We develop a new approach for approximating large independent sets when the input graph is a one-sided spectral expander - that is, the uniform random walk matrix of the graph has the second eigenvalue bounded away from 1. Consequently, we obtain a polynomial time algorithm to find linear-sized independent sets in one-sided expanders that are almost $3$-colorable or are promised to contain an independent set of size $(1/2-\epsilon)n$. Our second result above can be refined to require only a weaker vertex expansion property with an efficient certificate. Somewhat surprisingly, we observe that the analogous task of finding a linear-sized independent set in almost $4$-colorable one-sided expanders (even when the second eigenvalue is $o_n(1)$) is NP-hard, assuming the Unique Games Conjecture. All prior algorithms that beat the worst-case guarantees for this problem rely on bottom eigenspace enumeration techniques (following the classical spectral methods of Alon and Kahale) and require two-sided expansion, meaning a bounded number of negative eigenvalues of magnitude $\Omega(1)$. Such techniques naturally extend to almost $k$-colorable graphs for any constant $k$, in contrast to analogous guarantees on one-sided expanders, which are Unique Games-hard to achieve for $k \geq 4$. Our rounding builds on the method of simulating multiple samples from a pseudodistribution introduced by Barak et. al. for rounding Unique Games instances. The key to our analysis is a new clustering property of large independent sets in expanding graphs - every large independent set has a larger-than-expected intersection with some member of a small list - and its formalization in the low-degree sum-of-squares proof system.

翻译：我们提出了一种新方法，用于在输入图为一侧谱扩展图（即图的均匀随机游走矩阵的第二特征值远离1）时近似大独立集。因此，我们获得了一个多项式时间算法，可以在几乎3-可着色或保证包含大小为$(1/2-\epsilon)n$的独立集的一侧扩展图中找到线性规模的独立集。上述第二个结果可以细化为仅需较弱的顶点扩展性质，并具有高效证书。令人惊讶的是，我们观察到，在几乎4-可着色的一侧扩展图中（即使第二特征值为$o_n(1)$）寻找线性规模独立集的类似任务是NP难的（假设唯一博弈猜想）。所有先前在该问题上超越最坏情况保证的算法都依赖于底部特征空间枚举技术（遵循Alon和Kahale的经典谱方法），并需要两侧扩展，即数量为$\Omega(1)$量级的负特征值有界。这些技术自然扩展到任意常数$k$的几乎$k$-可着色图，这与几乎$k$-可着色一侧扩展图上的类似保证形成对比，后者在$k \geq 4$时是唯一博弈难的。我们的舍入方法基于Barak等人为舍入唯一博弈实例引入的从伪分布模拟多个样本的技术。我们分析的关键是扩展图中大独立集的新聚类性质——每个大独立集与一个小列表中的某个成员的交集大于预期——及其在低阶平方和证明系统中的形式化。