Local Computation Algorithms for Graphs of Non-Constant Degrees

TL;DR

This paper introduces local computation algorithms for graph problems with degrees that can grow with the number of vertices, achieving subexponential dependence on degree while maintaining efficient dependence on graph size.

Contribution

It presents new randomized LCAs for maximal independent sets and approximate maximum matching with complexities subexponential in degree and polylogarithmic in graph size.

Findings

LCA for maximal independent set with quasi-polynomial complexity in degree

LCA for approximate maximum matching with polynomial complexity in degree

Efficient algorithms for graphs with non-constant degrees

Abstract

In the model of \emph{local computation algorithms} (LCAs), we aim to compute the queried part of the output by examining only a small (sublinear) portion of the input. Many recently developed LCAs on graph problems achieve time and space complexities with very low dependence on $n$, the number of vertices. Nonetheless, these complexities are generally at least exponential in $d$, the upper bound on the degree of the input graph. Instead, we consider the case where parameter $d$ can be moderately dependent on $n$, and aim for complexities with subexponential dependence on $d$, while maintaining polylogarithmic dependence on $n$. We present: a randomized LCA for computing maximal independent sets whose time and space complexities are quasi-polynomial in $d$ and polylogarithmic in $n$; for constant $\epsilon > 0$, a randomized LCA that provides a $(1-\epsilon)$-approximation to maximum matching whose time and space complexities are polynomial in $d$ and polylogarithmic in $n$.