Moderately Exponential Time Approximation Algorithms for the Maximum Bounded-degreed Set Problem

Given a graph G = (V,E), a vertex set S ⊆ V is called an s-plex if every vertex v ∈ S is of degree at least |S| − s in G[S]. The Maximum s-plex (Max s-plex) problem is to find an splex of maximum cardinality in an input graph. It has applications on finding cohesive subgroups in social networks. A bounded-degree-d set S in a graph G = (V,E) is a vertex subset of G such that the maximum degree in G[S] is at most d. The Maximum Bounded-Degree-d Set (Max d-bds) problem is to find a bounded-degree-d set S of maximum cardinality in G. Both Max s-plex and Max d-bds are NP-hard problems. A vertex subset S is a bounded-degree-d set in G if and only if S is a (d+1)-plex in Ḡ. In this paper, we show that if P = NP , for all > 0, Max d-bds and Max s-plex cannot be approximated with a ratio greater than n −1 in polynomial time for bounded d ≥ 1 and s ≥ 2. Moreover, we design moderately exponential time 1/p-approximation algorithms to solve the Max d-bds problem where p is an integer satisfying p ≥ d + 1. For d ≥ 2, our 1/papproximation algorithms run in time faster than O(2) where n is the number of vertices in the input graph.