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A tight lower bound for the hardness of clutters

Author

Listed:
  • Vahan Mkrtchyan

    (Yerevan State University)

  • Hovhannes Sargsyan

    (Yerevan State University)

Abstract

A clutter (or antichain or Sperner family) L is a pair (V, E), where V is a finite set and E is a family of subsets of V none of which is a subset of another. Normally, the elements of V are called vertices of L, and the elements of E are called edges of L. A subset $$s_e$$ s e of an edge e of a clutter is recognizing for e, if $$s_e$$ s e is not a subset of another edge. The hardness of an edge e of a clutter is the ratio of the size of $$e\text {'s}$$ e 's smallest recognizing subset to the size of e. The hardness of a clutter is the maximum hardness of its edges. In this short note we prove a lower bound for the hardness of an arbitrary clutter. Our bound is asymptotically best-possible in a sense that there is an infinite sequence of clutters attaining our bound.

Suggested Citation

  • Vahan Mkrtchyan & Hovhannes Sargsyan, 2018. "A tight lower bound for the hardness of clutters," Journal of Combinatorial Optimization, Springer, vol. 35(1), pages 21-25, January.
    • Handle: RePEc:spr:jcomop:v:35:y:2018:i:1:d:10.1007_s10878-017-0151-z
    DOI: 10.1007/s10878-017-0151-z
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    References listed on IDEAS

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    1. Claussen, Jens Christian, 2007. "Offdiagonal complexity: A computationally quick complexity measure for graphs and networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 375(1), pages 365-373.
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