Approximating the \(k\)-Set Packing Problem by Local Improvements

  • Martin Fürer
  • Huiwen YuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8596)


We study algorithms based on local improvements for the \(k\)-Set Packing problem. The well-known local improvement algorithm by Hurkens and Schrijver [14] has been improved by Sviridenko and Ward [15] from \(\frac{k}{2}+\epsilon \) to \(\frac{k+2}{3}\), and by Cygan [7] to \(\frac{k+1}{3}+\epsilon \) for any \(\epsilon >0\). In this paper, we achieve the approximation ratio \(\frac{k+1}{3}+\epsilon \) for the \(k\)-Set Packing problem using a simple polynomial-time algorithm based on the method by Sviridenko and Ward [15]. With the same approximation guarantee, our algorithm runs in time singly exponential in \(\frac{1}{\epsilon ^2}\), while the running time of Cygan’s algorithm [7] is doubly exponential in \(\frac{1}{\epsilon }\). On the other hand, we construct an instance with locality gap \(\frac{k+1}{3}\) for any algorithm using local improvements of size \(O(n^{1/5})\), where \(n\) is the total number of sets. Thus, our approximation guarantee is optimal with respect to results achievable by algorithms based on local improvements.


\(k\)-set packing Tail change Local improvement Color coding 


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringThe Pennsylvania State UniversityUniversity ParkUSA

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