Abstract
In this paper, we derive two optimal parallel time bounds for the maximum clique problem on intervals. Given a set S of intervals on a real line, a clique C is a subset of S such that any two intervals in C intersect. A maximum cardinality clique, or simply a maximum clique, is a clique with the maximum cardinality. The cardinality of the maximum clique is called the clique number. The maximum (cardinality) clique problem is to compute the clique number. If each interval is associated with a number called weight, then the weight of a clique is the summation of the weights of the intervals in the clique. A maximum weighted clique is a clique of the maximum weight. The maximum weighted clique problem is to compute the weight of such a clique. If the weight of each interval is 1, then the maximum weighted clique problem is equivalent to the maximum cardinality clique problem. Therefore, the maximum cardinality clique problem is a special case of the maximum weighted clique problem. A set of A of intervals is said to be equivalent to a set B of intervals if there exists a bijection f : A ~ B such that any two intervals x and y in
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