On the Integrality Ratio for the Asymmetric Traveling Salesman Problem

We improve the lower bound on the integrality ratio of the Held-Karp bound for asymmetric TSP with triangle inequality from 4/3 to 2. 1. Introduction. The traveling salesman problem (TSP)—the problem of finding a minimum cost tour through a set of cities—is the most celebrated combinatorial optimization problem. It is often used as a testbed for novel ideas, as was the case for Adleman's molecular computing (Adleman [1]), memetic algorithms (Moscato [23]), or ant colony optimization (Dorigo et al. [10]), to cite just a few examples. The TSP book (Lawler et al. [20]) provides a tour d'horizon of combinatorial optimization, illustrating all the concepts and techniques on the TSP. The traveling salesman problem comes in two variants. The symmetric version (STSP) assumes that the cost c ij of going from city i to city j is equal to c ji while the more general asymmetric version (ATSP) does not make this assumption. In both cases, it is usually assumed—and we make this assumption in the rest of this paper—that we are in the metric case, i.e., the costs satisfy the triangle inequality: c ij + c jk ≥ c ik for all i, j, k. Even though the TSP is the most studied combinatorial optimization problem, little progress has been made on its approximability in the general metric case in the last quarter of a century. Christofides in 1976 (Christofides [9]) discovered a