A tight lower bound for the hardness of clutters

Vahan Mkrtchyan () and Hovhannes Sargsyan ()
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Vahan Mkrtchyan: Yerevan State University
Hovhannes Sargsyan: Yerevan State University

Journal of Combinatorial Optimization, 2018, vol. 35, issue 1, No 3, 25 pages

Abstract: 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.

Keywords: Clutter; Hardness; Independent set; Maximal independent set; Primary 05C69; Secondary 05C70; 05C15 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10878-017-0151-z

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