Encoding cardinality constraints using multiway merge selection networks

Abstract

Boolean cardinality constraints (CCs) state that at most (at least, or exactly) k out of n propositional literals can be true. We propose a new, arc-consistent, easy to implement and efficient encoding of CCs based on a new class of selection networks. Several comparator networks have been recently proposed for encoding CCs and experiments have proved their efficiency (Abío et al. 2013, Asín et al. Constraints 12(2): 195-221, 2011, Codish and Zazon-Ivry 2010, Eén and Sörensson Boolean Modeling and Computation 2: 1 -26, 2006). In our construction we use the idea of the multiway merge sorting networks by Lee and Batcher (1995) that generalizes the technique of odd-even sorting ones by merging simultaneously more than two subsequences. The new selection network merges 4 subsequences in that way. Based on this construction, we can encode more efficiently comparators in the combine phase of the network: instead of encoding each comparator separately by 3 clauses and 2 additional variables, we propose an encoding scheme that requires 5 clauses and 2 variables on average for each pair of comparators. We also extend the model of comparator networks so that the basic components are not only comparators (2-sorters) but more general m-sorters, for m ∈ {2, 3, 4}, that can also be encoded efficiently. We show that with small overhead (regarding implementation complexity) we can achieve a significant improvement in SAT-solver runtime for many test cases. We prove that the new encoding is competitive to the other state-of-the-art encodings.

New Core-Guided and Hitting Set Algorithms for Multi-Objective Combinatorial Optimization

“…We are interested in removing from the feasible space solutions that are dominated by some other known feasible solution. In order to do this, we make use of unary counters [3,13,14] that have been used to implement efficient PB satisfiability solvers.…”

“…The Core-Guided algorithm proposed in Algorithm 1 uses the selection delimiter encoding [14] that has been shown to be more compact. Next, the selection delimiter encoding is extended to produce a unary encoding for each objective function.…”

Exact and approximate determination of the Pareto set using minimal correction subsets

“…Hence, in this paper, the actual encoding of the aforementioned equivalences is done only after encoding the value of the objective function f (x) into CNF. First, we use an encoding using selection networks (Karpinski and Piotrów, 2019) that has been shown to be more compact. Next, the selection networks encoding is extended such that a unary encoding is produced where…”

Abstract Cores in Implicit Hitting Set MaxSat Solving

“…We compare the new solvers to the base maxhs (MSE 2019 version) as well as to two other solvers: the MSE 2019 version of RC2 (rc2) [4,20], the best performing solver in both the weighted and unweighted track and a new solver in MSE 2019 called UWrMaxSat (UWr) [4,21]. Both implement the OLL algorithm [1,25] and differ mainly in how the cardinality constraints are encoded into cnf.…”