Space-Constrained Interval Selection

TL;DR

This paper introduces streaming algorithms for the interval selection problem, achieving near-optimal approximation ratios with limited space, and establishes lower bounds showing these are essentially the best possible.

Contribution

It presents the first deterministic 2-approximation streaming algorithm and a 3/2-approximation for proper intervals, along with tight space complexity bounds.

Findings

Deterministic 2-approximation algorithm developed

Improved 3/2-approximation for proper intervals

Proven space lower bounds close to the algorithms' performance

Abstract

We study streaming algorithms for the interval selection problem: finding a maximum cardinality subset of disjoint intervals on the line. A deterministic 2-approximation streaming algorithm for this problem is developed, together with an algorithm for the special case of proper intervals, achieving improved approximation ratio of 3/2. We complement these upper bounds by proving that they are essentially best possible in the streaming setting: it is shown that an approximation ratio of $2 - \epsilon$ (or $3 / 2 - \epsilon$ for proper intervals) cannot be achieved unless the space is linear in the input size. In passing, we also answer an open question of Adler and Azar \cite{AdlerAzar03} regarding the space complexity of constant-competitive randomized preemptive online algorithms for the same problem.