On the Computational Efficiency of Branch-and-bound Algorithms

The behavior of the number of partial problems T(A) which are decomposed in a branchand-bound algorithm A (T(A) may be taken as a measure for the computational efficiency of A) is investigated in a fairly general setting. The first result is that the mean number f(n) of T(A) when A is applied to problems of size n grows at least as fast as exponentially with n, under relatively mild conditions, if A uses only the lower bound test as most of the conventional branch-and-bound algorithms do. Then it is pointed out that a possible way to avoid this exponential growth is to use the dominance test together with the lower bound test. The dominance test is also interesting from the view point of unifying a wide variety of algorithms as branch-and-bound. These points are exemplified by the well known Dijkstra algorithm for the shortest path problem and the Johnson algorithm for the two-machine flow-shop scheduling problem, for which f(n) :s: n-1 holds by virtue of the dominance test.