In the classical online model, the maximum independent set problem admits an $Ω(n)$ lower bound on the competitive ratio even for interval graphs, motivating the study of the problem under additional assumptions. We first study the problem on graphs with a bounded independent kissing number $ζ$, defined as the size of the largest induced star in the graph minus one. We show that a simple greedy algorithm, requiring no geometric representation, achieves a competitive ratio of $ζ$. Moreover, this bound is optimal for deterministic online algorithms and asymptotically optimal for randomized ones. This extends previous results from specific geometric graph families to more general graph classes. Since this bound rules out further improvements through randomization alone, we investigate the power of randomization with access to geometric representation. When the geometric representation of the objects is known, we present randomized online algorithms with improved guarantees. For unit ball graphs in $\mathbb{R}^3$, we present an algorithm whose expected competitive ratio is strictly smaller than the deterministic lower bound implied by the independent kissing number. For $α$-fat objects and for axis-aligned hyper-rectangles in $\mathbb{R}^d$ with bounded diameters, we obtain algorithms with expected competitive ratios that depend polylogarithmically on the ratio between the maximum and minimum object diameters. In both cases, the randomized lower bound implied by the independent kissing number grows polynomially with the ratio between the maximum and minimum object diameters, implying substantial performance guarantees for our algorithms.

翻译：在经典在线模型中，最大独立集问题即使在区间图上也具有Ω(n)的竞争比下界，这促使我们在额外假设下研究该问题。我们首先研究具有有界独立亲吻数ζ的图上的问题，其中ζ定义为图中最大诱导星的大小减一。我们证明，一个无需几何表示的简单贪心算法可以达到ζ的竞争比。此外，这一界对确定性在线算法是最优的，对随机算法是渐近最优的。这将对特定几何图族的先前结果推广到更一般的图类。由于这一界排除了仅通过随机化进一步改进的可能性，我们研究了在可访问几何表示的情况下随机化的能力。当已知对象的几何表示时，我们提出了具有改进保证的随机在线算法。对于R³中的单位球图，我们提出了一种算法，其期望竞争比严格小于由独立亲吻数隐含的确定性下界。对于α-胖物体以及Rᵈ中具有有界直径的轴对齐超矩形，我们获得了期望竞争比随物体最大与最小直径之比呈多对数增长的算法。在这两种情况下，由独立亲吻数隐含的随机下界随物体最大与最小直径之比呈多项式增长，这意味着我们的算法具有显著性能保证。