Asymmetric Traveling Salesman Problem

The asymmetric traveling salesman problem (ATSP) is one of the most fundamental problems in combinatorial optimization. An instance of ATSP is a directed complete graph G = (V,E) with edge weights w : E → N that satisfy the triangle inequality, i. e., w(a, c) ≤ w(a, b) +w(b, c) for all distinct vertices a, b, c ∈ V . The goal is to find a Hamiltonian cycle of minimum weight. The weight of a Hamiltonian cycle (or, more general, of a subset of E) is the sum of the weight if its edges. A special case of ATSP is γ-ATSP for γ ∈ [12 , 1], which is ATSP restricted to instances that satisfy the γ-triangle inequality: w(a, c) ≤ γ · (