We take an algorithmic approach to studying the solution space geometry of relatively sparse random and bounded degree $k$-CNFs for large $k$. In the course of doing so, we establish that with high probability, a random $k$-CNF $\Phi$ with $n$ variables and clause density $\alpha = m/n \lesssim 2^{k/6}$ has a giant component of solutions that are connected in a graph where solutions are adjacent if they have Hamming distance $O_k(\log n)$ and that a similar result holds for bounded degree $k$-CNFs at similar densities. We are also able to deduce looseness results for random and bounded degree $k$-CNFs in a similar regime. Although our main motivation was understanding the geometry of the solution space, our methods have algorithmic implications. Towards that end, we construct an idealized block dynamics that samples solutions from a random $k$-CNF $\Phi$ with density $\alpha = m/n \lesssim 2^{k/75}$. We show this Markov chain can with high probability be implemented in polynomial time and by leveraging spectral independence, we also observe that it mixes relatively fast, giving a polynomial time algorithm to with high probability sample a uniformly random solution to a random $k$-CNF. Our work suggests that the natural route to pinning down when a giant component exists is to develop sharper algorithms for sampling solutions in random $k$-CNFs.

翻译：我们用一种算法方法来研究相对稀少的随机和约束度为$k-CNF的解决方案空间几何。 在这样做的过程中,我们以很高的概率确定,一个随机的美元-CNF$\Phi$,其中含有美元变量和条款密度$alpha = m/n\lessim 2 ⁇ k/6}美元,其中含有一个巨大的解决方案组成部分,这些解决方案在一张图中相联,如果解决方案的相邻位置是 $-k(gg) 美元(g) 美元,而且一个类似结果在相似的密度下,约束度为$-CNF$-CNF美元。在类似制度下,我们还能够得出随机和约束度为$k-CNF$的随机松散结果。虽然我们的主要动机是理解解决方案空间的几何位置,但我们的方法具有算法影响。 最终,我们构建了一个理想的块动态,即从一个随机的 $-CNF 美元 = m/n COMsimm $-climexal $ 2 ⁇ /75} 快速算算出一个快速的概率。 我们的链路路路段会显示一个快速的概率到一个直径直径直径直路的概率, 直径直到一个比。