Weighted Boolean Optimization (WBO)

• Weighted Boolean Optimization (WBO) is a unified framework that generalizes MaxSAT and pseudo-Boolean optimization by managing both hard and soft linear constraints over Boolean variables.
• It employs core-guided, unsatisfiability-based methods and dynamic programming to efficiently solve combinatorial optimization problems with diverse constraints.
• The w-system abstraction in WBO enhances expressiveness and scalability, paving the way for advances in parameterized complexity, solver design, and cross-paradigm optimization.

Weighted Boolean Optimization (WBO) is a unified optimization framework that generalizes and subsumes the pseudo-Boolean optimization (PBO) and (weighted) MaxSAT paradigms, supporting both hard and soft linear constraints over Boolean variables and allowing clause or constraint weighting. WBO provides a semantic and algorithmic foundation for modeling and solving a wide range of combinatorial and declarative search-optimization problems arising in artificial intelligence, verification, and operations research, and is the subject of several contemporary lines of research in parameterized complexity, solver design, and abstract optimization frameworks (0903.0843, Lierler, 2022, Bannach et al., 2022).

1. Formal Definition and Relationship to PBO and MaxSAT

A WBO instance consists of a set of Boolean variables $X = \{x_1, \ldots, x_n\}$, a set of hard pseudo-Boolean (PB) constraints, and a set of soft PB constraints with associated positive integer weights. Each PB constraint is a linear inequality of the form $\sum_j a_{ij} \ell_j \geq b_i$, with $a_{ij}, b_i \in \mathbb{N}$ and each literal $\ell_j$ being either $x_j$ or $\neg x_j$.

The goal is to find an assignment $\nu: X \to \{0,1\}$ that satisfies all hard constraints and minimizes the total weight of unsatisfied soft constraints: $$\min_{\nu} \sum_{(\omega_k, c_k) \in \varphi_s} c_k \cdot \operatorname{unsat}(\omega_k, \nu)$$ where $\operatorname{unsat}(\omega_k, \nu) = 1$ iff $\nu$ falsifies $\sum_j a_{ij} \ell_j \geq b_i$0 and $\sum_j a_{ij} \ell_j \geq b_i$1 otherwise (0903.0843).

WBO strictly generalizes both weighted partial MaxSAT and PBO:

• Weighted MaxSAT $\sum_j a_{ij} \ell_j \geq b_i$2 WBO: Each clause $\sum_j a_{ij} \ell_j \geq b_i$3 of weight $\sum_j a_{ij} \ell_j \geq b_i$4 is represented as a soft PB-constraint $\sum_j a_{ij} \ell_j \geq b_i$5 with weight $\sum_j a_{ij} \ell_j \geq b_i$6.
• PBO $\sum_j a_{ij} \ell_j \geq b_i$7 WBO: Every PBO instance is mapped by encoding its hard PB constraints as WBO hard constraints; the PBO objective $\sum_j a_{ij} \ell_j \geq b_i$8 becomes soft unit constraints $\sum_j a_{ij} \ell_j \geq b_i$9 with weight $a_{ij}, b_i \in \mathbb{N}$0.

Conversely, WBO instances can be projected to PBO via selection variables or to weighted MaxSAT by CNF encoding of PB constraints (0903.0843).

2. Abstract Semantics via Weight Systems

A general semantic foundation for WBO and related paradigms is provided by the weight system (“w-system”) abstraction. Here, a w-system is a pair $a_{ij}, b_i \in \mathbb{N}$1, where $a_{ij}, b_i \in \mathbb{N}$2 is a set of “hard” theories/modules and $a_{ij}, b_i \in \mathbb{N}$3 is a finite set of “soft” w-conditions, each a pair $a_{ij}, b_i \in \mathbb{N}$4 (theory $a_{ij}, b_i \in \mathbb{N}$5, integer weight $a_{ij}, b_i \in \mathbb{N}$6, level $a_{ij}, b_i \in \mathbb{N}$7).

The soft conditions can support multi-level (lexicographic) objectives, flattened to single-level via normalization (Lierler, 2022). This abstraction:

• Embeds the standard WBO as the special case $a_{ij}, b_i \in \mathbb{N}$8 = CNF hard clauses, $a_{ij}, b_i \in \mathbb{N}$9 = weighted soft clauses,
• Allows translation and normalization (e.g., 0-weight elimination, uniform scaling, level flattening) without loss of soundness or optimality,
• Provides a logic-agnostic setting for the integration and comparison of optimization paradigms (e.g., MaxSAT, Answer Set Programming with weak constraints) (Lierler, 2022).

3. Algorithmic Techniques

Core-Guided, Unsatisfiability-Based Methods

The dominant algorithmic paradigm for practical WBO is the core-guided, unsatisfiability-based approach, generalizing the successful MSU algorithm for MaxSAT (0903.0843). At each iteration, a (pseudo-)Boolean solver is queried:

• If the current working formula is satisfiable, the optimum is reached.
• If unsatisfiable, a minimum-weight unsatisfiable core among soft constraints is identified. Soft constraints in the core are relaxed (with auxiliary variables and weight adjustments), and an aggregate constraint (e.g., cardinality constraint enforcing at least one relaxation) is added.
• The process repeats, incrementally raising a lower bound on the optimum until satisfiability.

WBO solvers may handle PB constraints natively (supporting arbitrary cardinality and general PB constraints) or by CNF encoding and interface with existing MaxSAT engines.

Dynamic-Programming on Hybrid Constraints

For “hybrid” Boolean optimization—where constraints may mix CNF, XOR, and general PB inequalities—dynamic-programming solvers (e.g., DPO) employ algebraic decision diagrams (ADDs), graph decomposition into project-join trees, and max-projection operations. This enables exact solutions to literal-weighted CNF formulas even with XOR or PB constraints, subject to bounded treewidth (Phan et al., 2022).

4. Parameterized Complexity and Expressiveness

WBO admits a nuanced parameterized complexity landscape, especially when clause/constraint weights may be negative:

• Sum objective with arbitrary weights: Maximizing the sum (possibly signed) is W[1]-hard parameterized by $\ell_j$0, even for monotone DNF/CNF of bounded clause size (Bannach et al., 2022).
• Absolute-value objective: Maximizing $\ell_j$1 is fixed-parameter tractable in $\ell_j$2, where $\ell_j$3 is the imbalance threshold and $\ell_j$4 is the maximum clause size. Via kernelization, a kernel of size $\ell_j$5 is achieved by reduction rules exploiting combinatorial properties of d-uniform hypergraphs.
• These results systematically chart the impact of negative weights and alternative objectives, showing that “absolute imbalance” restores tractability broken by negative weights (Bannach et al., 2022).

5. Solver Architecture and Translational Techniques

The w-system abstraction enables generic solver architectures that are parameterizable by the logic of the hard fragment (SAT, ASP, SMT, etc.) and the type of soft optimization criteria. This covers translational approaches:

• ASP to MaxSAT/WBO translation: ASP programs with weak constraints (o-programs) can be embedded into WBO or weighted MaxSAT via translation through w-systems (clausification and complementation), facilitating the use of MaxSAT/WBO solvers for ASP optimization tasks (Lierler, 2022).

Algorithmic schemes from MaxSAT and PBO—such as branch-and-bound, core-guided search, binary search, cutting-plane refinement—naturally extend to WBO due to its expressive interface (supporting both soft/hard PB constraints and clause/weight manipulation) (0903.0843).

6. Empirical Results and Practical Considerations

Experimental evaluations on industrial benchmarks (e.g., partial weighted MaxSAT instances from IND, FIR, SYN) have demonstrated that unsatisfiability-based WBO solvers are highly competitive, often outperforming both pure-MaxSAT and pure-PBO solvers. The WBO solver introduced in (0903.0843) solved nearly as many instances as the top-performing MSU-based MaxSAT solver and significantly more than dedicated PBO engines.

Dynamic-programming-based WBO solvers (e.g., DPO (Phan et al., 2022)) show strong performance on constraint instances admitting low treewidth (such as Bayesian inference CNFs or random XOR-CNF benchmarks), matching or exceeding state-of-the-art hybrid solvers under suitable width constraints.

7. Theoretical Significance and Extensions

Weighted Boolean Optimization provides a powerful, minimal unifying setting for Boolean and pseudo-Boolean combinatorial optimization:

• It absorbs advances and techniques developed independently in MaxSAT, PBO, and ASP optimization.
• The abstraction as weight-systems provides a common substrate for general results in soundness, equivalence, and model transformation.
• Parameterized tractability and small kernels for the absolute-value variants (via hypergraph kernelization) address expressiveness and complexity even in the presence of negative or mixed weights (Bannach et al., 2022).
• The interface supports generic solver design and translational pipelines, catalyzing further advances in scalable optimization tools and cross-paradigm algorithm sharing (Lierler, 2022).

Weighted Boolean Optimization thus constitutes both a central modeling construct and a technical foundation for contemporary research in automated reasoning, modular knowledge representation, and constraint optimization.