Quadratic assignment problem: a landscape analysis

Abstract

The anatomy of the fitness landscape for the quadratic assignment problem is studied in this paper. We study the properties of both random problems, and real-world problems. Using auto-correlation as a measure for the landscape ruggedness, we study the landscape of the problems and show how this property is related to the problem matrices with which the problems are represented. Our main goal in this paper is to study new properties of the fitness landscape, which have not been studied before, and we believe are more capable of reflecting the problem difficulties. Using local search algorithm which exhaustively explore the plateaus and the local optima, we explore the landscape, store all the local optima we find, and study their properties. The properties we study include the time it takes for a local search algorithm to find local optima, the number of local optima, the probability of reaching the global optimum, the expected cost of the local optima around the global optimum and the basin of attraction of the global and local optima. We study the properties for problems of different sizes, and through extrapolations, we show how the properties change with the system size and why the problem becomes harder as the system size grows. In our study we show how the real-world problems are similar to, or different from the random problems. We also try to show what properties of the problem matrices make the landscape of the real problems be different from or similar to one another.