Abstract
We show, by a non-trivial application of the color-coding method of Alon et al.[2], that Budgeted Unique Coverage (a variant of Set Cover) is fixed-parameter tractable, answering an open problem posed in [13]. We also give improved fixed-parameter tractable algorithms for two special cases of Budgeted Unique Coverage: Unique Coverage (the unweighted version) and Budgeted Max Cut.
To derandomize our algorithms we use an interesting variation of k-perfect hash families known as (k,s)-hash families which were studied by Alon et al.[1] in the context of a class of codes called parent identifying codes [3]. In this setting, for every s-element subset S of the universe, and every k-element subset X of S, there exists a function that maps X injectively and maps the remaining elements of S into a different range.
We give several bounds on the size of (k,s)-hash families. We believe that our application of color-coding may be used for other problems and that this is the first application of (k,s)-hash families to a problem outside the domain of coding theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alon, N., Cohen, G., Krivelevich, M., Litsyn, S.: Generalized hashing and parent-identifying codes. Journal of Combinatorial Theory Series A 104(1), 207–215 (2003)
Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)
Barg, A., Cohen, G., Encheva, S., Kabatiansky, G., Zémor, G.: A hypergraph approach to the identifying parent property: The case of multiple parents. SIAM Journal of Discrete Mathematics 14(3), 423–431 (2001)
Chen, J., Lu, S., Sze, S.-H., Zhang, F.: Improved algorithms for path, matching, and packing problems. In: SODA, pp. 298–307 (2007)
Demaine, E.D., Hajiaghayi, M.T., Feige, U., Salavatipour, M.R.: Combination can be hard: approximability of the unique coverage problem. In: SODA, pp. 162–171 (2006)
Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, New York (1999)
Erlebach, T., van Leeuwen, E.J.: Approximating geometric coverage problems. In: SODA, pp. 1267–1276 (2008)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)
Guruswami, V., Trevisan, L.: The complexity of making unique choices: Approximating 1-in-k sat. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 99–110. Springer, Heidelberg (2005)
Khuller, S., Moss, A., Naor, J.: The budgeted maximum coverage problem. Inf. Process. Lett. 70(1), 39–45 (1999)
Mehlhorn, K.: On the program size of perfect and universal hash functions. In: Proceedings of the 23th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 170–175. IEEE, Los Alamitos (1982)
Moser, H., Misra, N., Raman, V., Saurabh, S., Sikdar, S.: The parameterized complexity of the Unique Coverage problem (2009) (manuscript)
Moser, H., Raman, V., Sikdar, S.: The parameterized complexity of the unique coverage problem. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 621–631. Springer, Heidelberg (2007)
Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford Lecture Series in Mathematics and its Applications, vol. 31. Oxford University Press, Oxford (2006)
Schmidt, J.P., Siegel, A.: The spatial complexity of oblivious k-probe hash functions. SIAM J. Computing 19(5), 775–786 (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Misra, N., Raman, V., Saurabh, S., Sikdar, S. (2009). The Budgeted Unique Coverage Problem and Color-Coding. In: Frid, A., Morozov, A., Rybalchenko, A., Wagner, K.W. (eds) Computer Science - Theory and Applications. CSR 2009. Lecture Notes in Computer Science, vol 5675. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03351-3_29
Download citation
DOI: https://doi.org/10.1007/978-3-642-03351-3_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03350-6
Online ISBN: 978-3-642-03351-3
eBook Packages: Computer ScienceComputer Science (R0)