Extended neighborhood: Definition and characterization

We consider neighborhood search defined on combinatorial optimization problems. Suppose that N is a Neighborhood for combinatorial optimization problem X. We say that N′ is LO-equivalent (locally optimal) to N if for any instance of X, the set of locally optimal solutions with respect to N and N′ are the same. The union of two LO-equivalent neighborhoods is itself LO-equivalent to the neighborhoods. The largest neighborhood that is LO-equivalent to N is called the extended neighborhood of N, and denoted as N∗. We analyze some basic properties of the extended neighborhood. We provide a geometric characterization of the extended neighborhood N∗ when the instances have linear costs defined over a cone. For the TSP, we consider 2-opt∗, the extended neighborhood for the 2-opt (i.e., 2-exchange) neighborhood structure. We show that number of neighbors of each tour T in 2-opt∗ is at least (n /2 -2)!. We show that finding the best tour in the 2-opt∗ neighborhood is NP-hard. We also show that the extended neighborhood for the graph partition problem is the same as the original neighborhood, regardless of the neighborhood defined. This result extends to the quadratic assignment problem as well. This result on extended neighborhoods relies on a proof that the convex hull of solutions for the graph partition problem has a diameter of 1, that is, every two corner points of this polytope are adjacent.