We study the relationship between the approximation factor for the Set-Cover problem and the parameters Δ : the maximum cardinality of any subset, and k : the maximum number of subsets containing any element of the ground set. We show an LP rounding based approximation of \((k-1)(1-e^{-\frac{\ln \Delta}{k-1}}) +1\), which is substantially better than the classical algorithms in the range k ≈ ln Δ, and also improves on related previous works [19,22]. For the interesting case when k = θ(logΔ) we also exhibit an integrality gap which essentially matches our approximation algorithm. We also prove a hardness of approximation factor of \(\Omega\left(\frac{\log \Delta}{(\log\log \Delta)^2}\right)\) when k = θ(logΔ). This is the first study of the hardness factor specifically for this range of k and Δ, and improves on the only other such result implicitly proved in [18].


Vertex Cover Linear Programming Relaxation Performance Guarantee Satisfying Assignment Hardness Factor 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Rishi Saket
    • 1
  • Maxim Sviridenko
    • 2
  1. 1.IBM T.J. Watson Research CenterUSA
  2. 2.Department of Computer ScienceUniversity of WarwickUK

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