AI 中文总结
本文研究k-Horn公式的布尔连通性问题{\textsc{Conn $k$-Horn}}的复杂性，提出无结构限制的指数级算法和两种多项式时间算法，还证明变量恰好出现三次时{\textsc{Conn $3$-Horn}}仍为coNP完全。
AI 中文摘要
布尔连通性问题询问给定布尔公式的满足赋值集合是否构成n维超立方体中的连通子图。Makino、Tamaki和Yamamoto已证明，即使限制在k≥3的k-Horn公式上，该问题也是coNP完全的。本文进一步研究了k-Horn公式的布尔连通性问题{\textsc{Conn $k$-Horn}}的计算复杂性，给出了该问题的算法结果和困难性结果。算法方面，首先提出了一种无任何结构限制的、适用于任意k的精确指数时间算法，该算法基于Paturi、Pudlák和Zane提出的确定性PPZ算法构建，运行时间为$O^*(2^{(1 - 1/2k)n})$，空间为多项式空间，相比Makino等人提出的k-CNF公式的布尔连通性问题的已知算法，实现了指数级改进；接着给出了适用于任意k的两种多项式时间算法，分别对应两种限制条件：（i）每个变量最多出现两次，（ii）每个子句长度恰好为k且每个变量最多出现k次。困难性方面，证明了即使每个变量恰好出现三次，{\textsc{Conn $3$-Horn}}仍然是coNP完全的。
英文摘要
The Boolean connectivity problem asks whether the set of satisfying assignments of a given Boolean formula forms a connected subgraph in the $n$-dimensional hypercube. This problem is known to be $\mathsf{coNP}$-complete, even when restricted to $k$-Horn formulas for $k \geq 3$, as shown by Makino, Tamaki, and Yamamoto. In this paper, we further investigate the computational complexity of {\sc Conn $k$-Horn}, the Boolean connectivity problem for $k$-Horn formulas. We provide algorithmic and hardness results for {\sc Conn $k$-Horn}. On the algorithmic side, we first present an exact exponential-time algorithm for arbitrary $k$ without any structural restrictions. Our algorithm builds on the deterministic PPZ algorithm proposed by Paturi, Pudlák, and Zane. It runs in $O^*(2^{(1 - 1/2k)n})$ time and polynomial space, achieving an exponential improvement over the previously known algorithm for the Boolean connectivity problem of $k$-CNF formulas, shown by Makino, Tamaki, and Yamamoto. We next give two polynomial-time algorithms for arbitrary $k$ under the following two restrictions: (i) each variable appears at most twice, and (ii) each clause has length exactly $k$ and each variable appears at most $k$ times. On the hardness side, we prove that {\sc Conn $3$-Horn} remains $\mathsf{coNP}$-complete even when each variable appears exactly three times.
