An analysis of neighborhood functions on generic solution spaces

The effectiveness of local search algorithms on discrete optimization problems is influenced by the choice of the neighborhood function. A neighborhood function that results in all local minima being global minima is said to have zero L-locals. A polynomially sized neighborhood function with zero L-locals would ensure that at each iteration, a local search algorithm would be able to find an improving solution or conclude that the current solution is a global minimum. This paper presents a recursive relationship for computing the number of neighborhood functions over a generic solution space that results in zero L-locals. Expressions are also given for the number of tree neighborhood functions with zero L-locals. These results provide a first step towards developing expressions that are applicable to discrete optimization problems, as well as providing results that add to the collection of solved graphical enumeration problems.