A hierarchy of local decision

Laurent Feuilloley,Pierre Fraigniaud,Juho Hirvonen

doi:10.1016/j.tcs.2020.12.017

Abstract

We extend the notion of distributed decision in the framework of distributed network computing, inspired by both the polynomial hierarchy for Turing machines and recent results on so-called distributed graph automata. We show that, by using distributed decision mechanisms based on the interaction between a prover and a disprover, the size of the certificates distributed to the nodes for certifying a given network property can be drastically reduced. For instance, we prove that minimum spanning tree (MST) can be certified with O(log⁡n)-bit certificates in n-node graphs, with just one interaction between the prover and the disprover, while it is known that certifying MST requires Ω(log2⁡n)-bit certificates if only the prover can act. The improvement can even be exponential for some simple graph properties. For instance, it is known that certifying the existence of a nontrivial automorphism requires Ω(n2) bits if only the prover can act. We show that there is a protocol with two interactions between the prover and the disprover that certifies nontrivial automorphism with O(log⁡n)-bit certificates. These results are achieved by defining and analyzing a local hierarchy of decision which generalizes the classical notions of proof-labeling schemes and locally checkable proofs.

Highlights

This paper is tackling the long-standing issue of characterizing the power of local computation in the framework of distributed network computing [27]
The minimum-weight spanning tree (MST) task consists, given the weights x(u) of all the incident edges of every node u, in computing a subset y(u) of edges incident to u such that the set {y(u), u ∈ V } forms a MST in G
In a seminal result Linial showed that the same is true for maximal independent set (MIS) [24]: there is no local algorithm for constructing an MIS, even on an n-node ring

Summary

Introduction

This paper is tackling the long-standing issue of characterizing the power of local computation in the framework of distributed network computing [27]. Our concern is the ability to design local algorithms, defined as distributed algorithms in which every node of a network (i.e., every computing entity in the system) can compute its output after having consulted only nodes in its vicinity. Communications proceed along the links of the network, and, in a local algorithm, every node must output after having exchanged information with nodes at constant distance only. A construction task consists, for the nodes of a network G = (V, E) where each node u is given an input x(u), to collectively and concurrently compute a collection. There are many construction tasks that can be solved locally, such as approximate solutions of NP-hard graph problems (see, e.g., [8, 21, 22, 23]). In general it is Turing-undecidable whether or not a construction task can be solved locally [26]

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