We consider the problem of enumerating all minimal transversals (also called minimal hitting sets) of a hypergraph $\mathcal{H}$. An equivalent formulation of this problem known as the \emph{transversal hypergraph} problem (or \emph{hypergraph dualization} problem) is to decide, given two hypergraphs, whether one corresponds to the set of minimal transversals of the other. The existence of a polynomial time algorithm to solve this problem is a long standing open question. In \cite{fredman_complexity_1996}, the authors present the first sub-exponential algorithm to solve the transversal hypergraph problem which runs in quasi-polynomial time, making it unlikely that the problem is (co)NP-complete. In this paper, we show that when one of the two hypergraphs is of bounded VC-dimension, the transversal hypergraph problem can be solved in polynomial time, or equivalently that if $\mathcal{H}$ is a hypergraph of bounded VC-dimension, then there exists an incremental polynomial time algorithm to enumerate its minimal transversals. This result generalizes most of the previously known polynomial cases in the literature since they almost all consider classes of hypergraphs of bounded VC-dimension. As a consequence, the hypergraph transversal problem is solvable in polynomial time for any class of hypergraphs closed under partial subhypergraphs. We also show that the proposed algorithm runs in quasi-polynomial time in general hypergraphs and runs in polynomial time if the conformality of the hypergraph is bounded, which is one of the few known polynomial cases where the VC-dimension is unbounded.

翻译：我们考虑枚举超图$\mathcal{H}$的所有最小横贯（亦称最小命中集）问题。该问题的等价形式称为\emph{横贯超图}问题（或\emph{超图对偶化}问题），即给定两个超图，判定其中一个是否对应另一个的最小横贯集。是否存在多项式时间算法求解该问题是一个长期悬而未决的开放性问题。在文献\cite{fredman_complexity_1996}中，作者提出了首个解决横贯超图问题的亚指数算法，该算法在拟多项式时间内运行，从而降低了该问题是(co)NP完全问题的可能性。本文证明，当两个超图中有一个具有有界VC维时，横贯超图问题可在多项式时间内求解；等价而言，若$\mathcal{H}$是具有有界VC维的超图，则存在增量多项式时间算法来枚举其最小横贯。该结果推广了文献中大多数已知的多项式时间可解情形，因为这些情形几乎都考虑有界VC维的超图类。作为推论，对于在部分子超图下封闭的任何超图类，超图横贯问题均可在多项式时间内求解。我们还证明了所提算法在一般超图中具有拟多项式时间复杂度，而在超图的共形性有界时具有多项式时间复杂度——这是VC维无界情况下少数已知的多项式时间可解情形之一。