关于有向超图中超路径与极小分隔符枚举的复杂度
On the Complexity of Hyperpath and Minimal Separator Enumeration in Directed Hypergraphs
AI总结：
本文研究有向超图中$s$-$t$超路径与极小分隔符的枚举复杂度，证明导出超路径与极小分隔符枚举除非P=NP否则无输出多项式时间算法，但B-hypergraph的超路径枚举可通过回溯法在多项式延迟内求解。
摘要：
在本文中，我们探讨了（导出）$s$-$t$路径与极小$s$-$t$分隔符的枚举问题。这些问题是最著名的经典枚举问题中的一部分，对于（无）有向图，可以通过简单的回溯法在多项式延迟内求解。作为这些问题的一般化推广，我们考虑了\emph{有向超图}中的（导出）$s$-$t$超路径与极小$s$-$t$分隔符枚举。我们证明了将这些经典枚举问题扩展到有向超图中会极大地改变其复杂度。更准确地说，除非$P = NP$，否则不存在用于导出$s$-$t$超路径与极小$s$-$t$分隔符枚举的输出多项式时间算法；并且，如果存在用于$s$-$t$超路径枚举的输出多项式时间算法，那么即使有向超图是$BF$-hypergraph，极小横截枚举也能在输出多项式时间内求解。由于极小横截枚举是否存在输出多项式时间算法已是一个悬而未决的问题超过45年，这表明$BF$-hypergraph的$s$-$t$超路径枚举并非易事。作为一个正面结果，$B$-hypergraph的$s$-$t$超路径枚举可以通过回溯法在多项式延迟内求解。
英文摘要：
In this paper, we address the enumeration of (induced) $s$-$t$ paths and minimal $s$-$t$ separators. These problems are some of the most famous classical enumeration problems that can be solved in polynomial delay by simple backtracking for a (un)directed graph. As a generalization of these problems, we consider the (induced) $s$-$t$ hyperpath and minimal $s$-$t$ separator enumeration in a \emph{directed hypergraph}. We show that extending these classical enumeration problems to directed hypergraphs drastically changes their complexity. More precisely, there are no output-polynomial time algorithms for the enumeration of induced $s$-$t$ hyperpaths and minimal $s$-$t$ separators unless $P = NP$, and if there is an output-polynomial time algorithm for the $s$-$t$ hyperpath enumeration, then the minimal transversal enumeration can be solved in output polynomial time even if a directed hypergraph is $BF$-hypergraph. Since the existence of an output-polynomial time algorithm for the minimal transversal enumeration has remained an open problem for over 45 years, it indicates that the $s$-$t$ hyperpath enumeration for a $BF$-hypergraph is not an easy problem. As a positive result, the $s$-$t$ hyperpath enumeration for a $B$-hypergraph can be solved in polynomial delay by backtracking.
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