On the hardness of finding balanced independent sets in random bipartite graphs

TL;DR

This paper investigates the computational difficulty of finding large balanced independent sets in sparse random bipartite graphs, revealing a fundamental gap between what exists and what efficient algorithms can find.

Contribution

It establishes a statistical–computational gap for finding large balanced independent sets in random bipartite graphs using local and low-degree algorithms.

Findings

Local algorithms cannot find balanced independent sets of density greater than (1+ε) log d / d.

Existence of larger balanced independent sets is shown to be computationally hard.

A simple 1-local algorithm can achieve the lower bound, highlighting the gap.

Abstract

We consider the algorithmic problem of finding large \textit{balanced} independent sets in sparse random bipartite graphs, and more generally the problem of finding independent sets with specified proportions of vertices on each side of the bipartition. In a bipartite graph it is trivial to find an independent set of density at least half (take one of the partition classes). In contrast, in a random bipartite graph of average degree $d$, the largest balanced independent sets (containing equal number of vertices from each class) are typically of density $(2+o_d(1)) \frac{\log d}{d}$. Can we find such large balanced independent sets in these graphs efficiently? By utilizing the overlap gap property and the low-degree algorithmic framework, we prove that local and low-degree algorithms (even those that know the bipartition) cannot find balanced independent sets of density greater than $(1+\epsilon) \frac{\log d}{d}$ for any $\epsilon>0$ fixed and $d$ large but constant. This factor $2$ statistical--computational gap between what exists and what local algorithms can achieve is analogous to the gap for finding large independent sets in (non-bipartite) random graphs. Our results therefor suggest that this gap is pervasive in many models, and that hard computational problems can lurk inside otherwise tractable ones. A particularly striking aspect of the gap in bipartite graphs is that the algorithm achieving the lower bound is extremely simple and can be implemented as a $1$-local algorithm and a degree-$1$ polynomial (a linear function).