Given a $k$-CNF formula and an integer $s$, we study algorithms that obtain $s$ solutions to the formula that are maximally dispersed. For $s=2$, the problem of computing the diameter of a $k$-CNF formula was initiated by Creszenzi and Rossi, who showed strong hardness results even for $k=2$. Assuming SETH, the current best upper bound [Angelsmark and Thapper '04] goes to $4^n$ as $k \rightarrow \infty$. As our first result, we give exact algorithms for using the Fast Fourier Transform and clique-finding that run in $O^*(2^{(s-1)n})$ and $O^*(s^2 |\Omega_{F}|^{\omega \lceil s/3 \rceil})$ respectively, where $|\Omega_{F}|$ is the size of the solution space of the formula $F$ and $\omega$ is the matrix multiplication exponent. As our main result, we re-analyze the popular PPZ (Paturi, Pudlak, Zane '97) and Sch\"{o}ning's ('02) algorithms (which find one solution in time $O^*(2^{\varepsilon_{k}n})$ for $\varepsilon_{k} \approx 1-\Theta(1/k)$), and show that in the same time, they can be used to approximate the diameter as well as the dispersion ($s>2$) problems. While we need to modify Sch\"{o}ning's original algorithm, we show that the PPZ algorithm, without any modification, samples solutions in a geometric sense. We believe that this property may be of independent interest. Finally, we present algorithms to output approximately diverse, approximately optimal solutions to NP-complete optimization problems running in time $\text{poly}(s)O^*(2^{\varepsilon n})$ with $\varepsilon<1$ for several problems such as Minimum Hitting Set and Feedback Vertex Set. For these problems, all existing exact methods for finding optimal diverse solutions have a runtime with at least an exponential dependence on the number of solutions $s$. Our methods find bi-approximations with polynomial dependence on $s$.

翻译：给定一个$k$-CNF公式与整数$s$，我们研究获取该公式$s$个最大分散解的算法。当$s=2$时，计算$k$-CNF公式直径的问题由Creszenzi与Rossi提出，他们证明了即使对于$k=2$该问题也具有强困难性。在SETH假设下，当前最佳上界[Angelsmark与Thapper '04]在$k \rightarrow \infty$时趋近于$4^n$。作为首个结果，我们提出基于快速傅里叶变换与团查找的精确算法，其运行时间分别为$O^*(2^{(s-1)n})$与$O^*(s^2 |\Omega_{F}|^{\omega \lceil s/3 \rceil})$，其中$|\Omega_{F}|$是公式$F$解空间的大小，$\omega$是矩阵乘法指数。作为主要结果，我们重新分析了经典的PPZ算法（Paturi, Pudlak, Zane '97）与Schöning算法（'02）（两者均能在$O^*(2^{\varepsilon_{k}n})$时间内找到一个解，其中$\varepsilon_{k} \approx 1-\Theta(1/k)$），并证明在相同时间内，它们也可用于近似求解直径问题及分散问题（$s>2$）。虽然需要对Schöning原始算法进行修改，但我们证明PPZ算法无需任何修改即可在几何意义上对解进行采样。我们认为这一性质可能具有独立的研究价值。最后，我们提出在$\text{poly}(s)O^*(2^{\varepsilon n})$时间内输出NP完全优化问题的近似多样化近似最优解的算法，其中$\varepsilon<1$，适用于最小击中集、反馈顶点集等多个问题。对于这些问题，现有寻找最优多样化解的所有精确方法在运行时间上至少对解的数量$s$具有指数依赖。我们的方法能以$s$的多项式依赖找到双近似解。