Encoding Multi-Valued Decision Diagram Constraints as Binary Constraint Trees

Abstract

Ordered Multi-valued Decision Diagram (MDD) is a compact representation used to model various constraints, such as regular constraints and table constraints. It can be particularly useful for representing ad-hoc problem specific constraints. Many algorithms have been proposed to enforce Generalized Arc Consistency (GAC) on MDD constraints. In this paper, we introduce a new compact representation called Binary Constraint Tree (BCT). We propose tree binary encodings to transform any MDD constraint into a BCT constraint. We also present a specialized algorithm enforcing GAC on the BCT constraint resulting from a MDD constraint. Experimental results on a large set of benchmarks show that the BCT GAC algorithm can significantly outperform state-of-the-art MDD as well as table GAC algorithms.

Encoding Constraints as Binary Constraint Networks Satisfying BTP

“…For example, we can design a GAC propagator based on Directional Arc Consistency (DAC) (Dechter and Pearl 1987), as BTPN is solvable by DAC, see Proposition 5.5 in (Cooper, Jeavons, and Salamon 2010). In addition, similar to other binary encodings (Wang and Yap 2019, 2022b, it has potential to design specialized AC and DAC propagators for BTPN.…”

“…For example, the Multi-valued Decision Diagram (MDD) can be exponentially smaller than its table representation. As such (Wang and Yap 2022b) proposed binary encodings to encode MDD constraints as BCTs. They showed that BCT is strictly more succinct than many ad-hoc constraints Yap 2022a, 2023).…”

“…, multi-valued variable diagram (MVD)(Amilhastre et al 2014), smart table (smaT)(Mairy, Deville, and Lecoutre 2015), basic smart table(Verhaeghe et al 2017), semi-MDD (sMDD)(Verhaeghe, Lecoutre, and Schaus 2018), segmented table (segT) (Audemard, Lecoutre, and Maamar 2020) and BCT(Wang and Yap 2022b) constraints. The modelling power of BTPN…”

The Expressive Power of Ad-Hoc Constraints for Modelling CSPs

“…Binary Constraint Tree. A Binary Constraint Tree (BCT) (Wang and Yap 2022b) is a normalized binary CSP which has a tree structured constraint graph (Dechter 1987(Dechter , 1990a.…”

“…A wide range of combinatorial problems in real life can be modelled with Constraint Satisfaction Problems (CSPs) using a diverse of constraints. We will focus on ad-hoc constraints (also called generic constraints) that can be defined with various representations, such as tables (Bessière and Régin 1997;Lecoutre 2011;Lecoutre, Likitvivatanavong, and Yap 2015;Wang et al 2016;Demeulenaere et al 2016;Yap 2019, 2020), automatas (Pesant 2004;Quimper and Walsh 2006;Cheng, Xia, and Yap 2012), context-free grammars (Sellmann 2006;Walsh 2006, 2007;Kadioglu and Sellmann 2008), decision diagrams (Cheng and Yap 2010;Gange, Stuckey, and Szymanek 2011;Perez and Régin 2014;Vion and Piechowiak 2018;Schaus 2018, 2019), and binary constraint trees (Wang and Yap 2022b). Ad-hoc constraints are very useful for modelling specific constraints which do not fit well with common global constraints.…”