A restricted Lagrangean approach to the traveling salesman problem

We describe an algorithm for the asymmetric traveling salesman problem (TSP) using a new, restricted Lagrangean relaxation based on the assignment problem (AP). The Lagrange multipliers are constrained so as to guarantee the continued optimality of the initial AP solution, thus eliminating the need for repeatedly solving AP in the process of computing multipliers. We give several polynomially bounded procedures for generating valid inequalities and taking them into the Lagrangean function with a positive multiplier without violating the constraints, so as to strengthen the current lower bound. Upper bounds are generated by a fast heuristic whenever possible. When the bound-strengthening techniques are exhausted without matching the upper with the lower bound, we branch by using two different rules, according to the situation: the usual subtour breaking disjunction, and a new disjunction based on conditional bounds. We discuss computational experience on 120 randomly generated asymmetric TSP's with up to 325 cities, the maximum time used for any single problem being 82 seconds. Though the algorithm discussed here is for the asymmetric TSP, the approach can be extended to the symmetric TSP by using the 2-matching problem instead of AP. University Libraries Carnegie Mellon University Pittsburgh, Pennsylvania 15213 1. Outline of the Approach The traveling salesman problem (TSP), i.e., the problem of finding a minimum-cost tour (or hamiltonian circuit) in a directed graph G = (N,A), can be formulated as the problem of minimizing