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Approximability and exact resolution of the multidimensional binary vector assignment problem

Author

Listed:
  • Marin Bougeret

    (LIRMM)

  • Guillerme Duvillié

    (LIRMM
    Université libre de Bruxelles)

  • Rodolphe Giroudeau

    (LIRMM)

Abstract

In this paper we consider the multidimensional binary vector assignment problem. An input of this problem is defined by m disjoint multisets $$V^1, V^2, \ldots , V^m$$ V 1 , V 2 , … , 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 $$V^i$$ 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 ).

Suggested Citation

  • Marin Bougeret & Guillerme Duvillié & Rodolphe Giroudeau, 2018. "Approximability and exact resolution of the multidimensional binary vector assignment problem," Journal of Combinatorial Optimization, Springer, vol. 36(3), pages 1059-1073, October.
    • Handle: RePEc:spr:jcomop:v:36:y:2018:i:3:d:10.1007_s10878-018-0276-8
    DOI: 10.1007/s10878-018-0276-8
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