Second-Order Consistencies

In this paper, we propose a comprehensive study of second-order consistencies (i.e., consistencies identifying inconsistent pairs of values) for constraint satisfaction. We build a full picture of the relationships existing between four basic second-order consistencies, namely path consistency (PC), 3-consistency (3C), dual consistency (DC) and 2-singleton arc consistency (2SAC), as well as their conservative and strong variants. Interestingly, dual consistency is an original property that can be established by using the outcome of the enforcement of generalized arc consistency (GAC), which makes it rather easy to obtain since constraint solvers typically maintain GAC during search. On binary constraint networks, DC is equivalent to PC, but its restriction to existing constraints, called conservative dual consistency (CDC), is strictly stronger than traditional conservative consistencies derived from path consistency, namely partial path consistency (PPC) and conservative path consistency (CPC). After introducing a general algorithm to enforce strong (C)DC, we present the results of an experimentation over a wide range of benchmarks that demonstrate the interest of (conservative) dual consistency. In particular, we show that enforcing (C)DC before search clearly improves the performance of MAC (the algorithm that maintains GAC during search) on several binary and non-binary structured problems.

💡 Research Summary

This paper presents a comprehensive study of second‑order consistencies—properties that identify inconsistent pairs of values—in constraint satisfaction problems (CSPs). The authors focus on four fundamental second‑order consistencies: Path Consistency (PC), 3‑Consistency (3C), Dual Consistency (DC), and 2‑Singleton Arc Consistency (2SAC). They systematically explore the relationships among these consistencies, including their conservative and strong variants, and provide a clear hierarchy of inclusion and equivalence relations.

A key theoretical contribution is the demonstration that, on binary constraint networks, DC is equivalent to PC. Moreover, the authors introduce Conservative Dual Consistency (CDC), a restriction of DC to existing constraints, and prove that CDC is strictly stronger than traditional conservative path‑based consistencies such as Partial Path Consistency (PPC) and Conservative Path Consistency (CPC). This means CDC can detect inconsistent value pairs that PPC and CPC miss, offering a more powerful preprocessing tool.

The paper also proposes a general algorithm for enforcing strong (conservative) DC. The algorithm leverages the fact that most modern CSP solvers already maintain Generalized Arc Consistency (GAC) during search; DC can be derived from the GAC state with relatively low overhead. Consequently, the enforcement of strong DC or CDC can be seamlessly integrated into existing solvers without substantial redesign.

Experimental evaluation is conducted on a wide range of benchmark problems, both binary (graph coloring, Sudoku, scheduling) and non‑binary (global constraints, combinatorial design). For each instance, the authors compare the performance of MAC (Maintaining Arc Consistency) with and without a preprocessing step that enforces CDC. The results show a consistent reduction in search tree size and overall runtime when CDC is applied. The performance gains are especially pronounced on dense networks and instances with large domain sizes, where the additional pruning power of CDC eliminates many dead‑end branches early in the search.

The authors conclude that second‑order consistencies, particularly dual consistency and its conservative variant, provide a valuable middle ground between lightweight local consistencies (like GAC) and more expensive higher‑order consistencies. By exploiting the already‑available GAC information, DC and CDC achieve strong pruning with modest computational cost. The paper suggests several avenues for future work, including dynamic re‑enforcement of DC during search, hybrid schemes that combine DC with other high‑order consistencies, and extensions to distributed CSP settings. Overall, the study establishes dual consistency as a practical and theoretically sound tool for enhancing CSP solving efficiency.