Small Space Stream Summary for Matroid Center
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In the matroid center problem, which generalizes the k-center problem, we need to pick a set of centers that is an independent set of a matroid with rank r. We study this problem in streaming, where elements of the ground set arrive in the stream. We first show that any randomized one-pass streaming algorithm that computes a better than Δ-approximation for partition-matroid center must use Ω(r^2) bits of space, where Δ is the aspect ratio of the metric and can be arbitrarily large. This shows a quadratic separation between matroid center and k-center, for which the Doubling algorithm gives an 8-approximation using O(k)-space and one pass. To complement this, we give a one-pass algorithm for matroid center that stores at most O(r^2(1/ε)/ε) points (viz., stream summary) among which a (7+ε)-approximate solution exists, which can be found by brute force, or a (17+ε)-approximation can be found with an efficient algorithm. If we are allowed a second pass, we can compute a (3+ε)-approximation efficiently. We also consider the problem of matroid center with z outliers and give a one-pass algorithm that outputs a set of O((r^2+rz)(1/ε)/ε) points that contains a (15+ε)-approximate solution. Our techniques extend to knapsack center and knapsack center with outliers in a straightforward way, and we get algorithms that use space linear in the size of a largest feasible set (as opposed to quadratic space for matroid center).
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