Proof-Labeling Schemes: Broadcast, Unicast and In Between

TL;DR

This paper investigates how limiting the number of distinct messages a node can send affects the complexity of proof-labeling schemes in distributed systems, revealing both bounds and insensitivities for various problems.

Contribution

It introduces a unified framework for analyzing message constraints in proof-labeling schemes and provides new bounds and techniques for verification complexity under these constraints.

Findings

Linear message compression in r for agreement problems

Verification complexity gap for maximal matching between r=1 and r>1

Verification complexity insensitivity for spanning-tree predicate

Abstract

We study the effect of limiting the number of different messages a node can transmit simultaneously on the verification complexity of proof-labeling schemes (PLS). In a PLS, each node is given a label, and the goal is to verify, by exchanging messages over each link in each direction, that a certain global predicate is satisfied by the system configuration. We consider a single parameter r that bounds the number of distinct messages that can be sent concurrently by any node: in the case r=1, each node may only send the same message to all its neighbors (the broadcast model), in the case r is at least Delta, where Delta is the largest node degree in the system, each neighbor may be sent a distinct message (the unicast model), and in general, for r between 1 and Delta, each of the r messages is destined to a subset of the neighbors. We show that message compression linear in r is possible for verifying fundamental problems such as the agreement between edge endpoints on the edge state. Some problems, including verification of maximal matching, exhibit a large gap in complexity between r=1 and r>1. For some other important predicates, the verification complexity is insensitive to r, e.g., the question whether a subset of edges constitutes a spanning-tree. We also consider the congested clique model. We show that the crossing technique for proving lower bounds on the verification complexity can be applied in the case of congested clique only if r=1. Together with a new upper bound, this allows us to determine the verification complexity of MST in the broadcast clique. Finally, we establish a general connection between the deterministic and randomized verification complexity for any given number r.