Local Computation: Lower and Upper Bounds

TL;DR

This paper establishes fundamental limits and new algorithms for what can be computed locally in distributed networks, providing bounds and schemes for a broad class of optimization problems.

Contribution

It presents the first poly-logarithmic lower bounds and a new distributed algorithm for local computation of key optimization problems in networks.

Findings

Poly-logarithmic lower bounds for local computation of several optimization problems.

A new distributed algorithm for solving covering and packing linear programs.

Bounds and algorithms together characterize local computability and approximability.

Abstract

The question of what can be computed, and how efficiently, are at the core of computer science. Not surprisingly, in distributed systems and networking research, an equally fundamental question is what can be computed in a \emph{distributed} fashion. More precisely, if nodes of a network must base their decision on information in their local neighborhood only, how well can they compute or approximate a global (optimization) problem? In this paper we give the first poly-logarithmic lower bound on such local computation for (optimization) problems including minimum vertex cover, minimum (connected) dominating set, maximum matching, maximal independent set, and maximal matching. In addition we present a new distributed algorithm for solving general covering and packing linear programs. For some problems this algorithm is tight with the lower bounds, for others it is a distributed approximation scheme. Together, our lower and upper bounds establish the local computability and approximability of a large class of problems, characterizing how much local information is required to solve these tasks.