Approximability of the distance independent set problem on regular graphs and planar graphs

This paper studies generalized variants of the maximum independent set problem, called the MAXIMUM DISTANCE-d INDEPENDENT SET problem (MaxD dIS for short). For an integer d >= 2, a distance-d independent set of an unweighted graph G = (V, E) is a subset S subset of V of vertices such that for any pair of vertices u, v is an element of S, the number of edges in any path between u and v is at least d in G. Given an unweighted graph.., the goal of MaxD..IS is to find a maximum-cardinality distance-.. independent set of G. In this paper, we analyze the (in)approximability of the problem on.. -regular graphs (r >= 3) and planar graphs, as follows: (1) For every fixed integers d >= 3 and r >= 3, MaxD..IS on r-regular graphs is APX-hard. (2) We design polynomial-time.. (r(d-1))-approximation and O(r(d-2)/d)-approximation algorithms for MaxDdIS on 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 MaxD.. IS admits a polynomial-time approximation scheme (PTAS) for planar graphs.