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.

翻译：在超矩形最大独立集问题中，给定一组$n$个（可能重叠的）$d$维轴对齐超矩形，目标是找到一个不重叠超矩形的最大基数子集。当$d=1$时，这对应于经典的区间调度问题，其中简单的贪心算法即可返回最优解。在离线设定下，对于$d$维超矩形，已知多项式时间$(\log n)^{O(d)}$近似算法。然而，该问题在在线设定中变得尤为困难，其中输入对象（超矩形）以对抗性顺序逐一出现，且算法需要在每个对象到达时立即做出不可撤销的决定，是否选择该对象同时保持可行性。即使对于区间调度，已知竞争比存在$\Omega(n)$下界。为规避这些负面结果，本文研究随机顺序到达模型下轴对齐超矩形的在线最大独立集问题，其中对手指定输入对象集合，这些对象随后以均匀随机顺序到达。从典型的秘书问题出发，随机顺序模型已受到广泛关注，用于研究超越最坏情况竞争分析的算法。令人惊讶的是，我们证明该问题在随机顺序模型下几乎匹配已知的最优离线近似保证（仅相差多对数因子）。具体而言，我们针对该模型中的$d$维超矩形给出一个简单的$(\log n)^{O(d)}$竞争比算法，其运行时间为$\tilde{O_d}(n)$。我们的方法还推广到更一般的对象，如$d$维胖对象和椭球体，在随机顺序模型中同样给出$(\log n)^{O(d)}$竞争比算法。