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Weighted Upper Edge Cover: Complexity and Approximability

  • Kaveh KhoshkhahEmail author
  • Mehdi Khosravian Ghadikolaei
  • Jérôme Monnot
  • Florian Sikora
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)

Abstract

Optimization problems consist of either maximizing or minimizing an objective function. Instead of looking for a maximum solution (resp. minimum solution), one can find a minimum maximal solution (resp. maximum minimal solution). Such “flipping” of the objective function was done for many classical optimization problems. For example, Minimum Vertex Cover becomes Maximum Minimal Vertex Cover, Maximum Independent Set becomes Minimum Maximal Independent Set and so on. In this paper, we propose to study the weighted version of Maximum Minimal Edge Cover called Upper Edge Cover, a problem having application in genomic sequence alignment. It is well-known that Minimum Edge Cover is polynomial-time solvable and the “flipped” version is NP-hard, but constant approximable. We show that the weighted Upper Edge Cover is much more difficult than Upper Edge Cover because it is not \(O(\frac{1}{n^{1/2-\varepsilon }})\) approximable, nor \(O(\frac{1}{\varDelta ^{1-\varepsilon }})\) in edge-weighted graphs of size n and maximum degree \(\varDelta \) respectively. Indeed, we give some hardness of approximation results for some special restricted graph classes such as bipartite graphs, split graphs and k-trees. We counter-balance these negative results by giving some positive approximation results in specific graph classes.

Keywords

Maximum minimal edge cover Graph optimization problem Computational complexity Approximability 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Kaveh Khoshkhah
    • 1
    Email author
  • Mehdi Khosravian Ghadikolaei
    • 2
  • Jérôme Monnot
    • 2
  • Florian Sikora
    • 2
  1. 1.Institute of Computer ScienceTartu UniversityTartuEstonia
  2. 2.Université Paris-Dauphine, PSL University, CNRS, LAMSADEParisFrance

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