Aircraft routing: periodicity and complexity

2025-08-07

AI Summary

This paper addresses two fundamental questions in aircraft routing: (1) whether strict periodicity—e.g., identical daily flight sequences—is necessary for feasibility under maintenance constraints, and (2) whether the aperiodic variant is NP-hard. Using graph-theoretic modeling, combinatorial optimization, and computational complexity analysis, we establish three key results: First, we construct weakly periodic solutions satisfying four-day maintenance cycles, disproving the implicit assumption that strong periodicity is necessary. Second, we provide the first rigorous proof that the aperiodic aircraft routing problem is NP-hard. Third, we identify a practically relevant subclass and devise a polynomial-time algorithm for it. Collectively, these contributions unify the theoretical foundations of periodic and aperiodic aircraft routing, extend the boundary of tractability, and offer new insights into the interplay between operational constraints and computational complexity in aviation logistics.

Technology Category

• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Exploring periodic instances and solutions in aircraft routing
• Proving NP-hardness for non-periodic aircraft routing problems
• Analyzing polynomiality in special natural cases

Innovation

• Proves periodic solutions exist for maintenance constraints
• Establishes NP-hardness for non-periodic instances
• Demonstrates polynomiality in special natural cases