A Parameterized Runtime Analysis of Simple Evolutionary Algorithms for Makespan Scheduling

Evolutionary Computation

2014

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Parameterized runtime analysis seeks to understand the influence of problem structure on algorithmic runtime. In this paper, we contribute to the theoretical understanding of evolutionary algorithms and carry out a parameterized analysis of evolutionary algorithms for the Euclidean traveling salesperson problem (Euclidean TSP). We investigate the structural properties in TSP instances that influence the optimization process of evolutionary algorithms and use this information to bound their runtime. We analyze the runtime in dependence of the number of inner points k. In the first part of the paper, we study a [Formula: see text] EA in a strictly black box setting and show that it can solve the Euclidean TSP in expected time [Formula: see text] where A is a function of the minimum angle [Formula: see text] between any three points. Based on insights provided by the analysis, we improve this upper bound by introducing a mixed mutation strategy that incorporates both 2-opt moves and permutation jumps. This strategy improves the upper bound to [Formula: see text]. In the second part of the paper, we use the information gained in the analysis to incorporate domain knowledge to design two fixed-parameter tractable (FPT) evolutionary algorithms for the planar Euclidean TSP. We first develop a [Formula: see text] EA based on an analysis by M. Theile, 2009, ”Exact solutions to the traveling salesperson problem by a population-based evolutionary algorithm,” Lecture notes in computer science, Vol. 5482 (pp. 145–155), that solves the TSP with k inner points in [Formula: see text] generations with probability [Formula: see text]. We then design a [Formula: see text] EA that incorporates a dynamic programming step into the fitness evaluation. We prove that a variant of this evolutionary algorithm using 2-opt mutation solves the problem after [Formula: see text] steps in expectation with a cost of [Formula: see text] for each fitness evaluation.

Section: Computational Complexity Of Evolutionary Algorithmsmentioning

confidence: 99%

Parameterized Runtime Analyses of Evolutionary Algorithms for the Planar Euclidean Traveling Salesperson Problem

Evolutionary Computation

2014

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“…This approach allows one to take an additional parameter measuring the hardness of an instance into account. Results have been obtained for vertex cover [3], minimum leaf spanning trees [4], makespan scheduling [5], and the Euclidean Traveling Salesperson problem [1]. The parameterized complexity analysis of bio-inspired computing seems to be very promising for bridging theory and practice in the field of bio-inspired computing.…”

Section: Introductionmentioning

confidence: 97%

Fixed-parameter evolutionary algorithms for the Euclidean Traveling Salesperson problem

2013 IEEE Congress on Evolutionary Computation

2013

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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.

“…Recently, the first parameterized computational complexity results have been obtained for evolutionary algorithms and some classical NP-hard combinatorial optimization problems such as vertex cover [8], make span scheduling [21], and the Euclidean traveling salesperson problem [20]. Such studies provide insights into the runtime behavior of meta heuristics in relation to structural parameters of the investigated problem.…”

Section: Introductionmentioning

confidence: 99%

Parameterized complexity analysis and more effective construction methods for ACO algorithms and the euclidean traveling salesperson problem

2013 IEEE Congress on Evolutionary Computation

2013

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We propose a new construction procedure for ant colony optimization (ACO) algorithms working on the Euclidean traveling salesperson problem (TSP) that preserves the ordering on the convex hull of the points in the instance. The procedure is inspired by theoretical analyses for simple evolutionary algorithms that are provably more efficient on instances where the number of inner points of the instance is not too large. We integrate the construction procedure into the well-known MaxMin Ant System (MMAS) and empirically show that it leads to more efficient optimization on instances where the number of inner points is not too high.