SIAM Journal on Computing, Volume 52, Issue 6, Page 1321-1368, December 2023.
Abstract. Suppose that you wish to sample a random graph [math] over [math] vertices and [math] edges conditioned on the event that [math] does not contain a “small” [math]-size graph [math] (e.g., clique) as a subgraph. Assuming that most such graphs are [math]-free, the problem can be solved by a simple rejected-sampling algorithm (that tests for [math]-cliques) with an expected running time of [math]. Is it possible to solve the problem in a running time that does not grow polynomially with [math]? In this paper, we introduce the general problem of sampling a “random looking” graph [math] with a given edge density that avoids some arbitrary predefined [math]-size subgraph [math]. As our main result, we show that the problem is solvable with respect to some specially crafted [math]-wise independent distribution over graphs. That is, we design a sampling algorithm for [math]-wise independent graphs that supports efficient testing for subgraph-freeness in time [math], where [math] is a function of [math] and the constant [math] in the exponent is independent of [math]. Our solution extends to the case where both [math] and [math] are [math]-uniform hypergraphs. We use these algorithms to obtain the first probabilistic construction of constant-degree polynomially unbalanced expander graphs whose failure probability is negligible in [math] (i.e., [math]). In particular, given constants [math], we output a bipartite graph that has [math] left nodes and [math] right nodes with right-degree of [math] so that any right set of size at most [math] expands by factor of [math]. This result is extended to the setting of unique expansion as well. We observe that such a negligible-error construction can be employed in many useful settings and present applications in coding theory (batch codes and low-density parity-check codes), pseudorandomness (low-bias generators and randomness extractors), and cryptography. Notably, we show that our constructions yield a collection of polynomial-stretch locally computable cryptographic pseudorandom generators based on Goldreich’s one-wayness assumption resolving a central open problem in the area of parallel-time cryptography (e.g., Applebaum, Ishai, and Kushilevitz [SIAM J. Comput., 36 (2006), pp. 845–888] and Ishai et al. [Proceedings of the 40th Annual ACM Symposium on Theory of Computing, ACM, 2008, pp. 433–442]).

中文翻译：

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没有禁止子图和误差可忽略不计的不平衡扩展器的采样图

SIAM 计算杂志，第 52 卷，第 6 期，第 1321-1368 页，2023 年 12 月。
摘要。假设您希望在 [math] 顶点和 [math] 边上对随机图 [math] 进行采样，条件是 [math] 不包含“小”[math] 大小的图 [math]（例如，clique ）作为子图。假设大多数此类图都与 [math] 无关，则可以通过简单的拒绝采样算法（测试 [math]-cliques）来解决该问题，预期运行时间为 [math]。是否有可能在不随[数学]多项式增长的运行时间内解决问题？在本文中，我们介绍了使用给定边缘密度对“随机查看”图 [math] 进行采样的一般问题，以避免某些任意预定义 [math] 大小的子图 [math]。作为我们的主要结果，我们表明该问题对于一些专门设计的图上的[数学]独立分布是可以解决的。也就是说，我们为 [math] 方向的独立图设计了一种采样算法，支持在时间 [math] 上有效测试子图自由度，其中 [math] 是 [math] 和指数中常数 [math] 的函数独立于[数学]。我们的解决方案扩展到 [math] 和 [math] 都是 [math] 一致超图的情况。我们使用这些算法来获得常度多项式不平衡扩展图的第一个概率构造，其故障概率在[math]（即[math]）中可以忽略不计。特别是，给定常量 [math]，我们输出一个二分图，其中具有 [math] 左节点和 [math] 右节点，右度为 [math]，因此任何右集的大小最多为 [math] 扩展因子[数学]的。这个结果也被推广到唯一展开的设置。我们观察到，这种可忽略错误的构造可以在许多有用的设置中使用，并在编码理论（批处理码和低密度奇偶校验码）、伪随机性（低偏差生成器和随机性提取器）和密码学中得到应用。值得注意的是，我们表明我们的构造产生了一组基于 Goldreich 的单向假设的多项式拉伸局部可计算密码伪随机生成器，解决了并行时间密码学领域的一个中心开放问题（例如，Applebaum、Ishai 和 Kushilevitz [SIAM] J. Comput.，36 (2006)，第 845–888 页] 和 Ishai 等人[第 40 届 ACM 计算理论年度研讨会论文集，ACM，2008 年，第 433–442 页]）。