N-level Modulo-Based CNF encodings of Pseudo-Boolean constraints for MaxSAT

SAT encodings for Pseudo-Boolean constraints together with at-most-one constraints

“…Clauses (6) state that i j=1 q j x j ≥ i−1 j=1 q j x j . Clauses (7) and (8) enforce that if a variable x i is true then its coefficient is added to the input of the next counter. Finally, Clauses (9) enforce that the sum never exceeds K.…”

“…In this section we first revisit the n-Level Modulo Totalizer encoding for PB constraints from [8] and in Subsection 7.2 we provide its generalization to encode PB(AMO) constraints. Subsection 7.3 gives some construction details.…”

“…The n-Level Modulo Totalizer (MTO) encoding generalises the Weighted Totalizer encoding which, as stated in [8], is essentially an equivalent definition to the GT encoding that was presented in a parallel work. For the sake of readability we unify the nomenclature of the MTO encoding with the one of the GT encoding, but the overall idea of the encoding summarised here and the resulting clauses are the same as in the original work.…”

“…Therefore, the GMTO encoding consists of the previous definition of the totalizer together with clauses ( 19)- (30). Note that assuming that an AMO constraint over each set X i is satisfied, the value of the leaf node associated to X i will be at least q j , where x j ∈ X i is the variable that is set to true, and therefore the encoding correctly evaluates n i=1 q i x i ≤ K. The MTO encoding requires O(nβλ 0 ) auxiliary variables and O(nβλ 2 0 ) clauses (assuming [8], while the GMTO encoding requires O(N βλ 0 ) auxiliary variables and O(N βλ 2 0 ) clauses.…”

“…There has been a great deal of work on encoding PB constraints to SAT, some of which is reviewed by Philipp and Steinke [3]. State-of-the-art encodings are based on Binary Decision Diagrams [4,5], Sequential Weight Counters [6], Generalized Totalizers [7,8], and Polynomial Watchdog schemes [9,10]. Atmost-one (AMO) constraints (i.e.…”