On Recognizing Staircase Compatibility

Abstract

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.

EETTlib—Energy‐efficient train timetabling library

“…We start by describing the feasible set of possible timetable adaptations. To model this set, we use a graph-based formulation as a clique problem with multiple-choice constraints (CPMC), as introduced in [1,13] and further investigated in [2,4,5,7]. CPMC consists in finding a clique of size m ∈ N in an m-partite graph G = (V, E).…”

“…Overall, the presented timetabling models (B2) and (B3) amount to optimizing a convex piecewise linear objective function over the CPMC-polytope. Efficient methods to solve the underlying two NP-hard problems have been proposed in [1,2,4,7], respectively.…”

“…FIGURE2 Schematic representation of the effect of choosing between one of the three driving modes "always driving as fast as possible" (black), "accelerating to maximum velocity followed by coasting" (blue), and "accelerating to below maximum velocity followed by extended coasting" (red) on a sample underground leg…”