A reaction cut is a set of chemical reactions whose deletion blocks the operation of given reactions or the production of given chemical compounds. In this paper, we study two problems ReactionCut and MD-ReactionCut for calculating the minimum reaction cut of a metabolic network under a Boolean model. These problems are based on the flux balance model and the minimal damage model respectively. We show that ReactionCut and MD-ReactionCut are NP-hard even if the maximum outdegree of reaction nodes (K_out) is one. We also present O(1.822ⁿ), O(1.959ⁿ) and o(2ⁿ) time algorithms for MD-ReactionCut with K_out = 2, 3, k respectively where n is the number of reaction nodes and k is a constant. The same algorithms also work for ReactionCut if there is no directed cycle. Furthermore, we present a 2^{O((log n)√n)} time algorithm, which is faster than O((1+ε)ⁿ) for any positive constant ε, for the planar case of MD-ReactionCut under a reasonable constraint utilizing Lipton and Tarjan's separator algorithm.