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In the field of autonomous driving, traffic‐light‐controlled intersections are of special interest. We analyze how much an optimized coordination of vehicles and infrastructure can contribute to efficient transit through these bottlenecks, depending on traffic density and certain regulations of traffic lights. To this end, we develop a mixed‐integer linear programming model to describe the interaction between traffic lights and discretized traffic flow. It is based on a microscopic traffic model with centrally controlled autonomous vehicles. We aim to determine a globally optimal traffic flow for given scenarios on a simple, but extensible, urban road network. The resulting models are very challenging to solve, in particular when involving additional realistic traffic‐light regulations such as minimum red and green times. While solving times exceed real‐time requirements, our model allows an estimation of the maximum performance gains due to improved communication and serves as a benchmark for heuristic and decentralized approaches.

For the problem to find an m-clique in an m-partite graph, staircase compatibility has recently been introduced as a polynomial-time solvable special case. It is a property of a graph together with an m-partition of the vertex set and total orders on each subset of the partition. In optimization problems involving m-cliques in m-partite graphs as a subproblem, it allows for totally unimodular linear programming formulations, which have shown to efficiently solve problems from different applications. In this work, we address questions concerning the recognizability of this property in the case where the m-partition of the graph is given, but suitable total orders are to be determined. While finding these total orders is NP-hard in general, we give several conditions under which it can be done in polynomial time. For bipartite graphs, we present a polynomial-time algorithm to recognize staircase compatibility and show that staircase total orders are unique up to a small set of reordering operations. On m-partite graphs, where the recognition problem is NP-complete in the general case, we identify a polynomially solvable subcase and also provide a corresponding algorithm to compute the total orders. Finally, we evaluate the performance of our ordering algorithm for m-partite graphs on a set of artificial instances as well as real-world instances from a railway timetabling application. It turns out that applying the ordering algorithm to the real-world instances and subsequently solving the problem via the aforementioned totally unimodular reformulations indeed outperforms a generic formulation which does not exploit staircase compatibility.