The Permutation in a Haystack Problem and the Calculus of Search Landscapes

The natural encoding for many search and optimization problems is the permutation, such as the traveling salesperson, vehicle routing, scheduling, assignment and mapping problems, among others. The effectiveness of a given mutation or crossover operator depends upon the nature of what the permutation represents. For some problems, it is the absolute locations of the elements that most directly influences solution fitness; while for others, element adjacencies or even element precedences are most important. Different permutation operators respect different properties. We aim to provide the genetic algorithm or metaheuristic practitioner with a framework enabling effective permutation search landscape analysis. To this end, we contribute a new family of optimization problems, the permutation in a haystack, that can be parameterized to the various types of permutation problem (e.g., absolute versus relative positioning). Additionally, we propose a calculus of search landscapes, enabling analysis of search landscapes through examination of local fitness rates of change. We use our approach to analyze the behavior of common permutation mutation operators on a variety of permutation in a haystack landscapes; and empirically validate the prescriptive power of the search landscape calculus via experiments with simulated annealing.