欺骗多对数 junta 与 Lovász 局部引理的确定性并行算法
Deterministic parallel algorithms for fooling polylogarithmic juntas and the Lovasz Local Lemma
AI总结：
本文提出确定性并行算法，通过去随机化变量划分和新代码构造，在 NC1 复杂度下欺骗多对数 junta，并将 Lovász 局部引理的 NC 算法推广到 polylog(n) 变量，应用于任意度图的缺陷顶点着色。
摘要：
许多随机化算法可以通过条件期望方法或具有低（几乎）独立性的概率空间来高效地去随机化。从 Luby (1993) 开始，并延续至 Berger & Rompel (1991) 和 Chari 等人 (2000) 的一系列论文表明，这些技术可以结合使用，为涉及 $w$-junta 之和的组合优化问题提供确定性并行算法。我们通过去随机化变量划分和一种新的用于在 $GF(2)$ 上欺骗傅里叶字符的代码构造来改进这些算法。这将处理器复杂度降低到基本上与 $w$ 无关，同时运行时间从 $w$ 的指数级减少到 $w$ 的线性级。\n\n作为关键子程序，我们给出了一种新算法来生成能够欺骗给定邻域集合的概率空间。Schulman (1992) 给出了一个 NC 算法来处理大小为 $w \leq O(\log n)$ 的邻域。我们的新算法是 NC1，当 $w = O(\log n)$ 时具有本质上最优的时间和处理器复杂度；对于 $w = \text{polylog}(n)$，它仍然是 NC。这回答了 Schulman 的一个开放问题。\n\n这些算法的一个主要应用是 Lovász 局部引理的 NC 算法。先前的 NC 算法，包括 Moser & Tardos (2010) 的开创性算法和 Chandrasekaran 等人 (2013) 的工作，要求（本质上）坏事件只能跨越 $O(\log n)$ 个变量；我们将其放宽到 $\text{polylog}(n)$ 个变量。我们将其用于缺陷顶点着色的 $\text{NC}^2$ 算法，该算法适用于任意度数的图。
英文摘要：
Many randomized algorithms can be derandomized efficiently using either the method of conditional expectations or probability spaces with low (almost-) independence. A series of papers, beginning with Luby (1993) and continuing with Berger & Rompel (1991) and Chari et al. (2000), showed that these techniques can be combined to give deterministic parallel algorithms for combinatorial optimization problems involving sums of $w$-juntas. We improve these algorithms through derandomized variable partitioning and a new code construction for fooling Fourier characters over $GF(2)$. This reduces the processor complexity to essentially independent of $w$ while the running time is reduced from exponential in $w$ to linear in $w$. As a key subroutine, we give a new algorithm to generate a probability space which can fool a given set of neighborhoods. Schulman (1992) gave an NC algorithm to do so for neighborhoods of size $w \leq O(\log n)$. Our new algorithm is NC1, with essentially optimal time and processor complexity, when $w = O(\log n)$; it remains NC up to $w = \text{polylog}(n)$. This answers an open problem of Schulman. One major application of these algorithms is an NC algorithm for the Lovász Local Lemma. Previous NC algorithms, including the seminal algorithm of Moser & Tardos (2010) and the work of Chandrasekaran et. al (2013), required that (essentially) the bad-events could span only $O(\log n)$ variables; we relax this to $\text{polylog}(n)$ variables. We use this for an $\text{NC}^2$ algorithm for defective vertex coloring, which works for arbitrary degree graphs.
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