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).

翻译：我们考虑在稀疏随机二分图中寻找大规模\textit{平衡}独立集的算法问题，更一般地，是寻找二分图两侧顶点具有特定比例分配的独立集问题。在二分图中，很容易找到密度至少为一半的独立集（取其中一个划分类即可）。相比之下，在平均度为$d$的随机二分图中，最大的平衡独立集（每类顶点数相等）的典型密度为$(2+o_d(1)) \frac{\log d}{d}$。我们能否高效地找到这些图中的大规模平衡独立集？通过利用重叠间隙性质和低度算法框架，我们证明：对于任意固定的$\epsilon>0$和大但恒定的$d$，局部和低度算法（即使知道二分划分）无法找到密度大于$(1+\epsilon) \frac{\log d}{d}$的平衡独立集。这种实际存在值与局部算法可实现值之间的两倍统计-计算差距，与随机（非二分）图中寻找大规模独立集的差距类似。因此，我们的结果暗示这种差距在许多模型中普遍存在，且困难的计算问题可能潜伏在原本易处理的问题内部。二分图中该差距的一个特别显著特征是：达到下界的算法极其简单，可作为$1$-局部算法和度数-$1$多项式（线性函数）实现。