On the Analysis of Independent Sets via Multilevel Splitting

Counting the number of independent sets is an important problem in graph theory, combinatorics, optimization, and social sciences. However, a polynomial-time exact calculation, or even a reasonably close approximation, is widely believed to be impossible, since their existence implies an efficient solution to various problems in the non-deterministic polynomial-time complexity class. To cope with the approximation challenge, we express the independent set counting problem as a rare-event estimation problem. We develop a multilevel splitting algorithm which has a provable probabilistic performance guarantee on its lower bound, and is generally capable of delivering accurate results while using workable computational effort, even when applied to large graphs. We apply the algorithm to both counting and optimization (finding a maximum independent set) problems, and show that it compares favourably with the existing state of the art.