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.
Highlights
Several hard decision problems can be efficiently reduced to the Boolean satisfiability (SAT) problem and tried to be solved by recently-developed SAT-solvers
We show that multi-column selection networks are superior to standard selection networks previously proposed in the literature, in context of translating cardinality constraints into propositional formulas
Example 3 In Fig. 2 we present a schema of 4-Odd-Even Selection Network, which selects 3 largest elements from the input 01100000001
Summary
Several hard decision problems can be efficiently reduced to the Boolean satisfiability (SAT) problem and tried to be solved by recently-developed SAT-solvers. Xn are Boolean literals (that is, variables or their negations), ∼ is a relation from the set {} and k ∈ N. Such cardinality constraints appear naturally in formulations of different real-world problems including cumulative scheduling [20], timetabling [3] or formal hardware verification [8]. In a direct encoding of a cardinality constraint x1 + x2 + · · · + xn < k one can take all subsets of X = {x1, . The direct encoding is quite efficient for very small values of k and n, but for larger parameters another approach should be used
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