[Paper Review] Optimally Solving the MCM Problem Using Pseudo-Boolean Satisfiability

Nuno P. Lopes, Levent Aksoy+2 more|arXiv (Cornell University)|Nov 11, 2010

Numerical Methods and Algorithms18 references3 citations

TL;DR

This paper presents the first formal encoding of the Multiple Constant Multiplication (MCM) problem as a Pseudo-Boolean Satisfiability (PBS) problem, using three distinct encoding schemes. It proposes an exact algorithm that leverages generic PBS solvers to find optimal solutions, demonstrating that minisat+ and wbo outperform other solvers, with the third encoding yielding the best performance on benchmark instances up to 11-bit constants.

ABSTRACT

In this report, we describe three encodings of the multiple constant multiplication (MCM) problem to pseudo-boolean satisfiability (PBS), and introduce an algorithm to solve the MCM problem optimally. To the best of our knowledge, the proposed encodings and the optimization algorithm are the first formalization of the MCM problem in a PBS manner. This report evaluates the complexity of the problem size and the performance of several PBS solvers over three encodings.

Motivation & Objective

• To address the NP-hard MCM problem by formalizing it as a Pseudo-Boolean Satisfiability (PBS) problem for the first time.
• To design an exact algorithm that uses generic PBS solvers to find optimal solutions to the MCM problem.
• To evaluate the scalability and performance of multiple PBS solvers across three distinct encoding schemes on MCM benchmark instances.
• To identify the most effective encoding and solver combination for solving MCM problems optimally.

Proposed method

• The MCM problem is encoded into PBS using three different schemes, each representing the problem’s constraints and objective function in Boolean logic with pseudo-boolean constraints.
• The encoding models the MCM problem as a directed acyclic graph (DAG) where intermediate constants are built via A-operations (addition/subtraction with shifts), starting from 1.
• The objective function minimizes the number of A-operations, equivalent to minimizing the number of intermediate constants in the DAG.
• The PBS formulation uses integer variables and linear constraints to represent the addition/subtraction and shift operations, with the assumption that shifts are free and signs are handled via sign bits.
• The algorithm uses off-the-shelf PBS solvers (bsolo, minisat+, wbo) to search for the minimal solution across the three encodings.
• A virtual best solver (VBS) is used as a theoretical upper bound to compare solver performance across benchmarks.

Experimental results

Research questions

• RQ1 Can the MCM problem be optimally solved using a formalization in Pseudo-Boolean Satisfiability (PBS)?
• RQ2 Which of the three proposed PBS encodings yields the most efficient performance in terms of solving time and scalability?
• RQ3 How do leading PBS solvers (bsolo, minisat+, wbo) compare in solving MCM benchmark instances?
• RQ4 What is the practical scalability limit of PBS-based MCM solving in terms of bit-width and number of operations?

Key findings

• The third encoding outperforms the other two in terms of solving time and scalability, particularly on larger MCM instances.
• minisat+ and wbo solve more problems than bsolo, with minisat+ showing better scaling behavior across benchmarks.
• wbo is the best overall performer, achieving the highest percentage of solved instances (62.1%) and the best average solving time on combined SAT/UNSAT tests.
• The PBS-based approach scales up to approximately 10–11-bit target constants with a moderate number of A-operations, but does not scale to larger, more realistic problems without further optimization.
• Trivial SAT/UNSAT cases were filtered out, with only 28% of the FIR filter benchmarks being non-trivial, indicating that encoding heuristics can reduce solver load.
• The virtual best solver (VBS) achieves the highest success rate (65.3%), showing that combining multiple solvers improves overall performance.