Population Diversity Measures Based on Variable-Order Markov Models for the Traveling Salesman Problem

This paper presents entropy-based population diversity measures that take into account dependencies between the variables in order to maintain genetic diversity in a GA for the traveling salesman problem. The first one is formulated as the entropy rate of a variable-order Markov process, where the probability of occurrence of each vertex is assumed to be dependent on the preceding vertices of variable length in the population. Compared to the use of a fixed-order Markov model, the variable-order model has the advantage of avoiding the lack of sufficient statistics for the estimation of the exponentially increasing number of conditional probability components as the order of the Markov process increases. Moreover, we develop a more elaborate population diversity measure by further reducing the problem of the lack of statistics