Algorithms for Weighted Boolean Optimization (0903.0843v2)

Abstract: The Pseudo-Boolean Optimization (PBO) and Maximum Satisfiability (MaxSAT) problems are natural optimization extensions of Boolean Satisfiability (SAT). In the recent past, different algorithms have been proposed for PBO and for MaxSAT, despite the existence of straightforward mappings from PBO to MaxSAT and vice-versa. This papers proposes Weighted Boolean Optimization (WBO), a new unified framework that aggregates and extends PBO and MaxSAT. In addition, the paper proposes a new unsatisfiability-based algorithm for WBO, based on recent unsatisfiability-based algorithms for MaxSAT. Besides standard MaxSAT, the new algorithm can also be used to solve weighted MaxSAT and PBO, handling pseudo-Boolean constraints either natively or by translation to clausal form. Experimental results illustrate that unsatisfiability-based algorithms for MaxSAT can be orders of magnitude more efficient than existing dedicated algorithms. Finally, the paper illustrates how other algorithms for either PBO or MaxSAT can be extended to WBO.

• The paper introduces a unified framework merging PBO and MaxSAT into WBO to effectively model complex Boolean optimization problems.
• It presents a novel unsatisfiability-based algorithm that leverages techniques from MaxSAT, significantly boosting computational efficiency.
• Empirical results demonstrate superior performance on industrial benchmarks, underscoring the approach's practical applicability.

Algorithms for Weighted Boolean Optimization

The paper "Algorithms for Weighted Boolean Optimization" addresses the important extension of Boolean Satisfiability (SAT) known as Pseudo-Boolean Optimization (PBO) and Maximum Satisfiability (MaxSAT). The authors propose a novel unified framework called Weighted Boolean Optimization (WBO), which combines and extends the methodologies of PBO and MaxSAT. The main contribution of this work is the introduction of a new unsatisfiability-based algorithm for WBO, drawing inspiration from recent developments in unsatisfiability-based MaxSAT algorithms.

Key Contributions

The authors present several noteworthy contributions:

1. Unified Framework: WBO aggregates the PBO and MaxSAT formulations, allowing modeling of linear optimization problems over Boolean domains effectively. This unified framework facilitates seamless integration of pseudo-Boolean constraints, providing a more compact and expressive modeling language than classical MaxSAT.
2. Unsatisfiability-Based Algorithm: A new unsatisfiability-based algorithm is proposed for solving WBO problems. This algorithm extends previous methods used for MaxSAT, incorporating techniques to solve weighted and pseudo-Boolean constraints, either natively or through translation to clausal form. Experimental results underscore that unsatisfiability-based algorithms surpass dedicated algorithms by orders of magnitude in efficiency.
3. Empirical Validation: The paper provides experimental validation of the proposed algorithm, demonstrating its superior performance on problem instances derived from practical applications. The empirical results highlight how unsatisfiability-based MaxSAT solvers are highly competitive compared to state-of-the-art MaxSAT and PBO solvers.

The innovative approach not only reinforces the known capabilities of solving MaxSAT problems but also opens avenues for designing algorithms generalizable to WBO, enhancing their practical applicability in diverse fields like scheduling, routing, and design automation.

Numerical Results and Implications

Significant numerical results include the demonstration of the new algorithm's capacity to outperform existing solvers in terms of computation efficiency, especially when dealing with industrial problem instances characterized by large weights within clauses. Such results imply profound computational benefits, suggesting a high potential for WBO to solve complex real-world optimization problems more efficiently.

The implications of this research extend both practically and theoretically. With WBO, practitioners can tackle problems involving both hard and soft constraints, providing solutions where minimizing unsatisfied constraint weights is critical. Theoretically, the foundational hybrid of SAT, PBO, and MaxSAT encourages the development of more sophisticated optimization techniques, fostering further exploration into hybrid models that incorporate diverse constraint types.

Future Developments

The paper hints at future research directions, notably addressing adaptations of existing algorithms to be more compatible with WBO frameworks. Specifically, further investigation into integrating PBO solvers and extending the MSU family of algorithms for WBO is suggested. Future advancements in this domain could lead to even more efficient and versatile solvers capable of handling complex optimization scenarios with unprecedented precision and speed.

In conclusion, the introduction of WBO represents a significant stride in optimization techniques over Boolean domains, promising enhanced methodologies for tackling complex optimization problems both in theory and practice. This paper serves as a critical reference point for researchers aiming to build upon the current understanding and capabilities of Boolean-based optimization procedures.