The Budgeted Unique Coverage Problem and Color-Coding (Extended Abstract)

We show, by a non-trivial application of the color-coding method of Alon et if. [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.