Aircraft routing: periodicity and complexity

TL;DR

The paper analyzes two forms of aircraft routing under maintenance constraints: a periodic instance and a finite-horizon variant. It proves that, for $\gamma \leq 4$, feasibility implies the existence of absolutely periodic solutions, making the periodic problem polynomial in this range, while showing NP-hardness for the finite-horizon version when $\gamma \geq 4$. A key methodological contribution is the introduction of an intermediary constrained path partition problem that underpins the hardness reductions and a polynomial-time solution approach for the quiet-night case with a fixed number of airplanes via a pebble-game framework. These results clarify the relationship between feasible schedules and periodic structures and delineate the boundary between tractable and intractable instances in practical airline planning.

Abstract

The aircraft routing problem is one of the most studied problems of operations research applied to aircraft management. It involves assigning flights to aircraft while ensuring regular visits to maintenance bases. This paper examines two aspects of the problem. First, we explore the relationship between periodic instances, where flights are the same every day, and periodic solutions. The literature has implicitly assumed-without discussion-that periodic instances necessitate periodic solutions, and even periodic solutions in a stronger form, where every two airplanes perform either the exact same cyclic sequence of flights, or completely disjoint cyclic sequences. However, enforcing such periodicity may eliminate feasible solutions. We prove that, when regular maintenance is required at most every four days, there always exist periodic solutions of this form. Second, we consider the computational hardness of the problem. Even if many papers in this area refer to the NP-hardness of the aircraft routing problem, such a result is only available in the literature for periodic instances. We establish its NP-hardness for a non-periodic version. Polynomiality of a special but natural case is also proven.

Figures (3)

• Figure 1: On the left: an Eulerian directed graph $D$ with the black vertices representing the vertices of $B$. On the right: examples of periodic, absolutely periodic, and strongly periodic solutions. Bold vertices denote period start. (1) is not strongly periodic because $e$ is visited twice in every period of $W_2$. (3) is not absolutely periodic as $W_1$ and $W_2$ are neither disjoint nor identical.
• Figure 2: On the left, an instance of the periodic aircraft routing problem; on the right, the corresponding graph $H$.
• Figure 3: An example of an instance of the constrained path partition problem. The arcs in $M$ are in red. A possible configuration of the "pebble game" is also illustrated, with three pebbles in $X_3$, and possible values of their counters after a few moves.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Lemma 2.2
• proof : Proof of Lemma \ref{['lem:bip']}, 'if' direction
• proof : Proof of Theorem \ref{['thm:k4']}, case $\gamma=1$
• proof : Proof of Theorem \ref{['thm:k4']}, case $\gamma=2$
• proof : Proof of Theorem \ref{['thm:k4']}, case $\gamma=3$
• proof : Proof of Theorem \ref{['thm:k4']}, case $\gamma=4$