Partition-k-AC: An Efficient Filtering Technique Combining Domain Partition and Arc Consistency

A Framework for Decision-Based Consistencies

“…A value (x, a) of P is 1-AC-consistent iff (x, a) is SAC-consistent and ∀y ∈ vars(P ) \ {x}, ∃b ∈ dom(y) | (x, a) ∈ GAC (P | y=b ) [2]. A value (x, a) of P is BiSAC-consistent iff GAC (P ia | x=a ) = ⊥ where P ia is the CN obtained after removing every value (y, b) of P such that y = x and (x, a) / ∈ GAC(P | y=b ) [4].…”

“…Note that the CN P 4 is also obtained when setting ∆ to ∆ = . In [2], it is also shown that inferences regarding values may be obtained by considering the result of several decision-based checks. This is generalized below.…”

“…We can show that (z, a) is E GAC ∆ -inconsistent because (z, a) / ∈ AC(P | x∈{a,c} ) and (z, a) / ∈ AC(P | x∈{b,d} ). The consistency P-k-AC, introduced in [2], corresponds to S φ ∆ +E φ ∆ where φ = AC and ∆ necessarily corresponds to a partition of each domain into pieces of size at most k.…”

“…Note that there exist consistencies φ and decision mappings ∆ such that B φ ∆ is strictly stronger (⊲) than S φ ∆ and E φ ∆ (and also S φ ∆ +E φ ∆ ). For example, when φ = AC and ∆ = ∆ = , we have B φ ∆ = BiSAC, S φ ∆ = SAC and S φ ∆ + E φ ∆ = 1-AC, and we know that BiSAC ⊲ 1-AC [4], and 1-AC ⊲ SAC [2].…”

“…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].…”