New Inference Rules for Max-SAT (1111.0040v1)

Abstract: Exact Max-SAT solvers, compared with SAT solvers, apply little inference at each node of the proof tree. Commonly used SAT inference rules like unit propagation produce a simplified formula that preserves satisfiability but, unfortunately, solving the Max-SAT problem for the simplified formula is not equivalent to solving it for the original formula. In this paper, we define a number of original inference rules that, besides being applied efficiently, transform Max-SAT instances into equivalent Max-SAT instances which are easier to solve. The soundness of the rules, that can be seen as refinements of unit resolution adapted to Max-SAT, are proved in a novel and simple way via an integer programming transformation. With the aim of finding out how powerful the inference rules are in practice, we have developed a new Max-SAT solver, called MaxSatz, which incorporates those rules, and performed an experimental investigation. The results provide empirical evidence that MaxSatz is very competitive, at least, on random Max-2SAT, random Max-3SAT, Max-Cut, and Graph 3-coloring instances, as well as on the benchmarks from the Max-SAT Evaluation 2006.

• The paper introduces novel inference rules that simplify Max-SAT instances while preserving equivalence.
• The approach extends unit resolution by efficiently replacing almost common and complementary unit clauses.
• Empirical evaluations with the MaxSatz solver show enhanced performance on benchmark Max-SAT problems.

An Analysis of New Inference Rules for Max-SAT

The paper "New Inference Rules for Max-SAT" introduces a collection of novel inference rules specifically designed to address the Max-SAT problem, a known NP-hard problem of significant interest in various applications of artificial intelligence and theoretical computer science. This paper focuses on refining the approach towards solving Max-SAT instances, emphasizing the development of efficient inference strategies that preserve the equivalence between original and transformed problem instances while simplifying the task.

Key Contributions

The authors present several inference rules that extend traditional SAT techniques, such as unit resolution, to the Max-SAT domain. The rules proposed expand upon existing methodologies by converting Max-SAT instances into equivalent forms that are simpler and more computationally feasible. The primary inference rules outlined include:

1. Replacement of Almost Common Clauses – This rule efficiently replaces two clauses yielding a shorter resolvent in Max-SAT, unlike general resolution which is not applicable for Max-SAT.
2. Complementary Unit Clause Rule – Replaces complementary unit clauses with an empty clause, directly contributing to increased lower bounds beneficial for pruning during search.
3. Linear Resolution Patterns – These include rules for resolving chains of implications and extending them, such as transforming three specified clauses into a set that includes the empty clause alongside other supporting clauses.
4. Nonlinear Resolution Patterns – New rules designed to handle more complex logical structures, providing better management of complementary and overlapping literals.

Practical Implications and Evaluation

The effectiveness of these rules is empirically validated through the implementation of a Max-SAT solver, MaxSatz, which incorporates them. The experimental investigation reported in the paper demonstrates substantial improvements in solving instances of Random Max-2SAT, Max-3SAT, Max-Cut, and Graph 3-coloring problems. Notably, MaxSatz outperforms several contemporary solvers on benchmark datasets, a testament to the efficacy of the proposed inference rules.

The results underscore the potential for these rules to reduce the search space efficiently and improve the precision of lower bounds in branch-and-bound algorithms. This not only accelerates solution discovery but also enhances the solver's capability to handle more complex or larger problem instances effectively.

Theoretical Implications and Future Work

The paper's contribution extends beyond immediate practical improvements, offering a foundational structure for further extension in Max-SAT and related domains. The introduction of inference rules adaptable to weighted Max-SAT problems marks a significant advancement, laying the groundwork for addressing even broader classes of constraint satisfaction problems.

Future development of these techniques could explore how different variable ordering strategies during unit propagation might enhance rule application and inference capability. Similarly, extending the applicability of these rules to address ternary and higher-order clauses could significantly improve their versatility.

In conclusion, the paper represents a substantive advancement in Max-SAT solving strategies, with implications not only for practical applications but also for deeper theoretical understanding. The adaptability and demonstrated performance benefits of the proposed rules offer a robust framework for further innovation and application in artificial intelligence problem-solving contexts.