Exact and Approximate Stability of Solutions to Traveling Salesman Problems

• Published in: IEEE transactions on cybernetics Vol. 48; no. 2; pp. 583 - 595
• Main Authors: Niendorf, Moritz, Girard, Anouck R.
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.02.2018; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Approximation; Graph theory; Numerical stability; Optimization; Robustness; Robustness (mathematics); Routing; Stability analysis; Stability criteria; Traveling salesman problem; traveling salesman problem (TSP); Traveling salesman problems; Vehicles
• ISSN: 2168-2267; 2168-2275
• DOI: 10.1109/TCYB.2016.2647440

This paper presents the stability analysis of an optimal tour for the symmetric traveling salesman problem (TSP) by obtaining stability regions. The stability region of an optimal tour is the set of all cost changes for which that solution remains optimal and can be understood as the margin of optimality for a solution with respect to perturbations in the problem data. It is known that it is not possible to test in polynomial time whether an optimal tour remains optimal after the cost of an arbitrary set of edges changes. Therefore, this paper develops tractable methods to obtain under and over approximations of stability regions based on neighborhoods and relaxations. The application of the results to the two-neighborhood and the minimum 1 tree (M1T) relaxation are discussed in detail. For Euclidean TSPs, stability regions with respect to vertex location perturbations and the notion of safe radii and location criticalities are introduced. Benefits of this paper include insight into robustness properties of tours, minimum spanning trees, M1Ts, and fast methods to evaluate optimality after perturbations occur. Numerical examples are given to demonstrate the methods and achievable approximation quality.