Abstract
The conflict-driven clause learning (CDCL) paradigm has revolutionized
SAT solving over the last two decades. Extending this approach to
pseudo-Boolean (PB) solvers doing 0-1 linear programming holds the
promise of further exponential improvements in theory, but
intriguingly such gains have not materialized in practice. Also
intriguingly, most PB extensions of CDCL use not the division rule in
cutting planes as defined in [Cook et al., '87] but instead the
so-called saturation rule. To the best of our knowledge, there has
been no study comparing the strengths of division and saturation in
the context of conflict-driven PB learning, when all linear
combinations of inequalities are required to cancel variables.
We show that PB solvers with division instead of saturation can be
exponentially stronger. In the other direction, we prove that
simulating a single saturation step can require an exponential number
of divisions. We also perform some experiments to see whether these
phenomena can be observed in actual solvers. Our conclusion is that a
careful combination of division and saturation seems to be crucial to
harness more of the power of cutting planes.
Keywords
- Knowledge Representation and Reasoning: Computational Complexity of Reasoning
- Heuristic Search and Game Playing: Combinatorial Search and Optimisation
- Constraints and SAT: Constraints: Evaluation and Analysis
- Constraints and SAT: SAT: Evaluation and Analysis
