Space Constrained Dynamic Covering

In this paper, we identify a fundamental algorithmic problem that we term space-constrained dynamic covering (SCDC), arising in many modern-day web applications, including ad-serving and online recommendation systems in eBay and Netflix. Roughly speaking, SCDC applies two restrictions to the well-studied Max-Coverage problem [9]: Given an integer k, X = {1, 2, . . . , n} and I = {S1, . . . , Sm}, Si ⊆ X , find J ⊆ I, such that |J | ≤ k and ( ⋃ S∈J S) is as large as possible. The two restrictions applied by SCDC are: (1) Dynamic: At query-time, we are given a query Q ⊆ X , and our goal is to find J such that Q ⋂ ( ⋃ S∈J S) is as large as possible; (2) Spaceconstrained: We don’t have enough space to store (and process) the entire input; specifically, we have o(mn), sometimes even as little as O((m + n)polylog(mn)) space. The goal of SCDC is to maintain a small data structure so as to answer most dynamic queries with high accuracy. We present algorithms and complexity results for SCDC . We present deterministic and probabilistic neartight upper and lower bounds on the approximation ratio of SCDC as a function of the amount of space available. Our lower bound results show that to obtain constantfactor approximations we need Ω(mn) space. Fortunately, our upper bounds present an explicit tradeoff between space and approximation ratio, allowing us to determine the amount of space needed to guarantee certain accuracy. Finally, we study a practical subclass of SCDC allowing more accurate solutions despite limited storage space.