On a Lower Bound on the Computational Complexity of a Parallel Implementation of the Branch-and-bound Method

We study parallel complexity of the branch-and-bound method for optimization problems. We consider a standard implementation scheme for the branch-and-bound method on a parallel system, in which first only one processor is working, and then the resulting subtasks are given out to other processors. For this scheme, we give a lower bound on the parallel complexity independent of the problem. We study the complexity of this scheme for the Boolean knapsack problem. For a classical algorithmically hard example, we obtain parallel complexity bounds and show that these bounds coincide in order with each other and with the common lower bound on parallel complexity. Thus, we show that the common lower bound is achieved, in the order, for some optimization problems.