Compact Distributed Interactive Proofs for the Recognition of Cographs and Distance-Hereditary Graphs

Pedro Montealegre, Diego Ramírez-Romero, Ivan Rapaport

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We present compact distributed interactive proofs for the recognition of two important graph classes, well-studied in the context of centralized algorithms, namely complement reducible graphs and distance-hereditary graphs. Complement reducible graphs (also called cographs) are defined as the graphs not containing a four-node path P4 as an induced subgraph. Distance-hereditary graphs are a super-class of cographs, defined as the graphs where the distance (shortest paths) between any pair of vertices is the same on every induced connected subgraph. First, we show that there exists a distributed interactive proof for the recognition of cographs with two rounds of interaction. More precisely, we give a dAM protocol with a proof size of O(log n) bits that recognizes cographs with high probability. Moreover, our protocol can be adapted to verify any Turing-decidable predicate restricted to cographs in dAM with certificates of size O(log n). Second, we give a three-round, dMAM interactive protocol for the recognition of distance-hereditary graphs, still with a proof size of O(log n) bits. Finally, we show that any one-round (denoted dM ) or two-round, dMA protocol for the recognition of cographs or distance-hereditary graphs requires certificates of size Ω(log n) bits. Moreover, we show that any constant-round dAM protocol using shared randomness requires certificates of size Ω(log log n).

Original languageEnglish
Title of host publicationStabilization, Safety, and Security of Distributed Systems - 23rd International Symposium, SSS 2021, Proceedings
EditorsColette Johnen, Elad Michael Schiller, Stefan Schmid
PublisherSpringer Science and Business Media Deutschland GmbH
Pages395-409
Number of pages15
ISBN (Print)9783030910808
DOIs
StatePublished - 2021
Externally publishedYes
Event23rd International Symposium on Stabilization, Safety, and Security of Distributed Systems, SSS 2021 - Virtual, Online
Duration: 17 Nov 202120 Nov 2021

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume13046 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference23rd International Symposium on Stabilization, Safety, and Security of Distributed Systems, SSS 2021
CityVirtual, Online
Period17/11/2120/11/21

Keywords

  • Cographs
  • Distance hereditary graph
  • Distributed interactive proofs
  • Distributed recognition of graph classes

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