Problem
Research questions and friction points this paper is trying to address.
Boolean connectivity problem
k-Horn formulas
computational complexity
The Complexity of Boolean Connectivity Problem of $k$-Horn Formulas
🤖 AI Summary
研究了k-Horn公式布尔连通性问题的计算复杂度，提出了一种精确指数时间算法和两种多项式时间算法，并证明了当每个变量恰好出现三次时，3-Horn公式的连通性问题仍为coNP-完全。
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Boolean connectivity
k-Horn formulas
exponential-time algorithm
PPZ algorithm
coNP-complete
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