Recovery of disrupted airline operations using k-maximum matching in graphs

Abstract

When an aircraft is approaching an airport, it gets a short time interval called a slot by Air Traffic Control that it can use to land. If the landing of the aircraft is delayed for some reason, it loses its slot and Airline Operation Controllers have to assign it a new one. However, landing slots are a scarce resource of the airports and, in order to avoid that an aircraft waits too long, Airline Operation Controllers have to regularly modify the assignment of slots to the aircraft. Due to the system implemented to exchange slots, controllers can modify the slot-assignment using only two kinds of operations: either assign a slot that was free to aircraft A, or give the slot of another aircraft B to A and assign a free slot to B. The problem can then be modelled as follows.Let k≥1 be an odd integer, let G be a graph and M be a matching in G, i.e., a set of pairwise disjoint edges. A k-maximum matching is a largest possible matching that can be obtained from M by using only augmenting paths of length at most k. This problem has already been studied in the context of wireless networks, mainly because it provides a simple approximation for the maximum matching problem. This paper provides a polynomial-time algorithm when k≤3. We then prove that the problem is NP-hard in planar bipartite graphs with maximum degree at most 3 for any odd integer k≥5.