Maximal independent sets

Abstract

Abstract The line graph of G is a graph L whose vertices are edges of G. Two vertices are neighbors in L if they have a common endpoint (as edges) in G. Then a set M of edges of G is a matching in G iff it is independent (as a set of vertices) in L. The NC-construction of a maximal independent set is also related to matchings in the following way: the basic part of the algorithm is a parallel construction of a matching with sufficiently small weight. In this chapter we show how to construct in parallel a maximal independent set. Of course construction of a maximum cardinality independent set is a much harder problem, since it is NP-complete. We present two very simple randomized algorithms due to Luby [186], one of which can be derandomized using the technique from Chapter 4. However, the number of processors of a derandomized version is too large. We then show a completely different and more efficient NC-algorithm, due to Goldberg and Spencer. We also mention some extensions to hypergraphs. The maximal independent sets problem is much harder for hypergraphs than for standard graphs and no NC-algorithm is known to solve it. Another generalisation of maximal independent sets is the notion of a k-dependent set, we cover this subject at the end of the chapter.