An Algorithmic Study of the Hypergraph Tur\'{a}n Problem

We propose an algorithmic version of the hypergraph Tur\'{a}n problem (AHTP): given a $t$-uniform hypergraph $H=(V,E)$, the goal is to find the smallest collection of $(t-1)$-element subsets of $V$ such that every hyperedge $e \in E$ contains one of these subsets. In addition to its inherent combinatorial interest---for instance, the $t=3$ case is connected to Tuza's famous conjecture on covering triangles of a graph with edges---variants of AHTP arise in recently proposed reductions to fundamental Euclidean clustering problems. AHTP admits a trivial factor $t$ approximation algorithm as it can be cast as an instance of vertex cover on a structured $t$-uniform hypergraph that is a ``blown-up'' version of $H$. Our main result is an approximation algorithm with ratio $\frac{t}{2}+o(t)$. The algorithm is based on rounding the natural LP relaxation using a careful combination of thresholding and color coding. We also present results motivated by structural aspects of the blown-up hypergraph. The blown-up is a $\textit{simple}$ hypergraph with hyperedges intersecting in at most one element. We prove that vertex cover on simple $t$-uniform hypergraphs is as hard to approximate as general $t$-uniform hypergraphs. The blown-up hypergraph further has many forbidden structures, including a ``tent'' structure for the case $t=3$. Whether a generalization of Tuza's conjecture could also hold for tent-free $3$-uniform hypergraphs was posed in a recent work. We answer this question in the negative by giving a construction based on combinatorial lines that is tent-free, and yet needs to include most of the vertices in a vertex cover.