Lower bounds to randomized algorithms for graph properties

Abstract

For any property P on n-vertex graphs, let C(P) be the minimum number of edges that need to be examined by any decision tree algorithm for determining P. In 1975 Rivest and Vuillemin settled the Aanderra-Rosenberg Conjecture, proving that C(P) = Ω(n2) for every nontrivial monotone graph property P. An intriguing open question is whether the theorem remains true when randomized algorithms are allowed. In this paper we report progress on this problem, showing that Ω(n(log n)1/12) edges must be examined by a randomized algorithm for determining any nontrivial monotone graph property.