Locally Restricted Proof Labeling Schemes
Introduced by Korman, Kutten, and Peleg (PODC 2005), a proof labeling scheme (PLS) is a distributed verification system dedicated to evaluating if a given configured graph satisfies a certain property. It involves a centralized prover, whose role is to provide proof that a given configured graph is a yes-instance by means of assigning labels to the nodes, and a distributed verifier, whose role is to verify the validity of the given proof via local access to the assigned labels. In this paper, we introduce the notion of a locally restricted PLS in which the prover’s power is restricted to that of a LOCAL algorithm with a polylogarithmic number of rounds. To circumvent inherent impossibilities of PLSs in the locally restricted setting, we turn to models that relax the correctness requirements by allowing the verifier to accept some no-instances as long as they are not "too far" from satisfying the property in question. To this end, we evaluate (1) distributed graph optimization problems (OptDGPs) based on the notion of an approximate proof labeling scheme (APLS) (analogous to the type of relaxation used in sequential approximation algorithms); and (2) configured graph families (CGFs) based on the notion of a testing proof labeling schemes (TPLS) (analogous to the type of relaxation used in property testing algorithms). The main contribution of the paper comes in the form of two generic compilers, one for OptDGPs and one for CGFs: given a black-box access to an APLS (resp., PLS) for a large class of OptDGPs (resp., CGFs), the compiler produces a locally restricted APLS (resp., TPLS) for the same problem, while losing at most a (1 + ε) factor in the scheme’s relaxation guarantee. An appealing feature of the two compilers is that they only require a logarithmic additive label size overhead.
proof labeling schemes
generic compilers
SLOCAL algorithms
Theory of computation~Distributed algorithms
Theory of computation~Approximation algorithms analysis
20:1-20:22
Regular Paper
This work was supported in part by the Technion Hiroshi Fujiwara Cyber Security Research Center and the Israel National Cyber Directorate. In addition, the work of Shay Kutten was also supported in part by ISF grant 1346/22.
https://arxiv.org/abs/2208.08718
Yuval
Emek
Yuval Emek
Technion - Israel Institute of Technology, Haifa, Israel
Yuval
Gil
Yuval Gil
Technion - Israel Institute of Technology, Haifa, Israel
Shay
Kutten
Shay Kutten
Technion - Israel Institute of Technology, Haifa, Israel
10.4230/LIPIcs.DISC.2022.20
Noga Alon, Tali Kaufman, Michael Krivelevich, and Dana Ron. Testing triangle-freeness in general graphs. SIAM J. Discret. Math., 2008.
B. Awerbuch, B. Patt-Shamir, and G. Varghese. Self-stabilization by local checking and correction. In Proceedings 32nd Annual Symposium of Foundations of Computer Science, pages 268-277, 1991.
Baruch Awerbuch, Andrew V. Goldberg, Michael Luby, and Serge A. Plotkin. Network decomposition and locality in distributed computation. In 30th Annual Symposium on Foundations of Computer Science, Research Triangle Park, North Carolina, USA, 30 October - 1 November 1989, pages 364-369. IEEE Computer Society, 1989.
Nir Bacrach, Keren Censor-Hillel, Michal Dory, Yuval Efron, Dean Leitersdorf, and Ami Paz. Hardness of distributed optimization. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, 2019.
Lélia Blin, Pierre Fraigniaud, and Boaz Patt-Shamir. On proof-labeling schemes versus silent self-stabilizing algorithms. In Stabilization, Safety, and Security of Distributed Systems, pages 18-32, 2014.
Keren Censor-Hillel, Ami Paz, and Mor Perry. Approximate proof-labeling schemes. Theor. Comput. Sci., 811:112-124, 2020.
Yuval Emek and Yuval Gil. Twenty-two new approximate proof labeling schemes. In 34th International Symposium on Distributed Computing, DISC 2020, October 12-16, 2020, Virtual Conference, 2020.
Yuval Emek, Yuval Gil, and Shay Kutten. Locally restricted proof labeling schemes (full version), 2022. URL: http://arxiv.org/abs/2208.08718.
http://arxiv.org/abs/2208.08718
Laurent Feuilloley and Pierre Fraigniaud. Error-sensitive proof-labeling schemes. In 31st International Symposium on Distributed Computing, DISC 2017, October 16-20, 2017, Vienna, Austria, volume 91 of LIPIcs, pages 16:1-16:15, 2017.
Laurent Feuilloley, Pierre Fraigniaud, Juho Hirvonen, Ami Paz, and Mor Perry. Redundancy in distributed proofs. In 32nd International Symposium on Distributed Computing, DISC, volume 121 of LIPIcs, pages 24:1-24:18, 2018.
Laurent Feuilloley, Pierre Fraigniaud, Pedro Montealegre, Ivan Rapaport, Éric Rémila, and Ioan Todinca. Compact distributed certification of planar graphs. Algorithmica, 83(7):2215-2244, 2021.
Pierre Fraigniaud, Amos Korman, and David Peleg. Local distributed decision. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS, pages 708-717, 2011.
Mohsen Ghaffari, Fabian Kuhn, and Yannic Maus. On the complexity of local distributed graph problems. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 784-797. ACM, 2017.
Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653-750, 1998.
Shafi Goldwasser, Guy N. Rothblum, and Yael Tauman Kalai. Delegating computation: Interactive proofs for muggles. Electron. Colloquium Comput. Complex., page 108, 2017.
Mika Göös and Jukka Suomela. Locally checkable proofs in distributed computing. THEORY OF COMPUTING, 12:1-33, 2016.
Gillat Kol, Rotem Oshman, and Raghuvansh R. Saxena. Interactive distributed proofs. In PODC 2018 - Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, pages 255-264, July 2018.
Amos Korman and Shay Kutten. Distributed verification of minimum spanning trees. Distributed Comput., 20(4):253-266, 2007.
Amos Korman, Shay Kutten, and David Peleg. Proof labeling schemes. Distributed Comput., 22(4):215-233, 2010.
E. Kushilevitz and N. Nisan. Communication Complexity. Cambridge University Press, 2006.
Nathan Linial. Distributive graph algorithms-global solutions from local data. In 28th Annual Symposium on Foundations of Computer Science, Los Angeles, California, USA, 27-29 October 1987, pages 331-335. IEEE Computer Society, 1987.
Nathan Linial. Locality in distributed graph algorithms. SIAM J. Comput., 21(1):193-201, 1992.
Moni Naor and Larry J. Stockmeyer. What can be computed locally? SIAM J. Comput., 24(6):1259-1277, 1995.
Boaz Patt-Shamir and Mor Perry. Proof-labeling schemes: Broadcast, unicast and in between. In Stabilization, Safety, and Security of Distributed Systems - 19th International Symposium, SSS, volume 10616, pages 1-17. Springer, 2017.
David Peleg. Distributed Computing: A Locality-Sensitive Approach. Society for Industrial and Applied Mathematics, USA, 2000.
Omer Reingold, Guy N. Rothblum, and Ron D. Rothblum. Constant-round interactive proofs for delegating computation. SIAM J. Comput., 50(3), 2021.
Václav Rozhon and Mohsen Ghaffari. Polylogarithmic-time deterministic network decomposition and distributed derandomization. In Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 350-363. ACM, 2020.
Yuval Emek, Yuval Gil, and Shay Kutten
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode