An O(1.4658)-time exact algorithm for the maximum bounded-degree-1 set problem

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-Degree1 Set 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 a 2-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 NPhard. We give a simple branch-and-reduce algorithm using branching strategies with at most three branches for the Maximum Bounded-Degree1 Set problem. We analyze the running time of the algorithm using measure-and-conquer and show that it runs in time O(1.4658) which is faster than previous exact algorithms. This research is partially supported by the National Science Council of Taiwan under grants NSC 101–2221–E– 241–019–MY3 and NSC 102–2221–E–241–007–MY3. Ling-Ju Hung (corresponding author) is supported by the National Science Council of Taiwan under grant NSC 102–2811–E–241–001.