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

Overview & Synopsis

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 $t \geq 2$, it admits no algorithm with running time $\poly(|\mathcal{G}|)(s+1)^c$ for dimension at most $2t$ and any $c