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This paper addresses the multiplication of one data sample with multiple constants using addition/subtraction and shift operations, i.e., the multiple constant multiplications (MCM) problem. The MCM problem finds itself and its variants in many applications, such as digital finite impulse response (FIR) filters, linear signal transforms, and computer arithmetic. Although many efficient algorithms have been proposed to implement the MCM using the fewest number of operations, due to the NP-hardness of the problem, they have been heuristics, i.e., they cannot guarantee the minimum solution. In this work, we propose an exact algorithm based on the breadth-first search that finds the minimum number of operations solution of mid-size MCM instances in a reasonable time. The proposed exact algorithm has been tested on a set of instances including FIR filter and randomly generated instances, and compared with the previously proposed efficient heuristics. It is observed from the experimental results that, even though the previously proposed heuristics obtain similar results with the minimum number of operations solutions, there are instances for which the exact algorithm finds better solutions than the prominent heuristics.