In this paper we present a new approach to solve the Minimum Independent Dominating Setproblem in general graphs which is one of the hardest optimization problem. We propose amethod using a clique partition of the graph, partition that can be obtained greedily. Weprovide conditions under which our method has a better complexity than the complexity ofthe previously known algorithms. Based on our theoretical method, we design in the secondpart of this paper an efficient algorithm by including cuts in the search process. We thenexperiment it and show that it is able to solve almost all instances up to 50 vertices inreasonable time and some instances up to several hundreds of vertices. To go further andto treat larger graphs, we analyze a greedy heuristic. We show that it often gives good(sometimes optimal) results in large instances up to 60 000 vertices in less than 20 s.That sort of heuristic is a good approach to get an initial solution for our exact method.We also describe and analyze some of its worst cases.
