Inapproximability of Minimum Vertex Cover on k-Uniform k-Partite Hypergraphs

We study the problem of computing the minimum vertex cover on $k$-uniform $k$-partite hypergraphs when the $k$-partition is given. On bipartite graphs (k=2), the minimum vertex cover can be computed in polynomial time. For $k \ge 3,$ this problem is known to be NP-hard. For general $k$, the problem was studied by Lovasz, who gave a $\frac{k}{2}$-approximation based on the standard LP relaxation. Subsequent work by Aharoni, Holzman, and Krivelevich showed a tight integrality gap of $(\frac{k}{2} - o(1))$ for the LP relaxation. We further investigate the inapproximability of minimum vertex cover on $k$-uniform $k$-partite hypergraphs and present the following results (here $\varepsilon > 0$ is an arbitrarily small constant): NP-hardness of obtaining an approximation factor of $(\frac{k}{4} - \varepsilon)$ for even $k$ and $(\frac{k}{4} - \frac{1}{4k} - \varepsilon)$ for odd $k$, NP-hardness of obtaining a nearly optimal approximation factor of $(\frac{k}{2}-1+\frac{1}{2k}-\varepsilon)$, and an optimal uniqu...