Lower Bounds for Minimal-Transversal Enumeration of Bounded-Dimension Hypergraphs and the maximum Minimal Hitting Set Problem

We present lower bounds for the maximum minimal hitting set problem and for the problem of enumeration of minimal transversals in bounded-dimension hypergraphs. Our starting point is the recent reduction of Kobayashi, Kurita, and Wasa (2026), developed to establish ETH-based hardness of hypergraph dualization. Adapting their construction, we show that, under ETH, deciding whether a hypergraph $\mathcal{G}$ with $n$ vertices and $m$ hyperedges has a minimal transversal of size at least $k$ cannot be done in time $n^{f(k)}{m}^{o(k)}$, for an arbitrary computable $f$. We transfer this bound to the following decision problem: given a dimension-at-most-$r$ hypergraph and a family of $s$ of its minimal transversals, decide whether another exists. Under ETH, this problem admits no uniform algorithm with running time $n^{f(r)}(s+1{)}^{o(r)}$. Under SETH, for every fixed $t\ge 2$, it admits no algorithm with running time $\text{poly}(\mathrm{\mid }\mathcal{G}\mathrm{\mid })(s+1{)}^c$ for dimension at most $2t$ and any $c<t$. As corollaries, the same exponent obstructions hold for the total time of any output-polynomial enumeration algorithm, with $s$ replaced by the total number of minimal transversals. In particular, polynomial delay is excluded under SETH already for dimension at most four.