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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 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.
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.
In particular, the result holds for any \(0.53183\le \alpha < \frac{2}{3}\).
- 4.
ProcedureA could also be developed without using recursion; however, relying on the bounded search tree technique simplifies the presentation.
- 5.
This claim follows from the definition of x and since \(\alpha <\displaystyle {\frac{1}{2-\frac{1}{M+1}}}\).
References
Balasubramanian, R., Fellows, M., Raman, V.: An improved fixed-parameter algorithm for vertex cover. Inf. Process. Lett. 65(3), 163–168 (1998)
Bonnet, E., Lampis, M., Paschos, V.T.: Time-approximation trade-offs for inapproximable problems. CoRR abs/1502.05828 (2015)
Bonnet, E., Paschos, V.T.: Sparsification and subexponential approximation. CoRR abs/1402.2843 (2014)
Boria, N., Della Croce, F., Paschos, V.T.: On the max min vertex cover Problem. In: Kaklamanis, C., Pruhs, K. (eds.) WAOA 2013. LNCS, vol. 8447, pp. 37–48. Springer, Heidelberg (2014)
Bourgeois, N., Escoffier, B., Paschos, V.T.: Approximation of max independent set, min vertex cover and related problems by moderately exponential algorithms. Discrete Appl. Math. 159(17), 1954–1970 (2011)
Brankovic, L., Fernau, H.: A novel parameterised approximation algorithm for minimum vertex cover. Theor. Comput. Sci. 511, 85–108 (2013)
Buss, J., Goldsmith, J.: Nondeterminism within P. SIAM J. Comput. 22(3), 560–572 (1993)
Chandran, L.S., Grandoni, F.: Refined memorization for vertex cover. Inf. Process. Lett. 93(3), 123–131 (2005)
Chapelle, M., Liedloff, M., Todinca, I., Villanger, Y.: Treewidth and pathwidth parameterized by the vertex cover number. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 232–243. Springer, Heidelberg (2013)
Chen, J., Kanj, I.A., Jia, W.: Vertex cover: Further observations and further improvements. J. Algorithms 41(2), 280–301 (2001)
Chen, J., Kanj, I.A., Xia, G.: Labeled search trees and amortized analysis: improved upper bounds for NP-hard problems. Algorithmica 43(4), 245–273 (2005)
Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theor. Comput. Sci. 411(40–42), 3736–3756 (2010)
Chen, J., Liu, L., Jia, W.: Improvement on vertex cover for low degree graphs. Networks 35(4), 253–259 (2000)
ChlebÃk, M., ChlebÃová, J.: Crown reductions for the minimum weighted vertex cover problem. Discrete Appl. Math. 156(3), 292–312 (2008)
Cygan, M., Dell, H., Lokshtanov, D., Marx, D., Nederlof, J., Okamoto, Y., Paturi, R., Saurabh, S., Wahlström, M.: On problems as hard as CNF-SAT. In: CCC, pp. 74–84 (2012)
Downey, R., Fellows, M.: Fundamentals of parameterized complexity. Springer, Heidelberg (2013)
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness II: on completeness for W[1]. Theor. Comput. Sci. 141(1–2), 109–131 (1995)
Downey, R.G., Fellows, M.R., Stege, U.: Parameterized complexity: a framework for systematically confronting computational intractability. DIMACS 49, 49–99 (1999)
Fellows, M.R., Jansen, B.M.P., Rosamond, F.A.: Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity. Eur. J. Comb. 34(3), 541–566 (2013)
Fellows, M.R., Kulik, A., Rosamond, F., Shachnai, H.: Parameterized approximation via fidelity preserving transformations. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 351–362. Springer, Heidelberg (2012)
Fomin, F.V., Gaspers, S., Saurabh, S.: Branching and treewidth based exact algorithms. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 16–25. Springer, Heidelberg (2006)
Fomin, F.V., Gaspers, S., Saurabh, S., Stepanov, A.A.: On two techniques of combining branching and treewidth. Algorithmica 54(2), 181–207 (2009)
Fomin, F.V., Liedloff, M., Montealegre, P., Todinca, I.: Algorithms parameterized by vertex cover and modular width, through potential maximal cliques. In: Ravi, R., Gørtz, I.L. (eds.) SWAT 2014. LNCS, vol. 8503, pp. 182–193. Springer, Heidelberg (2014)
Hurink, J., Nieberg, T.: Approximating minimum independent dominating sets in wireless networks. Inf. Process. Lett 109(2), 155–160 (2008)
Issac, D., Jaiswal, R.: An \(O^*(1.0821^n)\)-time algorithm for computing maximum independent set in graphs with bounded degree 3. CoRR abs/1308.1351 (2013)
Jansen, B.M.P., Bodlaender, H.L.: Vertex cover kernelization revisited - upper and lower bounds for a refined parameter. Theory Comput. Syst. 53(2), 263–299 (2013)
Chapelle, M., Liedloff, M., Todinca, I., Villanger, Y.: Treewidth and pathwidth parameterized by the vertex cover number. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 232–243. Springer, Heidelberg (2013)
Niedermeier, R., Rossmanith, P.: Upper bounds for vertex cover further improved. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 561–570. Springer, Heidelberg (1999)
Niedermeier, R., Rossmanith, P.: On efficient fixed-parameter algorithms for weighted vertex cover. J. Algorithms 47(2), 63–77 (2003)
Peiselt, T.: An iterative compression algorithm for vertex cover. Ph.D. thesis Friedrich-Schiller-Universität Jena, Germany (2007)
Razgon, I.: Faster computation of maximum independent set and parameterized vertex cover for graphs with maximum degree 3. JDA 7(2), 191–212 (2009)
Shachnai, H., Zehavi, M.: A multivariate framework for weighted FPT algorithms. CoRR abs/1407.2033 (2014)
Xiao, M.: A note on vertex cover in graphs with maximum degree 3. In: Thai, M.T., Sahni, S. (eds.) COCOON 2010. LNCS, vol. 6196, pp. 150–159. Springer, Heidelberg (2010)
Zehavi, M.: Maximization problems parameterized using their minimization versions: the case of vertex cover. CoRR abs/1503.06438 (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-662-48054-0_49
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-48053-3
Online ISBN: 978-3-662-48054-0
eBook Packages: Computer ScienceComputer Science (R0)