Interval Selection in Data Streams: Weighted Intervals and the Insertion-Deletion Setting

Abstract

We study the Interval Selection problem in data streams: Given a stream of n intervals on the line, the objective is to compute a largest possible subset of non-overlapping intervals using O(|OPT|) space, where |OPT| is the size of an optimal solution. Previous work gave a 3/2-approximation for unit-length and a 2-approximation for arbitrary-length intervals [Emek et al., ICALP'12]. We extend this line of work to weighted intervals as well as to insertion-deletion streams. Our results include:  
1) When considering weighted intervals, a (3/2+ε)-approximation can be achieved for unit intervals, but any constant factor approximation for arbitrary-length intervals requires space Ω(n).
2) In the insertion-deletion setting where intervals can both be added and deleted, we prove that, even without weights, computing a constant factor approximation for arbitrary-length intervals requires space Ω(n), whereas in the weighted unit-length intervals case a (2+ε)-approximation can be obtained.  Our lower bound results are obtained via reductions to the recently introduced Chained-Index communication problem, further demonstrating the strength of this problem in the context of streaming geometric independent set problems.