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

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Stabilization, Safety, and Security of Distributed Systems (SSS 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13046))

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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 \(P_4\) 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 \(\mathsf {dAM}\) protocol with a proof size of \(\mathcal {O}(\log n)\) bits that recognizes cographs with high probability. Moreover, our protocol can be adapted to verify any Turing-decidable predicate restricted to cographs in \(\mathsf {dAM}\) with certificates of size \(\mathcal {O}(\log n)\).

Second, we give a three-round, \(\mathsf {dMAM}\) interactive protocol for the recognition of distance-hereditary graphs, still with a proof size of \(\mathcal {O}(\log n)\) bits.

Finally, we show that any one-round (denoted \(\mathsf {dM}\)) or two-round, \(\mathsf {dMA}\) protocol for the recognition of cographs or distance-hereditary graphs requires certificates of size \(\varOmega (\log n)\) bits. Moreover, we show that any constant-round \(\mathsf {dAM}\) protocol using shared randomness requires certificates of size \(\varOmega (\log \log n)\).

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.) and FONDECYT 11190482 (P.M.) and PAI + Convocatoria Nacional Subvención a la Incorporación en la Academia Año 2017 + PAI77170068 (P.M.).

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Authors and Affiliations

  1. Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile

    Pedro Montealegre

  2. Departamento de Ingeniería Matemática, Universidad de Chile, Santiago, Chile

    Diego Ramírez-Romero

  3. DIM-CMM (UMI 2807 CNRS), Universidad de Chile, Santiago, Chile

    Diego Ramírez-Romero & Ivan Rapaport

Authors
  1. Pedro Montealegre
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  2. Diego Ramírez-Romero
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  3. Ivan Rapaport
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Correspondence to Pedro Montealegre .

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Editors and Affiliations

  1. University of Bordeaux, Bordeaux, France

    Colette Johnen

  2. Chalmers University of Technology, Gothenburg, Sweden

    Elad Michael Schiller

  3. University of Vienna, Vienna, Austria

    Stefan Schmid

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Montealegre, P., Ramírez-Romero, D., Rapaport, I. (2021). Compact Distributed Interactive Proofs for the Recognition of Cographs and Distance-Hereditary Graphs. In: Johnen, C., Schiller, E.M., Schmid, S. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2021. Lecture Notes in Computer Science(), vol 13046. Springer, Cham. https://doi.org/10.1007/978-3-030-91081-5_26

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  • DOI: https://doi.org/10.1007/978-3-030-91081-5_26

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