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 is generally capable of delivering accurate results, while using a manageable 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 favorably with the existing state of the art.

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