  In this paper we deal with the signed Roman domination and signed total Roman
domination problems. For each problem we propose two integer linear programming
(ILP) formulations, the constraint programming (CP) formulation and variable
neighborhood search (VNS) method. We present proofs for the correctness of the
ILP formulations and a polyhedral study in which we show that the polyhedrons
of the two model relaxations are equivalent.
  VNS uses specifically designed penalty function that allows the appearance of
slightly infeasible solutions. The acceptance of these solutions directs the
overall search process to the promising areas in the long run.
  All proposed approaches are tested on the large number of instances.
Experimental results indicate that all of them reach optimal solutions for the
most of small and middle scale instances. Both ILP models have proven to be
more successful than the other two methods.
