Deterministic parallel algorithms for fooling polylogarithmic juntas and the Lovasz Local Lemma

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 work by Luby (1988) and continuing with Berger & Rompel (1991) and Chari et al. (1994), 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. This reduces the processor complexity to essentially independent of $w$ while the running time is reduced from exponential in $w$ to linear in $w$. For example, we improve the time complexity of an algorithm of Berger & Rompel (1991) for rainbow hypergraph coloring by a factor of approximately $\log^2 n$ and the processor complexity by a factor of approximately $m^{\ln 2}$. As a major application of this, we give an NC algorithm for the Lov\'{a}sz 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 allowing $\text{polylog}(n)$ variables. As two applications of our new algorithm, we give algorithms for defective vertex coloring and domatic graph partition. One main sub-problem encountered in these algorithms is to generate a probability space which can "fool" a given list of $GF(2)$ Fourier characters. Schulman (1992) gave an NC algorithm for this; we dramatically improve its efficiency to near-optimal time and processor complexity and code dimension. This leads to a new algorithm to solve the heavy-codeword problem, introduced by Naor & Naor (1993), with a near-linear processor complexity $(mn)^{1+o(1)}$.