Decision problem for Perfect Matchings in Dense k-uniform Hypergraphs

Abstract

For any $\gamma>0$, Keevash, Knox and Mycroft constructed a polynomial-time algorithm to determine the existence of perfect matchings in any $n$-vertex $k$-uniform hypergraph whose minimum codegree is at least $n/k+\gamma n$. We prove a structure theorem that enables us to determine the existence of a perfect matching for any $k$-uniform hypergraph with minimum codegree at least $n/k$. This solves a problem of Karpi\'nski, Ruci\'nski and Szyma\'nska completely. Our proof uses a lattice-based absorbing method.