We study the algorithmic problem of finding a large independent set in the Erd{ö}s-Rényi random graph $G(n,p)$. For constant $p$ and $b=1/(1-p)$, the largest independent set has size $2\log_b n$, while a simple greedy algorithm - revealing vertices sequentially and making decisions based only on previously seen vertices - finds an independent set of size $\log_b n$. In his seminal 1976 paper, Karp challenged to either improve this guarantee or establish its hardness. Decades later, this problem remains open - one of the most prominent algorithmic problems in the theory of random graphs. In this paper, we establish that a broad class of online algorithms fails to find an independent set of size $(1+ε)\log_b n$ whp. This class includes Karp's algorithm as a special case, and extends it by allowing the algorithm to query exceptional edges, not yet "seen" by the algorithm. Our lower bound holds for $p\in [d/n,1-n^{-1/d}]$. In the dense regime (constant $p$), we also prove that our result is asymptotically tight with respect to the number of exceptional edges queried, by designing an online algorithm which beats the half-optimality threshold when the number of exceptional edges slightly exceeds our bound. Our result provides evidence for the algorithmic hardness of Karp's problem, by supporting the conjectured optimality of the greedy algorithm and establishing it within the class of online algorithms. Our proof relies on a refined analysis of the geometric structure of large independent sets, establishing a variant of the Overlap Gap Property (OGP). While OGP has predominantly served as a barrier to stable algorithms, online algorithms are inherently unstable, necessitating new ideas. Our proof refines the OGP framework by incorporating several new ideas (including temporal interpolation paths and stopping-times) that we expect to be useful for other online models.

翻译：我们研究了在Erdős-Rényi随机图$G(n,p)$中寻找大规模独立集的算法问题。对于常数$p$和$b=1/(1-p)$，最大独立集的规模为$2\log_b n$，而一个简单的贪心算法——按顺序揭示顶点并仅基于已见顶点进行决策——能找到规模为$\log_b n$的独立集。在其1976年的开创性论文中，Karp提出挑战：要么改进这一保证，要么证明其硬度。数十年后，该问题仍然悬而未决，成为随机图理论中最突出的算法问题之一。本文证明，一大类在线算法以高概率无法找到规模为$(1+ε)\log_b n$的独立集。该类算法以Karp算法为特例，并通过允许算法查询尚未被算法"观测"的特殊边进行扩展。我们的下界适用于$p\in [d/n,1-n^{-1/d}]$。在稠密区域（常数$p$），我们还通过设计一种在线算法证明，当特殊边数量略微超过我们的界限时，该算法能够突破半最优性阈值，从而表明我们的结果在查询特殊边数量方面是渐近紧的。我们的结果为Karp问题的算法硬度提供了证据，支持了贪心算法最优性的猜想，并在在线算法类中确立了该结论。证明基于对大规模独立集几何结构的精细分析，建立了重叠间隙性质（OGP）的一个变体。虽然OGP主要作为稳定算法的障碍，但在线算法本质是不稳定的，这需要新的思路。我们的证明通过引入若干新思想（包括时间插值路径和停时）改进了OGP框架，这些思想预期对其他在线模型具有借鉴价值。