The maximum independent set problem is a classical NP-hard problem in theoretical computer science. In this work, we study a special case where the family of graphs considered is restricted to intersection graphs of sets of axis-aligned hyperrectangles and the input is provided in an online fashion. We prove bounds on the competitive ratio of an optimal online algorithm under the adaptive offline, adaptive online, and oblivious adversary models, for several classes of hyperrectangles and restrictions on the order of the input. We are the first to present results on this problem under the oblivious adversary model. We prove bounds on the competitive ratio for unit hypercubes, $\sigma$-bounded hypercubes, unit-volume hypercubes, arbitrary hypercubes, and arbitrary hyperrectangles, in both arbitrary and non-dominated order. We are also the first to present results under the adaptive offline and adaptive online adversary models with input in non-dominated order, proving bounds on the competitive ratio for the same classes of hyperrectangles; for input in arbitrary order, we present the first results on $\sigma$-bounded hypercubes, unit-volume hyperrectangles, arbitrary hypercubes, and arbitrary hyperrectangles. For input in dominating order, we show that the performance of the naive greedy algorithm matches the performance of an optimal offline algorithm in all cases. We also give lower bounds on the competitive ratio of a probabilistic greedy algorithm under the oblivious adversary model. We conclude by discussing several promising directions for future work.

翻译：最大独立集问题是理论计算机科学中的经典NP难问题。本文研究了一类特殊情况：所考虑的图族被限制为轴对齐超矩形交集图，且输入以在线方式提供。针对自适应离线、自适应在线和 oblivious 对手模型，我们证明了多种超矩形类别及输入顺序限制下最优在线算法的竞争比界。我们是首个在 oblivious 对手模型下给出该问题结果的研究。我们证明了单位超立方体、σ-有界超立方体、单位体积超立方体、任意超立方体和任意超矩形在任意顺序和非支配顺序下的竞争比界。我们也是首个在输入为非支配顺序时，针对自适应离线与自适应在线对手模型给出结果的研究，证明了相同超矩形类别的竞争比界；对于任意顺序输入，我们首次给出了σ-有界超立方体、单位体积超矩形、任意超立方体和任意超矩形的结果。对于支配顺序输入，我们证明朴素贪心算法的性能在所有情况下均与最优离线算法匹配。我们还给出了 oblivious 对手模型下概率贪心算法竞争比的下界。最后，我们讨论了若干有前景的未来研究方向。