  We propose a new methodology to develop heuristic algorithms using tree
decompositions. Traditionally, such algorithms construct an optimal solution of
the given problem instance through a dynamic programming approach. We modify
this procedure by introducing a parameter $W$ that dictates the number of
dynamic programming states to consider. We drop the exactness guarantee in
favour of a shorter running time. However, if $W$ is large enough such that all
valid states are considered, our heuristic algorithm proves optimality of the
constructed solution. In particular, we implement a heuristic algorithm for the
Maximum Happy Vertices problem using this approach. Our algorithm more
efficiently constructs optimal solutions compared to the exact algorithm for
graphs of bounded treewidth. Furthermore, our algorithm constructs higher
quality solutions than state-of-the-art heuristic algorithms Greedy-MHV and
Growth-MHV for instances of which at least 40\% of the vertices are initially
coloured, at the cost of a larger running time.
