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 $\Omega(n)$ 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.

翻译：在超矩形最大独立集问题中，我们给定一组包含 $n$ 个（可能重叠的）$d$ 维轴对齐超矩形，目标是找到具有最大基数的互不重叠超矩形子集。当 $d=1$ 时，该问题对应于经典的区间调度问题，其中简单的贪心算法即可获得最优解。在离线环境下，对于 $d$ 维超矩形，已知存在多项式时间的 $(\log n)^{O(d)}$ 近似算法。然而，该问题在在线环境下变得极具挑战性——输入对象（超矩形）以对抗性顺序逐个出现，且每个对象到达时，算法必须立即且不可撤销地决定是否选择该对象，同时保持解的可行性。即使对于区间调度问题，竞争比也存在 $\Omega(n)$ 的下界。为规避这些负面结果，本研究在随机顺序到达模型中探讨在线轴对齐超矩形最大独立集问题，其中对抗方指定输入对象集合，而对象以均匀随机顺序到达。从经典的秘书问题出发，随机顺序模型在突破最坏情况竞争分析的研究中受到广泛关注。令人惊讶的是，我们发现该问题在随机顺序模型中的表现几乎能达到已知最优离线近似保证，仅相差多对数因子。具体而言，我们针对该模型中的 $d$ 维超矩形提出了一个简单的 $(\log n)^{O(d)}$ 竞争算法，其时间复杂度为 $\tilde{O_d}(n)$。我们的方法还能为随机顺序模型中更一般的对象（如 $d$ 维胖对象和椭球体）生成 $(\log n)^{O(d)}$ 竞争算法。此外，我们的保证以高概率成立。