Simple and Efficient Greedy Algorithms for Hamilton Cycles in Random Intersection Graphs

Christoforos L. Raptopoulos, Paul G. Spirakis

Abstract: In this work we consider the problem of finding Hamilton Cycles in graphs derived from the uniform random intersection graphs model G n, m, p . In particular, (a) for the case m = n α, α> 1 we give a result that allows us to apply (with the same probability of success) any algorithm that finds a Hamilton cycle with high probability in a G n, k graph (i.e. a graph chosen equiprobably form the space of all graphs with k edges), (b) we give an expected polynomial time algorithm for the case p = constant and \(m \leq \alpha {\sqrt{{n}\over {{\rm log}n}}}\) for some constant α, and (c) we show that the greedy approach still works well even in the case \(m = o({{n}\over{{\rm log}n}})\) and p just above the connectivity threshold of G n, m, p (found in [21]) by giving a greedy algorithm that finds a Hamilton cycle in those ranges of m, p with high probability.