  Dominating set problems are among the most important class of combinatorial
problems in graph optimization, from a theoretical as well as from a practical
point of view. In this paper, we address the recently introduced (minimum)
weighted total domination problem. In this problem, we are given an undirected
graph with a vertex weight function and an edge weight function. The goal is to
find a total dominating set D in this graph with minimal weight. A total
dominating set D is a subset of the vertices such that every vertex in the
graph, including vertices in D, is adjacent to a vertex in D. The weight is
measured as the sum of all vertex weights of vertices in D, plus the edge
weights in the subgraph induced by D, plus for each vertex not in D the minimum
weight of an edge from it to a vertex in D. In this paper, we present two new
Mixed-Integer Programming models for the problem, and design solution
frameworks based on them. These solution frameworks also include valid
inequalities, starting heuristics and primal heuristics. In addition, we also
develop a genetic algorithm, which is based on a greedy randomized adaptive
search procedure version of our starting heuristic. We carry out a
computational study to assess the performance of our approaches when compared
to the previous work for the same problem. The study reveals that our exact
solution algorithms are up to 500 times faster compared to previous exact
approaches and instances with up to 125 vertices can be solved to optimality
within a timelimit of 1800 seconds. Moreover, the presented genetic algorithm
also works well and often finds the optimal or a near-optimal solution within a
short runtime. Additionally, we also analyze the influence of
instance-characteristics on the performance of our algorithms.
