Distributed Interactive Proofs for Planarity with Log-Star Communication

2025-10-21

AI Summary

This work addresses the efficient verification of graph planarity under the Distributed Interactive Proofs (DIP) model, aiming to minimize both the number of communication rounds and per-round message size. Methodologically, it introduces a multi-round randomized challenge-response protocol integrated with local consistency checks, enabling local verification of global structural properties within the DIP framework. The proposed protocol achieves $O(log^* n)$ rounds—the first to attain iterated-logarithmic round complexity for planarity—supporting both general planar graphs and embedded planar graphs. Proof size is compressed to $O(1)$ or $O(lceil log Delta / log^* n ceil)$, constituting the current state-of-the-art. Moreover, the protocol offers tunable trade-offs between verification accuracy and efficiency. By drastically reducing round complexity and communication overhead, this result advances the practical applicability of DIPs for verifying fundamental graph structural properties.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $O(lceillog Δ/log ^{*}n ceil)$, respectively. In fact, this result can be generalized as follows. For any $1leq rleq log^{*}n$, 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.

Problem

• Design efficient distributed interactive proofs for graph planarity
• Achieve logarithmic-star communication complexity in verification
• Enable compact proof sizes for embedded and general planarity

Innovation

• Distributed interactive proofs for graph planarity verification
• Log-star communication rounds with constant proof size
• Generalized protocol with adjustable rounds and proof sizes