Experiments with Reduction Finding

Abstract

Reductions are perhaps the most useful tool in complexity theory and, naturally, it is in general undecidable to determine whether a reduction exists between two given decision problems. However, asking for a reduction on inputs of bounded size is essentially a $\Sigma^p_2$ problem and can in principle be solved by ASP, QBF, or by iterated calls to SAT solvers. We describe our experiences developing and benchmarking automatic reduction finders. We created a dedicated reduction finder that does counter-example guided abstraction refinement by iteratively calling either a SAT solver or BDD package. We benchmark its performance with different SAT solvers and report the tradeoffs between the SAT and BDD approaches. Further, we compare this reduction finder with the direct approach using a number of QBF and ASP solvers. We describe the tradeoffs between the QBF and ASP approaches and show which solvers perform best on our $\Sigma^p_2$ instances. It turns out that even state-of-the-art solvers leave a large room for improvement on problems of this kind. We thus provide our instances as a benchmark for future work on $\Sigma^p_2$ solvers.