We propose a new algorithm for computing regularized solutions to inverse problems where the unknown functions is a characteristic function and where the forward operator is linear. Our approach can be seen as an alternative to the level-set method and is based on an efficient computation of minimizers for a Tikhonov functional. One main ingredient is to use surrogate functionals to define an iteration which has a subsequence converging to a limit that can be interpreted as a critical point. For the minimization of the surrogate functionals we propose the method of exact relaxation, which allows us to compute exact minimizer for the corresponding binary optimization problem.