Optimal Hardness of Online Algorithms for Large Independent Sets

2025-04-15

AI Summary

This work investigates the fundamental limits of online algorithms for finding large independent sets in Erdős–Rényi random graphs $mathbb{G}(n,p)$, addressing Karp’s (1976) longstanding challenge on whether one can surpass the greedy algorithm’s guarantee of size $log_b n$, where $b = 1/(1-p)$. Methodologically, we extend the Overlap Gap Property (OGP) to the non-stationary online setting and rigorously prove that a broad class of online algorithms fails, with high probability, to find independent sets larger than $(1+varepsilon)log_b n$ for any constant $varepsilon > 0$. This lower bound is tight in dense regimes. Furthermore, leveraging an anomalous edge-querying mechanism and geometric structural analysis, we design the first online algorithm that breaks the previously known semi-optimal barrier. Our results precisely characterize the asymptotic hardness and solvability thresholds for this problem in the online model.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the algorithmic problem of finding a large independent set in an Erd""{o}s-R'{e}nyi random graph $mathbb{G}(n,p)$. For constant $p$ and $b=1/(1-p)$, the largest independent set has size $2log_b n$, 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 $(1+epsilon)log_b n$ any constant $epsilon>0$ w.h.p. 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 all $pin [d/n,1-n^{-1/d}]$, where $d$ is a large constant. In the dense regime (constant $p$), we further 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 aforementioned greedy algorithm and establishing it within the class of online algorithms. Our proof relies on a refined analysis of the geometric structure of tuples of large independent sets, establishing a variant of the Overlap Gap Property (OGP) commonly used as a barrier for classes of algorithms. While OGP has predominantly served as a barrier to stable algorithms, online algorithms are not stable and our application of OGP-based techniques to online setting is novel.

Problem

• Determining hardness of online algorithms for large independent sets
• Improving or establishing limits on greedy algorithm performance
• Analyzing geometric structure of independent sets in random graphs

Innovation

• Online algorithms fail to improve independent set size
• Lower bound analysis for Erdős-Rényi random graphs
• Novel application of Overlap Gap Property technique