Near-Resolution of the Tradeoff Conjecture in Distributed Proof Labeling Schemes

2026-05-18

AI Summary

This work investigates the Tradeoff Conjecture concerning the relationship between label size and verification radius in distributed proof-labeling schemes (t-PLS). By leveraging graph-theoretic arguments and label compression techniques, the authors constructively design efficient t-PLS protocols for general graphs and for graph classes excluding a fixed minor. Their main contributions include an $O(t \log n)$-PLS for general graphs with label length $O(\lceil p/t \rceil)$, nearly confirming the original conjecture up to a logarithmic factor, and a refined scheme for minor-free graph classes achieving label size $O(\lceil p/t \rceil + \log n)$, thereby refuting a stronger variant of the conjecture.

Abstract

In the $t$-Proof Labeling Scheme model ($t$-PLS model), our goal is to certify that a network of nodes satisfies a given property $P$. A prover assigns a label to each node, and each node decides to accept or reject based on its labeled $t$-hop neighborhood. If $P$ holds, there exists a labeling that makes all nodes accept. If $P$ does not hold, in all labelings at least one node rejects. The cost of a scheme is its maximum label size. The Tradeoff Conjecture [Feuilloley, Fraigniaud, Hirvonen, Paz, and Perry, DISC 18, Dist. Comput.~21] hypothesizes that the existence of a $1$-PLS for a property $P$ with cost $p$ implies the existence of a $t$-PLS for $P$ with cost $O(\lceil p/t \rceil)$. The conjecture was initially shown to hold for specific graph classes, such as trees, cycles, and grids. Later, a weaker $\widetilde{O}(\lceil Δp/\sqrt{t} \rceil)$ cost was shown for fixed minor-free graphs, where $Δ$ is the maximum degree. In this work we resolve the Tradeoff Conjecture, up to a single logarithmic factor. In general graphs, we show that the existence of a $1$-PLS with cost $p$ implies the existence of an $O(t\log{n})$-PLS with cost $O(\lceil p/t \rceil)$ for the same property. For fixed minor-free graphs (which include e.g. planar graphs), we show that the existence of a $1$-PLS with cost $p$ implies the existence of a $t$-PLS with cost $O(\lceil p/t \rceil+\log{n})$ for the same property. We also refute a previously suggested stronger variant of the Tradeoff Conjecture, and show that having very large $t$-hop neighborhoods is an insufficient condition for obtaining a tradeoff better than $O(\lceil p/t \rceil)$.

Problem

• Proof Labeling Schemes
• Tradeoff Conjecture
• distributed verification
• label size
• graph properties

Innovation

• Proof Labeling Schemes
• Tradeoff Conjecture
• Distributed Verification
• Minor-Free Graphs
• Label Size Complexity