We study a class of bilevel mixed-integer linear programs with the following restrictions: all upper level variables x are binary, the lower level variables y occur in exactly one upper level constraint γx + βy ≥ c, and the lower level objective function is min y βy. We propose a new cut generation algorithm to solve this problem class, based on two simplifying assumptions. We then propose a row-and-column generation algorithm which works independently of the assumptions. We apply our methods to two problems: one is related to the optimal placement of measurement devices in an electrical network, and the other is the minimum zero forcing set problem, a variant of the dominating set problem. We exhibit computational results of both methods on the application-oriented instances as well as on randomly generated instances.