Redundancy in distributed proofs

Abstract

Distributed proofs are mechanisms that enable the nodes of a network to collectively and efficiently check the correctness of Boolean predicates on the structure of the network (e.g., having a specific diameter), or on objects distributed over the nodes (e.g., a spanning tree). We consider well known mechanisms consisting of two components: a prover that assigns a certificate to each node, and a distributed algorithm called a verifier that is in charge of verifying the distributed proof formed by the collection of all certificates. We show that many network predicates have distributed proofs offering a high level of redundancy, explicitly or implicitly. We use this remarkable property of distributed proofs to establish perfect tradeoffs between the size of the certificate stored at every node, and the number of rounds of the verification protocol.

Introduction to local certification

“…The exact construction depends on the language studied, because the graph should encode the disjointness problem with gadgets depending on that language. To our knowledge, there are two examples of such proofs in the literature: an Ω(n 2 / log n) bound for non-3-colorability [45] and an Ω(n/k) bound for diameter at most k in [16] (see also [30]). (For an introduction to this type of techniques, see [25].…”

“…Still on the communication side, another line of work proposes to increase the view of the nodes beyond constant [59,30] (see also [51]). In other words, these papers study the impact of a larger verification radius on the certificate size.…”

“…On the one hand, the fact that increasing the radius allows to decrease the size of the proofs indicates that there is some redundancy in the certificates. For example, it is not the case that in total we need n × O(log n) bits to certify a spanning tree, because if we allow a view at logarithmic distance, [30] proved that we are fine with O(1)-bit certificates. On the other hand, this redundancy is not global, in the sense that it is not the case that the exact same information is copied in every certificate.…”

Local certification of MSO properties for bounded treedepth graphs

“…Despite a lot of e orts on this problem and the design of speci c tools (so-called expander decompositions), the best algorithm known uses a large number of rounds, Θ(n 1/3 ), and is randomized [9]. The suspected hardness of this problem suggests that the certi cation problem could also be di cult (that is, could require large certi cates), because insights and concrete lower bounds have been successfully transferred from the CONGEST model to local certi cation in the past [8,18].…”

Compact Distributed Certification of Planar Graphs

“…The first difference is that LCPs allow for verification procedures performing many rounds, while PLS are restricted to a single round of verification. Nevertheless, PLS can be easily extended to many rounds whenever the certificates are large enough to contain IDs, which enables to trade longer verification time for smaller certificates size (see, e.g., [20]). The second difference is more profound.…”