Geometric crossover is a representation-independent generalization of the traditional crossover defined using the distance of the solution space. By choosing a distance firmly rooted in the syntax of the solution representation as a basis for geometric crossover, one can design new crossovers for any representation. Using a distance tailored to the problem at hand, the formal definition of geometric crossover allows us to design new problem-specific crossovers that embed problem-knowledge in the search. The standard encoding for multiway graph partitioning is highly redundant: each solution has a number of representations, one for each way of labeling the represented partition. Traditional crossover does not perform well on redundant encodings. We propose a new geometric crossover for graph partitioning based on a labeling-independent distance that filters out the redundancy of the encoding. A correlation analysis of the fitness landscape based on this distance shows that it is well suited to graph partitioning. A second difficulty with designing a crossover for multiway graph partitioning is that of feasibility: in general recombining feasible partitions does not lead to feasible offspring partitions. We design a new geometric crossover for permutations with repetitions that naturally suits partition problems and we test it on the graph partitioning problem. We then combine it with the labeling-independent crossover and obtain a much superior geometric crossover inheriting both advantages.

In optimization problems, the contribution of a variable to fitness often depends on the states of other variables. This phenomenon is referred to as epistasis or linkage. In this paper, we show that a new theory of epistasis can be established on the basis of Shannon's information theory. From this, we derive a new epistasis measure called entropic epistasis and some theoretical results. We also provide experimental results verifying the measure and showing how it can be used for designing efficient evolutionary algorithms.