Performance of a greedy algorithm for edge covering by cliques in interval graphs

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In this paper a greedy algorithm to detect conflict cliques in interval graphs and circular-arc graphs is analyzed. In a graph, a stable set requires that at most one vertex is chosen for each edge. It is equivalent to requiring that at most one vertex for each maximal clique is chosen. We show that this algorithm finds all maximal cliques for interval graphs, i.e. it can compute the convex hull of the stable set polytope. In case of circular-arc graphs, the algorithm is not able to detect all maximal cliques, yet remaining correct. This problem occurs in the context of railway scheduling. A train requests the allocation of a railway infrastructure resource for a specific time interval. As one is looking for conflict-free train schedules, the used resource allocation intervals in a schedule must not overlap. The conflict-free choices of used intervals for each resource correspond to stable sets in the interval graph associated to the allocation time intervals.