Moderately exponential time algorithms for the maximum bounded-degree-1 set problem

Abstract

A bounded-degree-1 set S in an undirected graph G=(V,E) is a vertex subset such that the maximum degree of G[S] is at most one. Given a graph G, the Maximum Bounded-Degree-1Set (Max 1-bds) problem is to find a bounded-degree-1 set S of maximum size in G.A notion related to bounded-degree sets is that of an s-plex used to define the cohesiveness of subgraphs in social networks. An s-plex S in a graph G=(V,E) is a vertex subset such that for each v∈S, degG[S](v)≥|S|−s. One can easily show that a graph G has a2-plex of size k iff the complement graph of G has a bounded-degree-1 set of size k. Both the Maximum 2-Plex problem and the Maximum Bounded-Degree-1 Set problem are NP-hard. We give a simple branch-and-reduce algorithm running in polynomial space and applying branching strategies with at most three branches for Max 1-bds. We analyze the running time of the algorithm using measure-and-conquer and show that it runs in time O∗(1.4613n) which is faster than previous exact algorithms. Moreover, we show that the Max 1-bds problem cannot be approximated to a ratio greater than nϵ−1 in polynomial time for all ϵ>0 unless P=NP. Some moderately exponential time approximation algorithms are given for Max 1-bds and its dual problem.