Satisfiability and Optimisation

Determining the satisfiability of propositional formulas—whether there exists an assignment of truth values to variables that renders a conjunction of clauses true—lies at the heart of modern computational logic and underpins a vast range of decision, verification and optimisation tasks. The canonical Boolean Satisfiability Problem (SAT) is NP-complete: no polynomial-time solution is known. Yet despite its worst-case intractability, practical SAT solvers routinely handle instances with millions of variables and clauses, thanks to advances in search strategies, unit-propagation mechanisms and learning from conflicts. Beyond simple satisfiability, optimisation variants such as Maximum Satisfiability (MaxSAT) and Minimum Satisfiability (MinSAT) seek assignments that maximise the number of satisfied clauses (or minimise the number of unsatisfied clauses) and generalise SAT to weighted, partial and soft‐constraint settings. These problems arise in hardware and software verification, combinatorial design, resource allocation, scheduling and knowledge discovery. Research efforts now focus on hybrid methodologies that combine systematic search, conflict-driven clause learning, local search metaheuristics and problem-specific oracles to explore vast combinatorial spaces efficiently, yielding both provable optimal solutions and high-quality approximations in time critical applications.

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A structured framework for Optimisation with Constraint Learning has recently been proposed to integrate data-derived constraints into traditional combinatorial models. By formalising a multi-step pipeline—model conceptualisation, data gathering and preprocessing, machine-learning of implicit feasibility constraints, solver integration and iterative model refinement—this approach enables systematic incorporation of empirical knowledge into optimisation, highlighting bottlenecks in learning accuracy and solver performance and charting directions for tighter synergy between statistical inference and exact search methods.

The Local Search with SAT Oracle paradigm recasts NP-hard combinatorial optimisation as a sequence of satisfiability subproblems solved by a modern SAT solver. Rather than relying on small, hand-crafted neighbourhoods, an algorithm formulates a neighbourhood constraint in conjunctive normal form that captures all candidate improvements beyond the current solution. A SAT solver is then invoked as an “oracle” to find any better neighbour, with learned clauses and restart strategies guiding the search towards global minima. This methodology has been successfully applied to cell-placement in VLSI design and extends generic local search techniques via off-the-shelf SAT technology.

Detailed Inference Calculi for MaxSAT and MinSAT have been developed to provide complete resolution-style and tableau-style proof systems for optimisation variants of SAT. These calculi introduce modified resolution rules with “compensation” clauses and conflict-driven clause learning mechanisms that preserve the count of unsatisfied clauses under derivations. Variable elimination algorithms and conflict analysis procedures yield sound and terminating decision methods for weighted and partial MaxSAT and MinSAT, underpinning the theoretical foundations of modern exact solvers.

Satisfiability and Optimisation publication trend

The graph below shows the total number of articles in satisfiability and optimisation across all publications each year (not limited to Nature Index journals).

Technical terms

Boolean Satisfiability (SAT): Decision problem of determining whether a propositional formula in conjunctive normal form can be made true by some assignment of its variables.

Maximum Satisfiability (MaxSAT): Optimisation variant of SAT seeking an assignment that satisfies the maximum number of clauses, or equivalently minimises the number of unsatisfied clauses.

Unit Propagation: Procedure whereby a clause with all but one literal falsified forces the remaining literal to be true, used to prune search in DPLL-based solvers.

Conflict-Driven Clause Learning (CDCL): Enhancement of DPLL in which, upon encountering a conflict, a new clause is derived by resolution from the conflicting clauses to prevent repeated exploration of the same infeasible region.

Constraint Learning: Process of inferring implicit constraints from data on feasible and infeasible states to augment or refine mathematical models prior to or during optimisation.

Local Search with SAT Oracle: Hybrid approach that encodes candidate improvement steps as SAT subproblems, using a SAT solver to navigate large neighbourhoods efficiently.

Resolution Rule for MaxSAT: Adapted resolution inference that introduces “compensation” clauses to ensure preservation of the minimal number of unsatisfied clauses during variable elimination.

References

1. Optimization with constraint learning: A framework and survey. European Journal of Operational Research (2024).
2. Local Search with a SAT Oracle for Combinatorial Optimization.