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A Lower Bound on CNF Encodings of the At-most-one Constraint

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    0494392 - ÚI 2020 RIV NL eng J - Journal Article
    Kučera, P. - Savický, Petr - Vorel, V.
    A Lower Bound on CNF Encodings of the At-most-one Constraint.
    Theoretical Computer Science. Roč. 762, March (2019), s. 51-73. ISSN 0304-3975. E-ISSN 1879-2294
    R&D Projects: GA ČR GBP202/12/G061
    Grant - others: GA ČR(CZ) GA15-15511S
    Institutional support: RVO:67985807
    Keywords : Knowledge compilation * Cardinality constraint * At most one constraint * Propagation complete encoding
    OECD category: Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
    Impact factor: 0.747, year: 2019
    Method of publishing: Limited access

    Constraint 'at most one' is a basic cardinality constraint which requires that at most one of its n boolean inputs is set to 1. This constraint is widely used when translating a problem into a conjunctive normal form (CNF) and we investigate its CNF encodings suitable for this purpose. An encoding differs from a CNF representation of a function in that it can use auxiliary variables. We are especially interested in propagation complete encodings which have the property that unit propagation is strong enough to enforce consistency on input variables. We show a lower bound on the number of clauses in any propagation complete encoding of the 'at most one' constraint. The lower bound almost matches the size of the best known encodings. We also study an important case of 2-CNF encodings where we show a slightly better lower bound. The lower bound holds also for a related 'exactly one' constraint.
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