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.

翻译：摘要：本文提出了一种利用树分解开发启发式算法的新方法论。传统上，此类算法通过动态规划方法构造给定问题实例的最优解。我们通过引入参数$W$（决定待考虑动态规划状态的数量）来改进这一过程。我们舍弃精确性保证以换取更短的运行时间。然而，若$W$足够大以涵盖所有有效状态，则我们的启发式算法可证明构造解的最优性。具体而言，我们采用此方法实现了一个针对最大幸福顶点问题的启发式算法。与针对有界树宽图的精确算法相比，我们的算法能更高效地构造最优解。此外，对于初始着色顶点占比至少为40%的实例，尽管运行时间更长，但我们的算法比当前最先进的启发式算法Greedy-MHV和Growth-MHV能构造出更高质量的解决方案。