Local certification of graphs with bounded genus

Naor, Parter, and Yogev [SODA 2020] recently designed a compiler for automatically translating standard centralized interactive protocols to distributed interactive protocols, as introduced by Kol, Oshman, and Saxena [PODC 2018]. In particular, by using this compiler, every linear-time algorithm for...

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Vydáno v:Discrete Applied Mathematics Ročník 325; s. 9 - 36
Hlavní autoři: Feuilloley, Laurent, Fraigniaud, Pierre, Montealegre, Pedro, Rapaport, Ivan, Rémila, Éric, Todinca, Ioan
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier B.V 30.01.2023
Elsevier
Témata:
Computer Science
Distributed graph algorithms
Distributed, Parallel, and Cluster Computing
Local certification
Locally checkable proofs
Proof-labeling scheme
Distributed graph algorithms
Proof-labeling scheme
Locally checkable proofs
Local certification
2012 ACM Subject Classification D.1.3 Concurrent Programming (Distributed programming)
proof-labeling scheme
locally checkable proofs
ISSN:0166-218X, 1872-6771
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Shrnutí:Naor, Parter, and Yogev [SODA 2020] recently designed a compiler for automatically translating standard centralized interactive protocols to distributed interactive protocols, as introduced by Kol, Oshman, and Saxena [PODC 2018]. In particular, by using this compiler, every linear-time algorithm for deciding the membership to some fixed graph class can be translated into a dMAM(O(logn)) protocol for this class, that is, a distributed interactive protocol with O(logn)-bit proof size in n-node graphs, and three interactions between the (centralized) computationally-unbounded but non-trustable prover Merlin, and the (decentralized) randomized computationally-limited verifier Arthur. As a corollary, there is a dMAM(O(logn)) protocol for recognizing the class of planar graphs, as well as for recognizing the class of graphs with bounded genus. We show that there exists a distributed interactive protocol for recognizing the class of graphs with bounded genus performing just a single interaction, from the prover to the verifier, yet preserving proof size of O(logn) bits. This result also holds for the class of graphs with bounded non-orientable genus, that is, graphs that can be embedded on a non-orientable surface of bounded genus. The interactive protocols described in this paper are actually proof-labeling schemes, i.e., a subclass of interactive protocols, previously introduced by Korman, Kutten, and Peleg [PODC 2005]. In particular, these schemes do not require any randomization from the verifier, and the proofs may often be computed a priori, at low cost, by the nodes themselves. Our results thus extend the recent proof-labeling scheme for planar graphs by Feuilloley et al. [PODC 2020], to graphs of bounded genus, and to graphs of bounded non-orientable genus.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2022.10.004