Partial MaxSAT Solving

• Partial MaxSAT Solving is a problem formulation that partitions clauses into hard constraints and soft clauses to maximize satisfiability, often with weights.
• Solvers apply diverse techniques such as stochastic local search, core-guided methods, and graph-based decompositions to efficiently navigate complex search spaces.
• Hybrid and mathematical programming approaches, including MILP and SDP relaxations, have enhanced scalability and performance in real-world applications like verification and optimization.

Partial MaxSAT Solving

Partial Maximum Satisfiability (Partial MaxSAT, PMS) extends the classical SAT and MaxSAT paradigms by partitioning a propositional formula's clauses into "hard" clauses, which must always be satisfied, and "soft" clauses, whose satisfaction is to be maximized (possibly weighted). This formalism and its weighted extension (WPMS) have pervasive applications in AI, verification, discrete optimization, and model selection. Techniques for solving PMS range from stochastic local search, core-guided complete algorithms, graph-structured decompositions, and hybrid methods integrating SAT, integer programming, and semidefinite programming relaxations.

1. Formal Definitions and Problem Structure

Let $V = \{ x_1, ..., x_n \}$ be Boolean variables and $F$ be a CNF formula with hard clauses $H$ and soft clauses $S$. Each soft clause $c \in S$ may carry an original static weight $w_\mathrm{org}(c) > 0$.

• Partial MaxSAT (PMS): Find a complete assignment $\alpha$ such that all hard clauses in $H$ are satisfied and the number of satisfied soft clauses in $S$ is maximized, or equivalently, $$\mathrm{Cost}(\alpha) = |\{ c \in S : \alpha \text{ falsifies } c \}|$$ is minimized. $F$0 is feasible if all hard clauses are satisfied, otherwise $F$1.
• Weighted Partial MaxSAT (WPMS): As above, but maximize the total original weight of satisfied soft clauses or minimize $F$2.

A solution $F$3 is strictly better than $F$4 if $F$5. This model subsumes SAT (all hard clauses) and (weighted) MaxSAT (all soft, no hard).

2. Algorithmic Paradigms and Clause Weighting

Contemporary PMS/WPMS solvers are broadly organized as follows:

• Stochastic Local Search (SLS): Modern SLS solvers (e.g., DeepDist, BandMaxSAT) assign dynamic weights to clauses to guide the search through the space of assignments. Clause weighting is central: weights are initialized and updated according to the clause role (hard/soft) and the current search state. Notable advances distinguish PMS from WPMS in their weighting and update rules:
  • In DeepDist (Jiang et al., 5 Dec 2025), weights for hard and soft clauses are treated according to different regimes after feasibility is achieved. For PMS, soft-clause weights are only updated if no improvement is observed near the best solution, encouraging escaping local optima; for WPMS, soft-clause weights are updated at every feasible local optimum to better respect original clause priorities.
  • Dynamic re-initialization includes a novel "zeroing" of hard weights after feasibility in PMS to shift search focus and enhance diversification.
  • Other SLS algorithms, such as BandMaxSAT, model clause selection as a multi-armed bandit, allowing adaptive selection of which soft clause to prioritize when the search is stuck (Zheng et al., 2022).
• Core-Guided and SAT-Reduction Approaches: Complete methods usually encode PMS/WPMS via SAT with relaxation variables on soft clauses, imposing cardinality/weighted constraints on the sum of violated soft clauses.
  • Standard forms utilize "relaxation variable" encoding—soft clause $F$6 becomes $F$7, $F$8 is penalized if set to true—and iteratively strengthen cardinality bounds until the minimal feasible cost is certified (Neves et al., 2015).
  • Partitioning the soft clauses into blocks (sometimes using graph structure such as the resolution graph or variable-incidence graph) enables incremental or modular solution: each partition is solved independently or in a merge schedule that honors their interdependencies (Neves et al., 2015, Orvalho et al., 2023).
  • Clause weighting and update strategies are critical also in core-guided solvers for the scheduling of relaxation and incremental strengthening of lower bounds.

3. Graph-Based Structural Decomposition and Symmetry Handling

• Partitioning and Graph Community Detection: Techniques such as UpMax (Orvalho et al., 2023) and the partitioned MSU3 algorithms (Neves et al., 2015) use resolution graphs or variable-incidence graphs to cluster tightly connected soft clauses. Community detection enables modular reasoning and improves scalability, especially when combined with efficient merge strategies and incremental propagation of lower bounds between merged blocks.
• Dynamic Programming for Structured Instances: On formulas of small "ps-width" (the number of precisely satisfiable subsets of clauses at any decomposition cut), bottom-up dynamic programming yields polynomial-time PMS/WPMS solving (Sæther et al., 2014). Such tractability is characterized via incidence graph parameters (e.g., interval/circular-arc bigraphs).
• Symmetry Breaking: The imposition of symmetry-breaking predicates (SBPs) is crucial for problems with many symmetric solutions. SBPs are constructed as lex-leader constraints for each generator of the formula's automorphism group and added as hard clauses, dramatically reducing the search effort for highly symmetric instances while preserving optimality (0804.0599).

4. Advanced Hybrid and Mathematical Programming Approaches

• Hybrid SLS+Complete Solvers: Leading SLS solvers like DeepDist and BandMaxSAT can be hybridized with complete solvers (e.g., TT-Open-WBO-Inc) to provide anytime performance and leverage the strengths of both paradigms. The best-known results in recent MaxSAT Evaluations have been achieved by such hybrids, outperforming both standalone local search and champion core-guided algorithms, especially in weighted and unweighted tracks (Jiang et al., 5 Dec 2025).
• MILP Formulations and Clause Cuts: Weighted PMS can be encoded as MILP with variables for assignment and relaxation, but straightforward formulations have weak relaxations. Clause Cuts are logically implied inequalities derived from unsatisfiable cores or learned clauses, which are injected into the MILP relaxation to significantly strengthen the formulation. Specialized separation routines (e.g., ICCA, LCCA) invoke a SAT-oracle to detect minimal unsatisfiable assumptions and yield short cuts, resulting in order-of-magnitude speedups over standard MILP (Gurobi) and outperforming dedicated MaxSAT solvers such as RC2 (Engelhardt et al., 25 Sep 2025).
• Sum-of-Squares and SDP Relaxations: Semidefinite programming relaxations for PMS/WPMS via sum-of-squares (SOS) polynomial lifting provide strong upper bounds and can be embedded within branch-and-bound. Parametric monomial bases and moment matrix representations, first-order SDP solvers, and hybrid branching heuristics yield competitive performance with state-of-the-art on dense or hard random families, extending Goemans–Williamson SDP methods for Max-2-SAT to the partial and weighted domain (Sinjorgo et al., 2023).

5. Performance, Ablation, and Theoretical Analyses

• Empirical Findings: State-of-the-art SLS solvers with differentiated weighting (e.g., DeepDist) significantly outperform previous best local search strategies and maintain a consistent lead, particularly on weighted instances. Ablation confirms that distinguishing PMS and WPMS in weighting and initialization, as well as incorporating specialized decimation for initial assignment construction, is indispensable for peak performance. Hybrid approaches strictly dominate the best standalone SLS or complete solvers on nearly all benchmark tracks (Jiang et al., 5 Dec 2025).
• Partitioning Impact: Decoupling of soft-clause partitioning from solver internals (as in UpMax) demonstrates robust performance improvements across a variety of problem families; graph-driven and user-guided partitions routinely enable existing solvers to solve an order of magnitude more instances, even with minimal infrastructure changes (Orvalho et al., 2023, Neves et al., 2015).
• Scalability and Parallelism: Distributed solvers (e.g., DistMS) using search-space and guiding-path splitting scale efficiently in practice. Different splitting and task assignment strategies suit different instance categories: search-space for industrial and guiding-path for synthetic/crafted problems (Neves et al., 2015).
• Tractability for Special Cases: For structured instances of bounded ps-width, dynamic programming yields polynomial-time PMS/WPMS solution. Classes like interval bigraphs and circular-arc bigraphs lead to small ps-width and tractable MaxSAT (Sæther et al., 2014).

6. Emerging Directions and Open Challenges

• Top-$F$9 Solutions and Diversification: The diversified top-$H$0 partial MaxSAT problem seeks $H$1 assignments covering a maximal set of satisfied soft clauses. A straightforward variable expansion encoding (EE) enables off-the-shelf partial MaxSAT solvers to address this challenge effectively. However, variable blow-up and lack of explicit diversity control indicate open ground for more compact, incremental, and diversity-sensitive encodings (Zhou et al., 2017).
• Extensions and Hybridizations: Further integration of cutting-plane, SDP, SAT, and SLS paradigms, as well as adaptive partition-merge procedures, and automated discovery of beneficial clause partitionings or SBPs, are active areas. Robustness across industrial/crafted/random tracks in MaxSAT Evaluations continues to be the primary metric for solver adoption.
• Theoretical Barriers: Full characterization of facets of the PMS polytope amenable to efficient clause-cut generation, as well as the effective application of chordal and block-diagonal term sparsity in SOS relaxations to industrial-scale formulas, remain open.

7. Application Domains and Real-World Impact

• Fault Tree Analysis and Beyond: PMS/WPMS serves as the backbone for reliability analysis (e.g., maximum probability minimal cut sets), model-based diagnosis, scheduling, and discrete optimization. In such applications, the translation to weighted partial MaxSAT is direct, and parallel portfolio approaches or core-guided complete methods are often immediately applicable (Barrère et al., 2020).
• Instance Structure Exploitation: Real-world optimization frequently presents formulas exhibiting cluster structure, modularity, or symmetry. Techniques exploiting these structural features—via partitioning, symmetry breaking, or parameterized tractability—are key to scaling MaxSAT technology across domains in verification, planning, and combinatorial design.

The present state of Partial MaxSAT solving amalgamates algorithm engineering, theoretical insights, graph and symmetry exploitation, and hybrid mathematical-programming approaches. The field continues to be driven by empirical evaluation on large, diverse tracks, and by innovations in structural decompositions, clause weighting, and solver modularity.

References: (Jiang et al., 5 Dec 2025, Zheng et al., 2022, Neves et al., 2015, Neves et al., 2015, Orvalho et al., 2023, 0804.0599, Engelhardt et al., 25 Sep 2025, Sinjorgo et al., 2023, Zhou et al., 2017, Sæther et al., 2014, Barrère et al., 2020)