Lower Bounds for Local Approximation

TL;DR

This paper demonstrates that for many classical graph problems, local algorithms can operate without unique node identifiers, relying only on port numbering and orientation, leading to tight bounds on approximation ratios.

Contribution

It proves that local algorithms for simple PO-checkable problems do not require unique identifiers, extending the understanding of symmetry breaking in anonymous networks.

Findings

Local algorithms with identifiers can be simulated without them for certain problems.

A tight lower bound on approximability of the minimum edge dominating set is derived.

Existence of homogeneous regular graphs with specific properties is established.

Abstract

In the study of deterministic distributed algorithms it is commonly assumed that each node has a unique $O(\log n)$-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient. Our result holds for so-called simple PO-checkable graph optimisation problems; this includes many classical packing and covering problems such as vertex covers, edge covers, matchings, independent sets, dominating sets, and edge dominating sets. We focus on the case of bounded-degree graphs and show that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks. As a corollary of our result and by prior work, we derive a tight lower bound on the local approximability of the minimum edge dominating set problem. Our main technical tool is an algebraic construction of homogeneously ordered graphs: We say that a graph is $(\alpha,r)$-homogeneous if its nodes are linearly ordered so that an $\alpha$ fraction of nodes have pairwise isomorphic radius-$r$ neighbourhoods. We show that there exists a finite $(\alpha,r)$-homogeneous $2k$-regular graph of girth at least $g$ for any $\alpha < 1$ and any $r$, $k$, and $g$.