A Study of Branch-and-Bound on the Asymmetric Traveling Salesman Problem

One of the approaches for optimally solving the asymmetric Traveling Salesman Problem (ATSP) is branch-and-bound subtour elimination using the assignment problem as a lower-bound cost function [10, 7]. This chapter studies the complexity of branch-and-bound subtour elimination on the asymmetric Traveling Salesman Problem. In Section 6.1, we first examine two-decade old arguments about the expected complexity of branch-and-bound subtour elimination, and shows that the branch-and-bound subtour elimination cannot find an optimal solution to the asymmetric Traveling Salesman Problem in polynomial time on average. This partially settles this long-standing debate on the expected complexity of branch-and-bound on the asymmetric Traveling Salesman Problem. Section 6.4 then compares depth-first branch-and-bound against local search, and shows that depth-first branch-and-bound significantly outperforms a local search algorithm.