Constant-factor approximation of domination number in sparse graphs

TL;DR

This paper presents a linear-time algorithm that approximates the k-domination number within a constant factor for sparse graph classes with bounded expansion, including minor-closed and surface-embeddable graphs.

Contribution

It introduces a novel linear-time approximation algorithm for k-domination in bounded expansion graphs, based on an approximate min-max characterization involving independent sets.

Findings

The algorithm achieves a constant-factor approximation for k-domination number.

It finds large 2k-independent sets and small k-dominating sets efficiently.

Results extend to all graphs with bounded arrangeability for the case k=1.

Abstract

The k-domination number of a graph is the minimum size of a set X such that every vertex of G is in distance at most k from X. We give a linear time constant-factor approximation algorithm for k-domination number in classes of graphs with bounded expansion, which include e.g. proper minor-closed graph classes, classes closed on topological minors or classes of graphs that can be drawn on a fixed surface with bounded number of crossings on each edge. The algorithm is based on the following approximate min-max characterization. A subset A of vertices of a graph G is d-independent if the distance between each pair of vertices in A is greater than d. Note that the size of the largest 2k-independent set is a lower bound for the k-domination number. We show that every graph from a fixed class with bounded expansion contains a 2k-independent set A and a k-dominating set D such that |D|=O(|A|), and these sets can be found in linear time. For domination number (k=1) the assumptions can be relaxed, and the result holds for all graph classes with arrangeability bounded by a constant.