We study the problem of finding the smallest set of nodes in a network whose removal results in an empty k-core, where the k-core is the subnetwork obtained after the iterative removal of all nodes of degree smaller than k. This problem is also known in the literature as finding the minimal contagious set. The main contribution of our work is an analysis of the performance of the recently introduced corehd algorithm [Zdeborová, Zhang, and Zhou, Sci. Rep. 6, 37954 (2016)10.1038/srep37954] on random graphs taken from the configuration model via a set of deterministic differential equations. Our analyses provide upper bounds on the size of the minimal contagious set that improve over previously known bounds. Our second contribution is a heuristic called the weak-neighbor algorithm that outperforms all currently known local methods in the regimes considered.