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

TL;DR

This paper proves the Tradeoff Conjecture in distributed proof labeling schemes up to a logarithmic factor, establishing a near-optimal relationship between label size and hop count for certifying network properties.

Contribution

It resolves the longstanding Tradeoff Conjecture for general and minor-free graphs, providing tight bounds and refuting a stronger previously conjectured variant.

Findings

Proves that 1-PLS with cost p implies t-PLS with cost O(t log n) in general graphs.

Shows that for minor-free graphs, 1-PLS with cost p implies t-PLS with cost O(ceil(p/t)+log n).

Refutes a stronger variant of the Tradeoff Conjecture, demonstrating limitations of large neighborhood size.

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 $\widetilde{O}(\lceil \Delta p/\sqrt{t} \rceil)$ cost was shown for fixed minor-free graphs, where $\Delta$ 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 $O(\lceil p/t \rceil)$.