Abstract
In this paper we consider the multidimensional binary vector assignment problem. An input of this problem is defined by m disjoint multisets \(V^1, V^2, \ldots , V^m\), 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 \(V^i\). 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
, and the restriction of this problem where every vector has at most c zeros by
.
was only known to be
-hard, even for 
. We show that, assuming the unique games conjecture, it is 
-hard to 
-approximate
for any fixed
and
. This result is tight as any solution is a
-approximation. We also prove without assuming UGC that
is
-hard even for
. Finally, we show that
is polynomial-time solvable for fixed
(which cannot be extended to
).
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Notes
i.e.admits an algorithm in
for an arbitrary function f.Incriminated problems are
and
for an arbitrary function f.
and
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This work is supported by the Interuniversity Attraction Poles Programme P7/36 \(\ll {\hbox {COMEX}} \gg \) initiated by the Belgian Science Policy Office.
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Bougeret, M., Duvillié, G. & Giroudeau, R. Approximability and exact resolution of the multidimensional binary vector assignment problem. J Comb Optim 36, 1059–1073 (2018). https://doi.org/10.1007/s10878-018-0276-8
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DOI: https://doi.org/10.1007/s10878-018-0276-8