Perfect Hashing and CNF Encodings of Cardinality Constraints

Detecting Cardinality Constraints in CNF

“…Well known and sophisticated encodings are the partial sum encoding [1], totalizer encoding [6], the sequential counter encoding [31], BDDs [13] or sorting networks [13], cardinality networks [5], as well as the perfect hashing encoding [9]. As shown in [26], these specialized encodings produce much smaller CNF formulas compared to the binomial encoding.…”

A Lower Bound on CNF Encodings of the At-Most-One Constraint

“…This can be explained as follows. If n = m 2 or n = m(m + 1), then ϕ p n (x) is given by (6). In this case, the size of the candidate for ϕ p n+1 (x) given by (6) is at least |ϕ Clearly, the size of ϕ p n is at most 3 + |ϕ p n−1 | which is the size of (5) and using this, one can prove by induction on n that for all n ≥ 3, we have…”

“…Moreover these two clauses are binary and the other literal in each of these clauses is on an auxiliary variable. The product encoding described by Chen [10] has this form, see also (6). The aim of this section is to show that if n is large enough and ϕ is a minimum size PCE of AMO * n which is not in regular form, then there is a PCE ϕ ′ of AMO * n−1 of size at most |ϕ| − 3.…”

Automated Reencoding of Boolean Formulas

“…Encoding the BDD into CNF has been done by the Tseitin transformation. There are two small sized encodings for cardinality constraints that do not provide arc consistency: the parallel counter [29] and the hybrid perfect hashing function based encoding [46]. Although the properties of the latter are very nice, it cannot guarantee arc consistency for all possible cardinality constraints.…”

“…Although the properties of the latter are very nice, it cannot guarantee arc consistency for all possible cardinality constraints. The arc consistent variant of perfect hashing function based encoding [46] uses slightly more clauses than the sequential counter, but needs less auxiliary variables. Since we focus on the number of clauses, we do not consider this encoding.…”