Improved Approximation Guarantees through Higher Levels of SDP Hierarchies

  • Authors:

  • Eden Chlamtac;Gyanit Singh

  • Affiliations:

  • Department of Computer Science, Princeton University, Princeton, USA NJ 08544;Departmen of Computer Science & Engineering, University of Washington, Seattle, USA WA 98195-2350

  • Venue:
  • APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
  • Year:
  • 2008

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Abstract

For every fixed 茂戮驴茂戮驴 0, we give an algorithm that, given an n-vertex 3-uniform hypergraph containing an independent set of size 茂戮驴n, finds an independent set of size $n^{\Omega(\gamma^2)}$. This improves upon a recent result of Chlamtac, which, for a fixed 茂戮驴 0, finds an independent set of size n茂戮驴in any 3-uniform hypergraph containing an independent set of size $(\frac12-\varepsilon)n$. The main feature of this algorithm is that, for fixed 茂戮驴, it uses the 茂戮驴(1/茂戮驴2)-level of a hierarchy of semidefinite programming (SDP) relaxations. On the other hand, we show that for at least one hierarchy which gives such a guarantee, 1/茂戮驴levels yield no non-trivial guarantee. Thus, this is a first SDP-based algorithm for which the approximation guarantee improves indefinitely as one uses progressively higher-level relaxations.