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Some Results on Incremental Vertex Cover Problem

  • Wenqiang Dai
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6124)

Abstract

In the classical k-vertex cover problem, we wish to find a minimum weight set of vertices that covers at least k edges. In the incremental version of the k-vertex cover problem, we wish to find a sequence of vertices, such that if we choose the smallest prefix of vertices in the sequence that covers at least k edges, this solution is close in value to that of the optimal k-vertex cover solution. The maximum ratio is called competitive ratio. Previously the known upper bound of competitive ratio was 4α, where α is the approximation ratio of the k-vertex cover problem. And the known lower bound was 1.36 unless P = NP, or 2 − ε for any constant ε assuming the Unique Game Conjecture. In this paper we present some new results for this problem. Firstly we prove that, without any computational complexity assumption, the lower bound of competitive ratio of incremental vertex cover problem is φ, where \(\phi=\frac{\sqrt{5}+1}{2}\approx 1.618\) is the golden ratio. We then consider the restricted versions where k is restricted to one of two given values(Named 2-IVC problem) and one of three given values(Named 3-IVC problem). For 2-IVC problem, we give an algorithm to prove that the competitive ratio is at most φα. This incremental algorithm is also optimal for 2-IVC problem if we are permitted to use non-polynomial time. For the 3-IVC problem, we give an incremental algorithm with ratio factor \((1+\sqrt{2})\alpha\).

Keywords

Approximation Algorithm Approximation Ratio Competitive Ratio Cover Problem Vertex Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Wenqiang Dai
    • 1
  1. 1.School of Management and EconomicsUniversity of Electronic Science and Technology of ChinaChengduP. R.  China

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