Proof-labeling schemes: Broadcast, unicast and in between

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 rthat 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 >= 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 1 <= r <= 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 spanningtree. We also consider the congested clique model. We show that the crossing technique [1] 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. (c) 2022 Elsevier B.V. All rights reserved.