Fixed-parameter evolutionary algorithms for the Euclidean Traveling Salesperson problem

Abstract

Recently, Sutton and Neumann [1] have studied evolutionary algorithms for the Euclidean traveling salesman problem by parameterized runtime analyses taking into account the number of inner points k and the number of cities n. They have shown that simple evolutionary algorithms are XPalgorithms for the problem, i. e., they obtain an optimal solution in expected time O(n g(k) ) where g(k) is a function only depending on k. We extend these investigations and design two evolutionary algorithms for the Euclidean Traveling Salesperson problem that run in expected time g(k) · poly(n) where k is a parameter denoting the number inner points for the given TSP instance, i. e., they are fixed-parameter tractable evolutionary algorithms for the Euclidean TSP parameterized by the number of inner points. While our first approach is mainly of theoretical interest, our second approach leverages problem structure by directly searching for good orderings of the inner points and provides a novel and highly effective way of tackling this important problem. Our experimental results show that searching for a permutation on the inner points is a significantly powerful practical strategy.

A Survey on Recent Progress in the Theory of Evolutionary Algorithms for Discrete Optimization

“…Afterwards, the ability of evolutionary algorithms to fill in the inner points correctly given that the points on the convex hull are in correct order has been examined. Ant colony optimization [NSN13b] and evolutionary algorithms [NSN13a,SNN14] have been investigated in this context. For ant colony optimization, the crucial aspect to obtain a runtime of O(n k ) is to construct solutions following the order on the convex hull.…”

Parameterized Complexity Analysis of Randomized Search Heuristics

“…Parameterized results for evolutionary algorithms for the Euclidean TSP have been developed in a series of papers [40,29,30,41] in the context of the inner-point parameterization of Deȋneko et al [9]. We also would like to mention that the generalized traveling salesperson problem has been investigated in the context of parameterized complexity.…”