Color spanning objects: Algorithms and hardness results

In this paper, we study the SHORTEST COLOR SPANNINGt- INTERVALS problem, and related generalizations, namely SMALLEST COLOR SPANNINGt- SQUARES and SMALLEST COLOR SPANNINGt- CIRCLES. The generic setting is the following: we are given n points in the plane (or on a line), each colored with one of k colors. For each color i we also have a demand s<inf>i</inf>. Given a budget t, we are required to find at most t objects (for example, intervals, squares, circles, etc.) that cover at least s<inf>i</inf> points of color i. Typically, the goal is to minimize the maximum perimeter or area. We provide exact algorithms for these problems for the cases of intervals, circles and squares, generalizing several known results. In the case of intervals, we provide a comprehensive understanding of the complexity landscape of the problem after taking several natural parameters into account. Given that the problem turns out to be W[1]-hard parameterized by the standard parameters, we introduce a new parameter, namely sparsity, and prove new hardness and tractability results in this context. For squares and circles, we use existing algorithms of one smallest color spanning object in order to design algorithms for getting t identical objects of minimum size whose union spans all the colors.