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International Symposium on Mathematical Foundations of Computer Science

MFCS 2015: Mathematical Foundations of Computer Science 2015 pp 589–600Cite as

Maximum Minimal Vertex Cover Parameterized by Vertex Cover

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9235))

Abstract

The parameterized complexity of problems is often studied with respect to the size of their optimal solutions. However, for a maximization problem, the size of the optimal solution can be very large, rendering algorithms parameterized by it inefficient. Therefore, we suggest to study the parameterized complexity of maximization problems with respect to the size of the optimal solutions to their minimization versions. We examine this suggestion by considering the Maximum Minimal Vertex Cover (MMVC) problem, whose minimization version, Vertex Cover, is one of the most studied problems in the field of Parameterized Complexity. Our main contribution is a parameterized approximation algorithm for MMVC, including its weighted variant. We also give conditional lower bounds for the running times of algorithms for MMVC and its weighted variant.

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Notes

  1. 1.

    For example, they show that one can guarantee the approximation ratios 0.1 and 0.4 in times \({\mathcal O}^*(1.162^{opt})\) and \({\mathcal O}^*(1.552^{opt})\), respectively.

  2. 2.

    Indeed, \(x \approx 0.11964\) if \(\frac{1}{x^x(1-x)^{1-x}}= 3^{\frac{1}{3}}\), and \(x=1-\frac{1-\alpha }{M(2\alpha -1)+1-\alpha }=\frac{1}{2}\) if \(\alpha =\frac{1}{2-\frac{1}{M+1}}\).

  3. 3.

    In particular, the result holds for any \(0.53183\le \alpha < \frac{2}{3}\).

  4. 4.

    ProcedureA could also be developed without using recursion; however, relying on the bounded search tree technique simplifies the presentation.

  5. 5.

    This claim follows from the definition of x and since \(\alpha <\displaystyle {\frac{1}{2-\frac{1}{M+1}}}\).

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Authors and Affiliations

  1. Department of Computer Science, Technion IIT, 32000, Haifa, Israel

    Meirav Zehavi

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  1. Meirav Zehavi
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Correspondence to Meirav Zehavi .

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Editors and Affiliations

  1. University of Rome "Tor Vergata", Rome, Italy

    Giuseppe F. Italiano

  2. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

  3. University of Edinburgh, Edinburgh, United Kingdom

    Donald T. Sannella

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Zehavi, M. (2015). Maximum Minimal Vertex Cover Parameterized by Vertex Cover. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_49

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  • DOI: https://doi.org/10.1007/978-3-662-48054-0_49

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