, Rotem Oshman, and Uri Zwick. Public vs. private randomness in simultaneous multi-party communication complexity. In Proc. of the International Colloquium on Structural Information and Communication Complexity, volume 9988 of Lecture Notes in Computer Science, pages 60-74, 2016.
Pierre Fraigniaud, Amos Korman, and David Peleg. Towards a complexity theory for local distributed computing. Journal of the ACM (JACM), 60(5):1-26, 2013.
Pierre Fraigniaud, Pedro Montealegre, Rotem Oshman, Ivan Rapaport, and Ioan Todinca. On Distributed Merlin-Arthur Decision Protocols. In International Colloquium on Structural Information and Communication Complexity, pages 230-245. Springer, 2019.
Pierre Fraigniaud, Boaz Patt-Shamir, and Mor Perry. Randomized proof-labeling schemes. Distributed Computing, 32(3):217-234, 2019.
Oded Goldreich, Silvio Micali, and Avi Wigderson. Proofs that yield nothing but their validity or all languages in np have zero-knowledge proof systems. Journal of the ACM (JACM), 38(3):690-728, 1991.
Shafi Goldwasser, Silvio Micali, and Charles Rackoff. The knowledge complexity of interactive proof systems. SIAM Journal on computing, 18(1):186-208, 1989.
Mika Göös and Jukka Suomela. Locally checkable proofs in distributed computing. Theory of Computing, 12(1):1-33, 2016.
Gillat Kol, Rotem Oshman, and Raghuvansh R Saxena. Interactive distributed proofs. In Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, pages 255-264. ACM, 2018.
Amos Korman and Shay Kutten. Distributed verification of minimum spanning trees. Distributed Computing, 20(4):253-266, 2007.
Amos Korman, Shay Kutten, and David Peleg. Proof labeling schemes. Distributed Computing, 22(4):215-233, 2010.
Ilan Kremer, Noam Nisan, and Dana Ron. On randomized one-round communication complexity. Computational Complexity, 8(1):21-49, 1999. URL: https://doi.org/10.1007/s000370050018.
https://doi.org/10.1007/s000370050018
Eyal Kushilevitz. Communication complexity. In Advances in Computers, volume 44, pages 331-360. Elsevier, 1997.
Moni Naor, Merav Parte, and Eylon Yogev. The power of distributed verifiers in interactive proofs. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1096-115. SIAM, 2020.
Moni Naor and Larry Stockmeyer. What can be computed locally? SIAM Journal on Computing, 24(6):1259-1277, 1995.
Ilan Newman and Mario Szegedy. Public vs. private coin flips in one round communication games. In Proc. of the 28th ACM Symposium on Theory of Computing, STOC '09, pages 561-570, 1996.
Andrew Chi-Chih Yao. Some complexity questions related to distributive computing (preliminary report). In Proceedings of the eleventh annual ACM symposium on Theory of computing, pages 209-213, 1979.
Pedro Montealegre, Diego Ramírez-Romero, and Ivan Rapaport
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