On a Polynomially Solvable Subclass of the Clique Problem with Applications in Energy-Efficient Timetabling

Abstract

The clique problem under multiple-choice constraints is an ${\mathcal {N}\mathcal {P}}$-hard variant of the general clique problem, which incorporates a structure commonly found in real-world applications like underground, railway or runway scheduling. It is relevant whenever there is a set of decisions with discrete options for each decision and possible conflicts between options. In this article, we identify a polynomial-time solvable subclass and determine its complete convex hull using graph-theoretic arguments related to perfect graphs. Since the convex hull can have exponentially many facets, we present criteria on how to more efficiently find the stable sets required to describe the convex hull as well as a polynomial-time separation algorithm. Finally, the theoretical results were successfully applied to energy-efficient underground and railway scheduling.