Approximability of the Distance Independent Set Problem on Regular Graphs and Planar Graphs

Abstract

This paper studies generalized variants of the maximum independent set problem, called the Maximum Distance-d Independent Set problem ($\mathsf{MaxD}d\mathsf{IS}$ for short). 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 $\mathsf{MaxD}d\mathsf{IS}$ is to find a maximum-cardinality distance-d independent set of G. In this paper, we analyze the (in)approximability of the problem on r-regular graphs ($r\ge 3$) and planar graphs, as follows: (1) For every fixed integers $d\ge 3$ and $r\ge 3$, $\mathsf{MaxD}d\mathsf{IS}$ on r-regular graphs is APX-hard. (2) We design polynomial-time $O(r^{d-1})$-approximation and $O(r^{d-2}/d)$-approximation algorithms for $\mathsf{MaxD}d\mathsf{IS}$ on r-regular graphs. (3) We sharpen the above $O(r^{d-2}/d)$-approximation algorithms when restricted to $d=r=3$, and give a polynomial-time 2-approximation algorithm for MaxD3IS on cubic graphs. (4) Finally, we show that $\mathsf{MaxD}d\mathsf{IS}$ admits a polynomial-time approximation scheme (PTAS) for planar graphs.

This work is partially supported by JSPS KAKENHI Grant Numbers JP15J05484, JP15H00849, JP16K00004, and JP26330017.