Second-Order Consistencies

Abstract

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

Revisiting Neighborhood Inverse Consistency on Binary CSPs

“…Strong consistency properties for binary CSPs that do not affect the topology of the constraint graph have been carefully reviewed, studied, and compared to each others [5,14]. Such properties include maxRPC [3], SAC [4], CDC [13], and, the strongest of them all, sCDC [14].…”

“…Such properties include maxRPC [3], SAC [4], CDC [13], and, the strongest of them all, sCDC [14]. Further, Lecoutre et al show that, on binary CSPs, strong Conservative Dual Consistency (sCDC) is equal SAC+CDC [14].…”

“…Such properties include maxRPC [3], SAC [4], CDC [13], and, the strongest of them all, sCDC [14]. Further, Lecoutre et al show that, on binary CSPs, strong Conservative Dual Consistency (sCDC) is equal SAC+CDC [14]. 5 While NIC was shown to be incomparable to SAC [5], its relationship to sCDC has not yet been addressed [13].…”

“…We show how the structure of the dual graph of a binary CSP affects the consistency level enforced by RNIC, characterizing the conditions where RNIC cannot be stronger than another relational consistency property that we had defined in [11] when both properties are enforced on binary CSPs. In order to characterize the effectiveness of RNIC on binary CSPs despite the identified limitation imposed by the structure, we turn our attention back to 'strong' consistency properties defined for binary CSPs, and compare RNIC, both theoretically and empirically, to NIC and strong Conservative Dual Consistency (sCDC) [14], showing that all three properties are incomparable.…”

A Framework for Decision-Based Consistencies

“…Among such decision-based consistencies, we find SAC (singleton arc consistency), partition-k-AC [2], weak-k-SAC [22], BiSAC [4], and DC (dual consistency) [15]. Besides, a partial form of SAC, better known as shaving, has been introduced for a long time [6,18] and is still an active subject of research [17,21]; when shaving systematically concerns the bounds of each variable domain, it is called BoundSAC [16].…”