In this work we deal with the problem of learning a maximum weighted (*k*+1)-order decomposable graph coarser than a given maximal *k*-order decomposable graph (also known as hypertree of tree-width *k*-1). An Integer Linear Programming formulation for the problem has recently been proposed and used in order to solve instances of the problem with a moderate number of vertices. However, as the problem is known to be NP-hard, it is of practical interest to develop approximate algorithms able to work with a limited amount of computational resources. In this paper we propose an approximate Integer Linear Programming formulation for the problem using a threshold distance which discards the edges that, on average, have a low probability of being contained in the solution. Experiments have been carried out with weighted graphs and probabilistic graphical models. Using the proposed formulation we have obtained results close to the optimum, even when most of the candidate edges were discarded using the distance criterion. The obtained good results indicate that the approximate formulation has important applications for learning probabilistic graphical models using decomposable scores, e.g., BDe.

This work deals with the so-called weighted independent domination problem, which is an *NP*-hard combinatorial optimization problem in graphs. In contrast to previous work, this paper considers the problem from a non-theoretical perspective. The first contribution consists in the development of three integer linear programming models. Second, two greedy heuristics are proposed. Finally, the last contribution is a population-based iterated greedy metaheuristic which is applied in two different ways: (1) the metaheuristic is applied directly to each problem instance, and (2) the metaheuristic is applied at each iteration of a higher-level framework{\textemdash}known as construct, merge, solve \& adapt{\textemdash}to sub-instances of the tackled problem instances. The results of the considered algorithmic approaches show that integer linear programming approaches can only compete with the developed metaheuristics in the context of graphs with up to 100 nodes. When larger graphs are concerned, the application of the populated-based iterated greedy algorithm within the higher-level framework works generally best. The experimental evaluation considers graphs of different types, sizes, densities, and ways of generating the node and edge weights.

This work deals with the so-called weighted independent domination problem, which is an *NP*-hard combinatorial optimization problem in graphs. In contrast to previous theoretical work from the literature, this paper considers the problem from an algorithmic perspective. The first contribution consists in the development of an integer linear programming model and a heuristic that makes use of this model. Second, two greedy heuristics are proposed. Finally, the last contribution is a population-based iterated greedy algorithm that takes profit from the better one of the two developed greedy heuristics. The results of the compared algorithmic approaches show that small problem instances based on random graphs are best solved by an efficient integer linear programming solver such as CPLEX. Larger problem instances are best tackled by the population-based iterated greedy algorithm. The experimental evaluation considers random graphs of different sizes, densities, and ways of generating the node and edge weights.