The game of Cops and Robbers is a well-known game played on graphs. In this paper, we consider the class of graphs of bounded diameter. We improve the strategy of cops and the previously used probabilistic method, which results in an improved upper bound for the cop number of graphs of bounded diameter. In particular, for graphs of diameter 4, we improve the upper bound from $n^{\frac{2}{3}+o(1)}$ to $n^{\frac{3}{5}+o(1)}$ and for diameter 3 from $n^{\frac{2}{3}+o(1)}$ to $n^{\frac{4}{7}+o(1)}$.

Let H be a k-graph on n vertices, with minimum codegree at least n/k + cn for some fixed c > 0. In this paper we construct a polynomial-time algorithm which finds either a perfect matching in H or a certificate that none exists. This essentially solves a problem of Karpinski, Rucinski and Szymanska, who previously showed that this problem is NP-hard for a minimum codegree of n/k - cn. Our algorithm relies on a theoretical result of independent interest, in which we characterise any such hypergraph with no perfect matching using a family of lattice-based constructions.