Distributed Interactive Proofs for Planarity with Log-Star Communication

TL;DR

This paper introduces highly communication-efficient distributed interactive proof protocols for planarity testing, achieving extremely low proof sizes and rounds, advancing distributed graph property verification.

Contribution

It presents the first distributed interactive proof protocols for planarity with logarithmic-star rounds and minimal proof sizes, generalizing to various round complexities.

Findings

O(log* n)-round protocol for embedded planarity with O(1) proof size

O(log* n)-round protocol for planarity with proof size depending on maximum degree

Generalized protocols for any 1 ≤ r ≤ log* n with adjustable rounds and proof sizes

Abstract

We provide new communication-efficient distributed interactive proofs for planarity. The notion of a \emph{distributed interactive proof (DIP)} was introduced by Kol, Oshman, and Saxena (PODC 2018). In a DIP, the \emph{prover} is a single centralized entity whose goal is to prove a certain claim regarding an input graph $G$. To do so, the prover communicates with a distributed \emph{verifier} that operates concurrently on all $n$ nodes of $G$. A DIP is measured by the amount of prover-verifier communication it requires. Namely, the goal is to design a DIP with a small number of interaction rounds and a small \emph{proof size}, i.e., a small amount of communication per round. Our main result is an $O(\log ^{*}n)$-round DIP protocol for embedded planarity and planarity with a proof size of $O(1)$ and $O(\lceil\log \Delta/\log ^{*}n\rceil)$, respectively. In fact, this result can be generalized as follows. For any $1\leq r\leq \log^{*}n$, there exists an $O(r)$-round protocol for embedded planarity and planarity with a proof size of $O(\log ^{(r)}n)$ and $O(\log ^{(r)}n+\log \Delta /r)$, respectively.