Note on Upper Bounds for TSP Domination Number

The Asymmetric Traveling Salesman Problem (ATSP) is stated as follows. Given a weighted complete digraph (K∗ n, w), find a Hamilton cycle (called a tour) in K∗ n of minimum cost. Here the weight function w is a mapping from A(K∗ n), the set of arcs in K∗ n, to the set of reals. The weight of an arc xy of K∗ n is w(x, y). The weight w(D) of a subdigraph D of K∗ n is the sum of the weights of arcs in D. It is well known that most combinatorial optimization problems including the ATSP are NP-hard. Due to the lack of polynomial time algorithms to solve NP-hard problems to optimality, researchers and practitioners often use various heuristics such as local search and genetic algorithms that usually provide good solutions for instances that arise in practice. Very often heuristics do not have any theoretical guarantee for the optimization problem under consideration, i.e., for some instances of the problem the value of heuristic solution is arbitrary far from the optimum. Hence, normally various heuristics for the same problem are compared in computational experiments. The outcomes of computational experiments heavily rely on the authors choice of families of instances and, thus, are non-objective. With this state of affairs in mind, Glover and Punnen [3] suggested a new approach for evaluation of heuristics that compares heuristics according to their so-called domination ratio. We define this notion only for the ATSP since its extension to other problems is obvious. The domination number, domn(A, n), of a heuristic A for the ATSP is the maximum integer d = d(n) such that, for every instance I of the ATSP on n cities, A produces a tour T which is not worse than at least d tours in I including T itself. The ratio domr(A, n) = domn(A, n)/(n − 1)!, i.e., the domination number divided by the total number of tours, is the domination ratio of A. It is known the nearest neighbor algorithm for the ATSP is of domination number 1 (first proved in [7]). This means that for every n ≥ 2, there is an instance of ATSP on n vertices, for which the nearest neighbor algorithm finds the unique worst possible tour. Since the number of distinct tours in an n-vertex complete digraph is (n− 1)!, we see that the nearest neighbor algorithm is of domination ratio 1/(n − 1)!. There are many ATSP algorithms of domination number at least (n− 2)! [8], i.e., in the worst case they guarantee that their tour is at least as good as (n− 2)!− 1 other tours. Clearly, the domination ratio is well defined for every heuristic and, for the same optimization problem, a heuristic with higher domination ratio may be considered a better choice than a heuristic with lower domination ratio. Ben-Arieh et al. [1] compared two heuristics for the generalized ATSP. The heuristics performed equally well in computational experiments, but it was proved that one of them has a significantly larger domination number. For the