The Power of Distributed Verifiers in Interactive Proofs

Abstract

We explore the power of interactive proofs with a distributed verifier. In this setting, the verifier consists of n nodes and a graph G that defines their communication pattern. The prover is a single entity that communicates with all nodes by short messages. The goal is to verify that the graph G belongs to some language in a small number of rounds, and with small communication bound, i.e., the proof size.This interactive model was introduced by Kol, Oshman and Saxena (PODC 2018) as a generalization of non-interactive distributed proofs. They demonstrated the power of interaction in this setting by constructing protocols for problems as Graph Symmetry and Graph Non-Isomorphism -both of which require proofs of Ω(n 2 )-bits without interaction.In this work, we provide a new general framework for distributed interactive proofs that allows one to translate standard interactive protocols (i.e., with a centralized verifier) to ones where the verifier is distributed with a proof size that depends on the computational complexity of the verification algorithm run by the centralized verifier. We show the following:• Every (centralized) computation that can be performed in time O(n) can be translated into three-round distributed interactive protocol with O(log n) proof size. This implies that many graph problems for sparse graphs have succinct proofs (e.g., testing planarity).• Every (centralized) computation implemented by either a small space or by uniform NC circuit can be translated into a distributed protocol with O(1) rounds and O(log n) bits proof size for the low space case and polylog(n) many rounds and proof size for NC.• We also demonstrate the power of our compilers for problems not captured by the above families. We show that for Graph Non-Isomorphism, one of the striking demonstrations of the power of interaction, there is a 4-round protocol with O(log n) proof size, improving upon the O(n log n) proof size of Kol et al.• For many problems we show how to reduce proof size below the naturally seeming barrier of log n. By employing our RAM compiler, we get a 5-round protocols with proof size O(log log n) for a family of problems including Fixed Automorphism, Clique and Leader Election (for the later two problems we actually get O(1) proof size).• Finally we discuss how to make these proofs non-interactive arguments via random oracles.Our compilers capture many natural problems and demonstrates the difficultly in showing lower bounds in these regimes.with the nodes of the network in rounds. In each round, a node u sends the prover a random challenge R u . Then, the prover responds by sending each node u its respond Y u . Nodes can exchange their proof Y u only with their immediate neighbors N(u) in the network in order to decide whether to accept the proof. For accepting a proof all nodes must accept and to reject it is enough that one node rejects. A simple example for a "distributed NP" proof is 3-coloring of a graph: the prover gives each node in the graph its color, and nodes exchange colors with their neigh...

Introduction to local certification

“…This model provides a better understanding of problems such as symmetric dumbbell graphs in Section 4.3, and follow-up works have shown that powerful tools can be designed for this setting, with trade-offs between the different resources (number of bits exchanged, number of prover-nodes communication rounds etc.) [18,57,55].…”

“…This is a recent and active area of research. We refer to [57] for a list of nice open questions. Let us just mention that an important question is to establish the best trade-off possible between interaction and communication.…”

Distributed Interactive Proofs for the Recognition of Some Geometric Intersection Graph Classes

“…It is always important to keep in mind the compiler defined in [40] which turns, automatically, any problem solved in NP in time τ (n) into a dMAM protocol that uses bandwidth τ (n) log n/n. Therefore, any class of sparse graphs that can be recognized in linear time, can also be recognized by a dMAM protocol with logarithmic-sized certificates.…”

Redundancy in distributed proofs

“…However, all predicates on labeled graphs can be certified by allowing randomization [23], or by allowing just one alternation of quantifiers (the analog of Π 2 in the polynomial hierarchy) [8]. A recent line of work studies a distributed variant of interactive proofs [15,33,40].…”