The complexity of perfect matchings and packings in dense hypergraphs

Abstract: Given two $k$-graphs $H$ and $F$, a perfect $F$-packing in $H$ is a collection of vertex-disjoint copies of $F$ in $H$ which together cover all the vertices in $H$. In the case when $F$ is a single edge, a perfect $F$-packing is simply a perfect matching. For a given fixed $F$, it is often the case that the decision problem whether an $n$-vertex $k$-graph $H$ contains a perfect $F$-packing is NP-complete. Indeed, if $k \geq 3$, the corresponding problem for perfect matchings is NP-complete whilst if $k=2$ the problem is NP-complete in the case when $F$ has a component consisting of at least $3$ vertices. In this paper we give a general tool which can be used to determine classes of (hyper)graphs for which the corresponding decision problem for perfect $F$-packings is polynomial time solvable. We then give three applications of this tool: (i) Given $1\leq \ell \leq k-1$, we give a minimum $\ell$-degree condition for which it is polynomial time solvable to determine whether a $k$-graph satisfying this condition has a perfect matching; (ii) Given any graph $F$ we give a minimum degree condition for which it is polynomial time solvable to determine whether a graph satisfying this condition has a perfect $F$-packing; (iii) We also prove a similar result for perfect $K$-packings in $k$-graphs where $K$ is a $k$-partite $k$-graph. For a range of values of $\ell,k$ (i) resolves a conjecture of Keevash, Knox and Mycroft whilst (ii) answers a question of Yuster in the negative. In many cases our results are best possible in the sense that lowering the minimum degree condition means that the corresponding decision problem becomes NP-complete.