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 its 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. In a surprising contrast to our algorithmic result, 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 pseudo-distribution introduced by Bafna 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：我们提出了一种新的近似方法，用于在输入图为单侧谱扩展图时寻找大独立集——即该图的均匀随机游走矩阵的第二特征值远离1。基于此，我们获得了一个多项式时间算法，可在几乎3-可着色或承诺包含大小为$(1/2-\epsilon)n$的独立集的单侧扩展图中找到线性规模的独立集。上述第二个结果可进一步优化，仅需具有高效可验证性的较弱顶点扩展性质。与我们的算法结果形成惊人对比的是，我们发现在几乎4-可着色单侧扩展图中寻找线性规模独立集的任务（即使第二特征值为$o_n(1)$）是NP-难的，前提是接受唯一游戏猜想。此前所有超越该问题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），并要求双侧扩展，即具有$\Omega(1)$量级的有限个负特征值。此类技术可自然推广至任意常数$k$的几乎$k$-可着色图，而在单侧扩展图上的类似保证对于$k \geq 4$的情况被证明是唯一游戏猜想下的困难问题。我们的舍入方法建立在Bafna等人为舍入唯一游戏实例而提出的伪分布多重样本模拟技术之上。分析的关键在于扩展图中大独立集的新聚类性质——每个大独立集与某个小规模列表中的元素存在超预期交集——及其在低阶平方和证明系统中的形式化表述。