Feature-Based Diversity Optimization for Problem Instance Classification.

Understanding the behaviour of heuristic search methods is a challenge. This even holds for simple local search methods such as 2-OPT for the Travelling Salesperson Problem (TSP). In this article, we present a general framework that is able to construct a diverse set of instances which are hard or easy for a given search heuristic. Such a diverse set is obtained by using an evolutionary algorithm for constructing hard or easy instances which are diverse with respect to different features of the underlying problem. Examining the constructed instance sets, we show that many combinations of two or three features give a good classification of the TSP instances in terms of whether they are hard to be solved by 2-OPT.

Expected Fitness Gains of Randomized Search Heuristics for the Traveling Salesperson Problem.

Randomized search heuristics are frequently applied to NP-hard combinatorial optimization problems. The runtime analysis of randomized search heuristics has contributed tremendously to our theoretical understanding. Recently, randomized search heuristics have been examined regarding their achievable progress within a fixed-time budget. We follow this approach and present a fixed-budget analysis for an NP-hard combinatorial optimization problem. We consider the well-known Traveling Salesperson Problem (TSP) and analyze the fitness increase that randomized search heuristics are able to achieve within a given fixed-time budget. In particular, we analyze Manhattan and Euclidean TSP instances and Randomized Local Search (RLS), (1+1) EA and (1+[Formula: see text]) EA algorithms for the TSP in a smoothed complexity setting, and derive the lower bounds of the expected fitness gain for a specified number of generations.

Parameterized runtime analyses of evolutionary algorithms for the planar euclidean traveling salesperson problem.

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.