A parallel algorithm for computing the critical independence number and related sets

Abstract

An independent set I c is a critical independent set if ∣ I c ∣ − ∣ N ( I c )∣ ≥ ∣ J ∣ − ∣ N ( J )∣ , for any independent set J . The critical independence number of a graph is the cardinality of a maximum critical independent set. This number is a lower bound for the independence number and can be computed in polynomial-time. The existing algorithm runs in O ( n 2. 5 √( m /log n )) time for a graph G with n = ∣ V ( G )∣ vertices and m edges. It is demonstrated here that there is a parallel algorithm using n processors that runs in O ( n 1. 5 √( m /log n )) time. The new algorithm returns the union of all maximum critical independent sets. The graph induced on this set is a König-Egerváry graph whose components are either isolated vertices or which have perfect matchings.