TY - GEN

T1 - Compact Distributed Interactive Proofs for the Recognition of Cographs and Distance-Hereditary Graphs

AU - Montealegre, Pedro

AU - Ramírez-Romero, Diego

AU - Rapaport, Ivan

N1 - Publisher Copyright:
© 2021, Springer Nature Switzerland AG.

PY - 2021

Y1 - 2021

N2 - 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).

AB - 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).

KW - Cographs

KW - Distance hereditary graph

KW - Distributed interactive proofs

KW - Distributed recognition of graph classes

UR - http://www.scopus.com/inward/record.url?scp=85119833555&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-91081-5_26

DO - 10.1007/978-3-030-91081-5_26

M3 - Conference contribution

AN - SCOPUS:85119833555

SN - 9783030910808

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 395

EP - 409

BT - Stabilization, Safety, and Security of Distributed Systems - 23rd International Symposium, SSS 2021, Proceedings

A2 - Johnen, Colette

A2 - Schiller, Elad Michael

A2 - Schmid, Stefan

PB - Springer Science and Business Media Deutschland GmbH

Y2 - 17 November 2021 through 20 November 2021

ER -