On Recognizing Staircase Compatibility

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