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Approximability and exact resolution of the multidimensional binary vector assignment problem

par Bougeret, Marin;Duvillié, Guillerme ;Giroudeau, Rodolphe
Référence Journal of combinatorial optimization, 36, 3, page (1059-1073)
Publication Publié, 2018-10

Article révisé par les pairs

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Résumé :

In this paper we consider the multidimensional binary vector assignment problem. An input of this problem is defined by m disjoint multisets V1, V2, … , Vm, each composed of n binary vectors of size p. An output is a set of n disjoint m-tuples of vectors, where each m-tuple is obtained by picking one vector from each multiset Vi. To each m-tuple we associate a p dimensional vector by applying the bit-wise AND operation on the m vectors of the tuple. The objective is to minimize the total number of zeros in these n vectors. We denote this problem by [InlineMediaObject not available: see fulltext.], and the restriction of this problem where every vector has at most c zeros by [InlineMediaObject not available: see fulltext.]. [InlineMediaObject not available: see fulltext.] was only known to be [InlineMediaObject not available: see fulltext.]-hard, even for [InlineEquation not available: see fulltext.]. We show that, assuming the unique games conjecture, it is [InlineEquation not available: see fulltext.]-hard to [InlineEquation not available: see fulltext.]-approximate [InlineMediaObject not available: see fulltext.] for any fixed [InlineMediaObject not available: see fulltext.] and [InlineMediaObject not available: see fulltext.]. This result is tight as any solution is a [InlineMediaObject not available: see fulltext.]-approximation. We also prove without assuming UGC that [InlineMediaObject not available: see fulltext.] is [InlineMediaObject not available: see fulltext.]-hard even for [InlineMediaObject not available: see fulltext.]. Finally, we show that [InlineMediaObject not available: see fulltext.] is polynomial-time solvable for fixed [InlineMediaObject not available: see fulltext.] (which cannot be extended to [InlineMediaObject not available: see fulltext.]).