Approximation Algorithm for the Distance-3 Independent Set Problem on Cubic Graphs

Abstract

For an integer \(d \ge 2\), a distance-d independent set of an unweighted graph \(G = (V, E)\) is a subset \(S \subseteq V\) of vertices such that for any pair of vertices \(u, v \in S\), the number of edges in any path between u and v is at least d in G. Given an unweighted graph G, the goal of Maximum Distance-d Independent Set problem (MaxD d IS) is to find a maximum-cardinality distance-d independent set of G. In this paper we focus on MaxD3IS on cubic (3-regular) graphs. For every fixed integer \(d\ge 3\), MaxD d IS is NP-hard even for planar bipartite graphs of maximum degree three. Furthermore, when \(d =3\), it is known that there exists no \(\sigma \)-approximation algorithm for MaxD3IS oncubic graphs for constant \(\sigma < 1.00105\). On the other hand, the previously best approximation ratio known for MaxD3IS on cubic graphs is 2. In this paper, we improve the approximation ratio into 1.875 for MaxD3IS on cubic graphs.