# Sharp Online Hardness for Large Balanced Independent Sets

## TL;DR

This paper investigates the size and algorithmic complexity of large balanced independent sets in dense bipartite graphs, establishing tight bounds and designing an online algorithm that nearly achieves the optimal size.

## Contribution

It introduces a precise size characterization for large balanced independent sets in dense bipartite graphs and develops an online algorithm that approaches this bound, supported by a new lower bound using the OGP framework.

## Key findings

- Largest $oldsymbol{ extgamma}$-balanced independent set size is $oldsymbol{rac{ ext{log}_b n}{ extgamma(1- extgamma)}}$ whp.
- Proposed online algorithm achieves size close to the upper bound with high probability.
- No online algorithm can surpass the bound by a factor of $(1+oldsymbol{ extepsilon})$, establishing tightness.

## Abstract

We study the algorithmic problem of finding large $\gamma$-balanced independent sets in dense random bipartite graphs; an independent set is $\gamma$-balanced if a $\gamma$ proportion of its vertices lie on one side of the bipartition. In the sparse regime, Perkins and Wang established tight bounds within the low-degree polynomial (LDP) framework, showing a factor-$1/(1-\gamma)$ statistical-computational gap via the Overlap Gap Property (OGP) framework tailored for stable algorithms. However, these techniques do not appear to extend to the dense setting. For the related large independent set problem in dense random graph, the best known algorithm is an online greedy procedure that is inherently unstable, and LDP algorithms are conjectured to fail even in the "easy" regime where greedy succeeds. We show that the largest $\gamma$-balanced independent set in dense random bipartite graphs has size $\alpha:=\frac{\log_b n}{\gamma(1-\gamma)}$ whp, where $n$ is the size of each bipartition, $p$ is the edge probability, and $b=1/(1-p)$. We design an online algorithm that achieves $(1-\epsilon)(1-\gamma)\alpha$ whp for any $\epsilon>0$. We complement this with a sharp lower bound, showing that no online algorithm can achieve $(1+\epsilon)(1-\gamma)\alpha$ with nonnegligible probability. Our results suggest that the same factor-$1/(1-\gamma)$ gap is also present in the dense setting, supporting its conjectured universality. While the classical greedy procedure on $G(n,p)$ is straightforward, our algorithm is more intricate: it proceeds in two stages, incorporating a stopping time and suitable truncation to ensure that $\gamma$-balancedness-a global constraint-is met despite operating with limited information. Our lower bound utilizes the OGP framework; we build on a recent refinement of this framework for online models and extend it to the bipartite setting.