Online Maximum $k$-Interval Coverage Problem

TL;DR

This paper investigates the online maximum coverage problem on a line, proposing algorithms with near-optimal competitive ratios and analyzing their performance bounds in various settings.

Contribution

It introduces a dynamic programming approach for the offline problem and develops single- and double-threshold algorithms for the online problem, improving worst-case performance.

Findings

The double-threshold algorithm DOA outperforms the single-threshold SOA.

Lower bounds on competitive ratios are established for different problem settings.

Deterministic multi-threshold algorithms cannot surpass SOA's performance.

Abstract

We study the online maximum coverage problem on a line, in which, given an online sequence of sub-intervals (which may intersect among each other) of a target large interval and an integer $k$, we aim to select at most $k$ of the sub-intervals such that the total covered length of the target interval is maximized. The decision to accept or reject each sub-interval is made immediately and irrevocably (no preemption) right at the release timestamp of the sub-interval. We comprehensively study different settings of this problem regarding both the length of a released sub-interval and the total number of released sub-intervals. We first present lower bounds on the competitive ratio for the settings concerned in this paper, respectively. For the offline problem where the sequence of all the released sub-intervals is known in advance to the decision-maker, we propose a dynamic-programming-based optimal approach as the benchmark. For the online problem, we first propose a single-threshold-based deterministic algorithm SOA by adding a sub-interval if the added length exceeds a certain threshold, achieving competitive ratios close to the lower bounds, respectively. Then, we extend to a double-thresholds-based algorithm DOA, by using the first threshold for exploration and the second threshold (larger than the first one) for exploitation. With the two thresholds solved by our proposed program, we show that DOA improves SOA in the worst-case performance. Moreover, we prove that a deterministic algorithm that accepts sub-intervals by multi non-increasing thresholds cannot outperform even SOA.