The lower bound on complexity of parallel branch-and-bound algorithm for subset sum problem

Abstract

The subset sum problem is a particular case of the Boolean knapsack problem where each item has the price equal to its weight. This problem can be informally stated as searching for most dense packing of a set of items into a box with limited capacity. Recently, coarse-grain parallelization approaches to Branch-and-Bound (B&B) method attracted some attention due to the growing popularity of weakly-connected distributed computing platforms. In this paper we consider one of such approaches for solving the subset sum problem. One of the processors (manager) performs some number of B&B steps on the first stage with generating some subproblems. On the second stage, the generated subproblems are sent to other processors, one subproblem per processor. The processors solve completely the received subproblems, the manager collects all the obtained solutions and chooses the optimal one. For this algorithm we formally define the parallel execution model (frontal scheme of parallelization) and the notion of the frontal scheme complexity. We study the frontal scheme complexity for a series of subset sum problems.