A local constant-factor approximation algorithm for MDS problem in anonymous network

TL;DR

This paper presents a distributed algorithm that approximates the Minimum Dominating Set problem in planar graphs without using node identifiers, relying only on port orderings, thus challenging the necessity of unique IDs in such networks.

Contribution

It introduces a constant-factor approximation algorithm for MDS in planar graphs that operates without node identifiers, using only port orderings in the CONGEST model.

Findings

Algorithm achieves constant approximation ratio

No unique identifiers needed for the algorithm

Applicable to planar graphs in distributed settings

Abstract

In research on distributed local algorithms it is commonly assumed that each vertex has a unique identifier in the entire graph. However, it turns out that in case of certain classes of graphs (for example not lift-closed bounded degree graphs) identifiers are unnecessary and only a port ordering is needed. One of the open issues was whether identifiers are essential in planar graphs. In this paper, we answer this question and we propose an algorithm which returns constant approximation of the MDS problem in CONGEST model. The algorithm doesn't use any additional information about the structure of the graph and the nodes don't have unique identifiers. We hope that this paper will be very helpful as a hint for further comparisons of the unique identifier model and the model with only a port numbering in other classes of graphs.