An O(qn) algorithm to q-color a proper family of circular arcs

A family of arcs on a circle is proper if no arc is properly contained within another. While general minimal arc coloring is NP-complete, Orlin et al. recently obtained on O(n²) algorithm for q-coloring a proper family of arcs by modeling proper arc coloring as a shortest path problem in an associated network (with negative edges). This paper simplifies Orlin's shortest-path network to obtain an O(qn) time algorithm.