SAT for Upward Book Embedding: An Empirical Study

Rustem Kakimov, Xing Tan

Abstract: The 2-page upward book embedding (2UBE) problem, recently proven NP-complete, is a fundamental problem in graph theory with various applications. Given its computational intractability, SAT-based approaches offer a promising direction for efficiently determining embeddability. In this paper, we present two practical SAT encodings, SAT-1 (largely adopted from the literature) and SAT-2 (specifically tailored for 2UBE). We also introduce a Constraint Programming (CP) formulation as an alternative approach. Our empirical evaluation on benchmark datasets demonstrates that SAT solvers significantly outperform CP in solving 2UBE instances. Additionally, we analyze the scalability of these methods on large grid graphs, revealing that SAT-2 achieves up to a 40% speedup over SAT-1. Using SAT-2, we discovered a phase transition in 2UBE, which occurs when the edge-to-node ratio (m/n) ≈ 1.5. In general, we have established SAT-based approaches as both a practical and scalable solution for book embedding problems and a tool for evaluating their structural properties.

External IDs:dblp:conf/ai/Kakimov025