Search algorithms for the multiple constant multiplications problem: Exact and approximate

This article addresses the multiplication of one data sample with multiple constants using addition/subtraction and shift operations, i.e., the multiple constant multiplications (MCM) operation. In the last two decades, many efficient algorithms have been proposed to implement the MCM operation using the fewest number of addition and subtraction operations. However, due to the NP-hardness of the problem, almost all the existing algorithms have been heuristics. The main contribution of this article is the proposal of an exact depth-first search algorithm that, using lower and upper bound values of the search space for the MCM problem instance, finds the minimum solution consuming less computational resources than the previously proposed exact breadth-first search algorithm. We start by describing the exact breadth-first search algorithm that can be applied on real mid-size instances. We also present our recently proposed approximate algorithm that finds solutions close to the minimum and is able to compute better bounds for the MCM problem. The experimental results clearly indicate that the exact depth-first search algorithm can be efficiently applied to large size hard instances that the exact breadth-first search algorithm cannot handle and the heuristics can only find suboptimal solutions.