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.
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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
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DOI: https://doi.org/10.1007/978-3-642-03351-3_29
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