Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams
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We study the Maximum Independent Set problem for geometric objects given in the data stream model. A set of geometric objects is said to be independent if the objects are pairwise disjoint. We consider geometric objects in one and two dimensions, i.e., intervals and disks. Let α be the cardinality of the largest independent set. Our goal is to estimate α in a small amount of space, given that the input is received as a one-pass turnstile stream. We also consider a generalization of this problem by assigning weights to each object and estimating β, the largest value of a weighted independent set. We provide the first algorithms for estimating α and β in turnstile streams. For unit-length intervals, we obtain a (2+ϵ)-approximation to α and β in poly((n)/ϵ) space. We also show a matching lower bound. For arbitrary-length intervals, we show any c-approximation to α, and thus also β, requires Ω(n^1/c/2^c) space. To this end, we introduce a new communication problem and lower bound its information complexity. In light of the lower bound we provide algorithms for estimating α for arbitrary-length intervals under a bounded intersection assumption. We also study the parameterized space complexity of estimating α and β, where the parameter is the ratio of maximum to minimum interval length. For unit-radius disks, we obtain a (8√(3)/π)-approximation to α and β in poly((n)/ϵ) space, which is closely related to the hexagonal circle packing constant.
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