On k-column sparse packing programs

  • Authors:

  • Nikhil Bansal;Nitish Korula;Viswanath Nagarajan;Aravind Srinivasan

  • Affiliations:

  • IBM T.J. Watson Research Center, Yorktown Heights, NY;Dept. of Computer Science, University of Illinois, Urbana, IL;IBM T.J. Watson Research Center, Yorktown Heights, NY;Dept. of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD

  • Venue:
  • IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
  • Year:
  • 2010

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Abstract

We consider the class of packing integer programs (PIPs) that are column sparse, where there is a specified upper bound k on the number of constraints that each variable appears in. We give an improved (ek+o(k))-approximation algorithm for k-column sparse PIPs. Our algorithm is based on a linear programming relaxation, and involves randomized rounding combined with alteration. We also show that the integrality gap of our LP relaxation is at least 2k−1; it is known that even special cases of k-column sparse PIPs are $\Omega(\frac{k}{\log k})$-hard to approximate. We generalize our result to the case of maximizing monotone submodular functions over k-column sparse packing constraints, and obtain an $\smash{\left(\frac{e^2k}{e-1} + o(k) \right)}$-approximation algorithm. In obtaining this result, we prove a new property of submodular functions that generalizes the fractionally subadditive property, which might be of independent interest.