Distributed Interactive Proofs for Planarity with Log-Star Communication

TL;DR

This work introducesDistributed Interactive Proofs (DIPs) for planarity that dramatically reduce prover–verifier communication. The authors develop the DoubleLog and IteratedLog protocols, combining LR-sorting reductions with self-reduction techniques to achieve $O( ext{log}^{*} n)$ rounds with constant per-round proof size for embedded planarity and sublogarithmic-style proofs for planarity with bounded degree. They further provide constant-round trade-offs, showing $O(d)$ rounds with $O( ext{log}^{(d+1)} n)$ proof size for embedded planarity and planarity, plus extensions to outerplanar, series-parallel, and treewidth-$2$ graphs. The approach hinges on partitioning into inner/outer blocks, reducing outer-block checks to polynomial identity/equality tests, and enabling efficient verification through carefully engineered encoding and self-reduction. The results push sublogarithmic communication in DIP planarity verification and raise open questions about constant-round possibilities and embeddings.

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 $O(\lceil\log Δ/\log ^{*}n\rceil)$, 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 $O(\log ^{(r)}n+\log Δ/r)$, respectively.

Figures (2)

• Figure 1: A valid instance of LR-sorting. The edges of the Hamiltonian path $H$ are solid and the edges of $E-H$ are dashed.
• Figure 2: A valid instance of equality. Here, $e^{*}$ is depicted by the dashed edge which connects between $P$ on the left and $P'$ on the right. The encoded bitstrings are $\alpha=\alpha'=1011$.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1: GilP2025
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6