Random-Order Online Independent Set of Intervals and Hyperrectangles

TL;DR

This work studies the online maximum independent set of axis-aligned hyperrectangles in the random-order arrival model, and gives a simple $(log n)^{O(d)}$-competitive algorithm for $d-dimensional hyperrectangles in this model, which runs in $\tilde{O_d}(n)$ time.

Abstract

In the Maximum Independent Set of Hyperrectangles problem, we are given a set of $n$ (possibly overlapping) $d$-dimensional axis-aligned hyperrectangles, and the goal is to find a subset of non-overlapping hyperrectangles of maximum cardinality. For $d=1$, this corresponds to the classical Interval Scheduling problem, where a simple greedy algorithm returns an optimal solution. In the offline setting, for $d$-dimensional hyperrectangles, polynomial time $(\log n)^{O(d)}$-approximation algorithms are known. However, the problem becomes notably challenging in the online setting, where the input objects (hyperrectangles) appear one by one in an adversarial order, and on the arrival of an object, the algorithm needs to make an immediate and irrevocable decision whether or not to select the object while maintaining the feasibility. Even for interval scheduling, an $Ω(n)$ lower bound is known on the competitive ratio. To circumvent these negative results, in this work, we study the online maximum independent set of axis-aligned hyperrectangles in the random-order arrival model, where the adversary specifies the set of input objects which then arrive in a uniformly random order. Starting from the prototypical secretary problem, the random-order model has received significant attention to study algorithms beyond the worst-case competitive analysis. Surprisingly, we show that the problem in the random-order model almost matches the best-known offline approximation guarantees, up to polylogarithmic factors. In particular, we give a simple $(\log n)^{O(d)}$-competitive algorithm for $d$-dimensional hyperrectangles in this model, which runs in $\tilde{O_d}(n)$ time. Our approach also yields $(\log n)^{O(d)}$-competitive algorithms in the random-order model for more general objects such as $d$-dimensional fat objects and ellipsoids. Furthermore, our guarantees hold with high probability.

Figures (6)

• Figure 1: Instance $\mathcal{I}\xspace$
• Figure 2: For $d=2$, $\Delta=3$, a piercing set of $\mathcal{H}\xspace$ can be formed by placing 36 points inside each $\mathcal{H}\xspace'$
• Figure 3: Values of $\sigma$ for some geometric objects (a) $\sigma=7/4$ for equilateral triangle (b) $\sigma=\sqrt{2}$ for ellipse (c) $\sigma=3$ for star with all boundary edges of equal length
• Figure 4: Figure for \ref{['lem:fatobjects']}
• Figure 5: Non-uniform scaling. The vertical bars on the axis represent the $t$ starting points of the intervals in $\mathcal{I}\xspace_0$. After scaling, they become evenly spaced.

Theorems & Definitions (31)

• Lemma 1: Hypergeometric concentration
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Definition 1
• Lemma 4
• proof
• proof : Proof of claim.