Transversal Rank, Conformality and Enumeration

TL;DR

This paper introduces new algorithms for recognizing hypergraphs with a given transversal rank, improving runtime dependence on the number of edges, and explores the complexity of related enumeration problems.

Contribution

It presents an algorithm with improved runtime for recognizing hypergraphs of a certain transversal rank and introduces a look-ahead method for enumerating minimal hitting sets.

Findings

New algorithm with runtime O(Δ^{k-2} mn^{k-1}) for recognizing hypergraphs with transversal rank ≥ k.

A look-ahead technique for finding higher-order extensions and enumerating minimal hitting sets.

Establishes equivalences between improving recognition algorithms and breakthroughs in hypergraph enumeration.

Abstract

The transversal rank of a hypergraph is the maximum size of its minimal hitting sets. Deciding, for an $n$-vertex, $m$-edge hypergraph and an integer $k$, whether the transversal rank is at least $k$ takes time $O(m^{k+1} n)$ with an algorithm that is known since the 70s. It essentially matches an $(m+n)^{\Omega(k)}$ ETH-lower bound by Ara\'ujo, Bougeret, Campos, and Sau [Algorithmica 2023] and Dublois, Lampis, and Paschos [TCS 2022]. Many hypergraphs seen in practice have much more edges than vertices, $m \gg n$. This raises the question whether an improvement of the run time dependency on $m$ can be traded for an increase in the dependency on $n$. Our first result is an algorithm to recognize hypergraphs with transversal rank at least $k$ in time $O(\Delta^{k-2} mn^{k-1})$, where $\Delta \le m$ is the maximum degree. Our main technical contribution is a ``look-ahead'' method that allows us to find higher-order extensions, minimal hitting sets that augment a given set with at least two more vertices. We show that this method can also be used to enumerate all minimal hitting sets of a hypergraph with transversal rank $k^*$ with delay $O(\Delta^{k^*-1} mn^2)$. We then explore the possibility of further reducing the running time for computing the transversal rank to $\textsf{poly}(m) \cdot n^{k+O(1)}$. This turns out to be equivalent to several breakthroughs in combinatorial algorithms and enumeration. Among other things, such an improvement is possible if and only if $k$-conformal hypergraphs can also be recognized in time $\textsf{poly}(m) \cdot n^{k+O(1)}$, and iff the maximal hypercliques/independent sets of a uniform hypergraph can be enumerated with incremental delay.