Formula partitioning revisited

Abstract

Dividing a Boolean formula into smaller independent sub-formulae can be a useful technique for accelerating the solution of Boolean problems, including SAT and #SAT. Nevertheless, and despite promising early results, formula partitioning is hardly used in state-of-the-art solvers. In this paper, we show that this is rooted in a lack of consistency of the usefulness of formula partitioning techniques. In particular, we evaluate two existing and a novel partitioning model, coupled with two existing and two novel partitioning algorithms, on a wide range of benchmark instances. Our results show that there is no one-size-fits-all solution: for different formula types, different partitioning models and algorithms are the most suitable. While these results might seem negative, they help to improve our understanding about formula partitioning; moreover, the findings also give guidance as to which method to use for what kinds of formulae.

Guiding SAT Solving by Formula Partitioning

“…Earlier, we have also conducted a set of experiments with formula partitioning approaches [41], putting more focus on aspects of partitioning the formulas than on integrating partitioning into SAT solving. In contrast to this research, that early study only examined the hard partitioning approach for using partitioning to solve SAT, and it deduced that the results with this method are inconclusive.…”

“…In our previous work [41], we investigated the use of various versions of the Fiduccia-Mattheyses (FM) heuristic for SAT partitioning on a diverse set of benchmark formulas. Our findings indicated that, although each considered partitioning method gave promising results on some problem instances, but none of them was consistently good on the set of benchmarks as a whole.…”

“…Step 5 Step 0 was covered in our earlier paper [41]; Steps 1 to 5 are the contributions of this paper.…”

Network Flow-Based Refinement for Multilevel Hypergraph Partitioning

“…In these cases, it is difficult to find meaningful vertex moves that improve the solution quality because large hyperedges are likely to have many vertices in multiple blocks [53]. Thus the gain of moving a single vertex to another block is likely to be zero [41].While finding balanced minimum cuts in hypergraphs is NP-hard, a minimum cut separating two vertices can be found in polynomial time using network flow algorithms and the well-known max-flow min-cut theorem [21]. Flow algorithms find an optimal min-cut and do not suffer the drawbacks of move-based approaches.…”

“…In these cases, it is difficult to find meaningful vertex moves that improve the solution quality because large hyperedges are likely to have many vertices in multiple blocks [53]. Thus the gain of moving a single vertex to another block is likely to be zero [41].…”