NC2 Algorithms regarding Hamiltonian paths and circuits in interval graphs

Abstract

This paper describes an O(log2n) time, n processor EREW PRAM algorithm to determine if there is a Hamiltonian path in an interval graph. If there is a Hamiltonian path, we can find it within the same resource bounds. Many graph theoretic problems including finding all maximal cliques, optimal coloring, minimum clique cover, minimum weight dominating set, and maximum independent set were previously known to be in NC when restricted to interval graphs. However, the Hamiltonian path problem was open and resisted classification until now. We also show that testing whether an interval graph has a Hamiltonian circuit can be done in NC2. If the intervals are presorted, our parallel approach leads to an O(nα(n)) sequential algorithm, where α(n) is the inverse of Ackermann's function. This improves on the previous bound of O(n log log n) for the sequential case with presorted intervals.