Evaluating the Difficulty of Instances of the Travelling Salesman Problem in the Nearby of the Optimal Solution Based on Random Walk Exploration

Abstract

Combinatorial optimization is one of the main research areas in Evolutionary Computing and Operational Research, and the Travelling Salesman Problem one of their most popular problems. The never ending quest of researchers for new and more difficult combinatorial problems to stress their evolutionary algorithms leads to investigate how to measure the difficulty of Travelling Salesman Problem instances. By developing methodologies for separating ease from difficult instances, researchers will be confident about the performance of their algorithms. In this proof-of-concept, a methodology for evaluating the difficulty of instances of the Travelling Salesman Problem in the nearby of the optimal solution is proposed. This methodology is based on the use of Random Walk to explore the closeness area of the optimal solution. Instances with a more pronounced gradient towards the optimal solution might be considered easier than instances exhibiting almost a null gradient. The exploration of this gradient is done by starting from the optimal tour and later modifying it with a Random Walk process. The aim is to propose a methodology to evaluate the difficulty of instances of Travelling Salesman Problem, which can be applied to other combinatorial-problems instances. As a consequence of this work, a methodology to evaluate the difficulty of Travelling Salesman Problem instances is proposed and confronted to a wide set of instances, and finally a rank of their difficulty is stated.