Search Results

We present algorithms for the Max Coverage and Max Unique Coverage problems in the data stream model. The input to both problems are m subsets of a universe of size n and a value k ∈ [m]. In Max Coverage, the problem is to find a collection of at most k sets such that the number of elements covered by at least one set is maximized. In Max Unique Coverage, the problem is to find a collection of at most k sets such that the number of elements covered by exactly one set is maximized. These problems are closely related to a range of graph problems including matching, partial vertex cover, and capacitated maximum cut. In the data stream model, we assume k is given and the sets are revealed online. Our goal is to design single-pass algorithms that use space that is sublinear in the input size. Our main algorithmic results are: - If the sets have size at most d, there exist single-pass algorithms using O(d^{d+1} k^d) space that solve both problems exactly. This is optimal up to polylogarithmic factors for constant d. - If each element appears in at most r sets, we present single pass algorithms using Õ(k² r/ε³) space that return a 1+ε approximation in the case of Max Coverage. We also present a single-pass algorithm using slightly more memory, i.e., Õ(k³ r/ε⁴) space, that 1+ε approximates Max Unique Coverage. In contrast to the above results, when d and r are arbitrary, any constant pass 1+ε approximation algorithm for either problem requires Ω(ε^{-2}m) space but a single pass O(ε^{-2}mk) space algorithm exists. In fact any constant-pass algorithm with an approximation better than e/(e-1) and e^{1-1/k} for Max Coverage and Max Unique Coverage respectively requires Ω(m/k²) space when d and r are unrestricted. En route, we also obtain an algorithm for a parameterized version of the streaming Set Cover problem.