Optimal Hardness of Online Algorithms for Large Independent Sets

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.

💡 Research Summary

The paper investigates the long‑standing problem posed by Karp (1976) of finding a large independent set in the dense Erdős–Rényi random graph G(n,p). In the dense regime (constant p) the maximum independent set has size about 2·log_b n (with b = 1/(1‑p)), while the classic greedy algorithm—revealing vertices one by one and making decisions based only on already seen edges—achieves only about log_b n, i.e., half of the optimum. The central question is whether any polynomial‑time algorithm can surpass this “half‑optimal” barrier and achieve (1+ε)·log_b n for some ε>0.

The authors focus on a broad class of online algorithms. An online algorithm processes vertices in order t=1,…,n; at step t it must decide immediately whether to include vertex t in the independent set, using only the subgraph induced by vertices {1,…,t} together with a limited set of “future‑edge” queries. The total number of queried edges whose both endpoints end up in the final independent set is constrained to be at most c·log_b n for a constant c. This model captures the greedy algorithm (c=0) and natural extensions that are allowed a modest look‑ahead.

Main results

1. Lower bound (Theorem 1.3). For any ε>0 there exists a sufficiently small constant c(ε) such that any online algorithm respecting the above query budget fails, with high probability, to produce an independent set larger than (1+ε)·log_b n. In other words, even with a limited amount of future information, the greedy threshold log_b n is essentially optimal for this class. The bound holds uniformly for edge probabilities p∈