Incrementally Finding the Vertices Absent from the Maximum Independent Sets

Abstract

A vertex v in a graph G is called an absent vertex if it is not in any maximum independent set of G. Absent vertex discovery is useful in various scenarios. For example, if G depicts a wireless communication interference graph, the existence of absent vertices in G may indicate network throughput bottlenecks. However, finding all the absent vertices is hard since it is at least as difficult as finding all the maximum independent sets, which is NP-hard. This paper focuses on a method that finds the absent vertices incrementally, in the hope of finding many such vertices quickly in the early incremental stages. The method iteratively invokes two polynomial-time algorithms to find the ‘easy’ absent vertices, and then the expensive exact maximum independent set solver to find the ‘difficult’ ones. At each iteration, the Mirror theorem is used to find extra absent vertices, and then all the absent vertices found so far are removed from the graph before going into the next iteration, until all absent vertices are found. Experimental results show that the above method can find most absent vertices much earlier than the baseline brute-force method on several widely-used datasets, showing its effectiveness.