In and out forests on combinatorial landscapes

Fitness landscape theory is a mathematical framework for numerical analysis of search algorithms on combinatorial optimization problems. We study a representation of fitness landscape as a weighted directed graph. We consider out forest and in forest structures in this graph and establish important relationships among the forest structures of a directed graph, the spectral properties of the Laplacian matrices, and the numbers of local optima of the landscape. These relationships provide a new approach for computing the numbers of local optima for various problem instances and neighborhood structures.