A heuristic algorithm using tree decompositions for the maximum happy vertices problem

TL;DR

This paper introduces a heuristic algorithm based on tree decompositions for the Maximum Happy Vertices problem, balancing solution quality and computational efficiency by adjusting a parameter that controls the dynamic programming state space.

Contribution

It presents a novel heuristic approach that leverages tree decompositions with a tunable parameter to improve solution quality and runtime over existing methods.

Findings

The heuristic constructs optimal solutions more efficiently for graphs of bounded treewidth.

It produces higher quality solutions than existing heuristics when many vertices are initially colored.

The algorithm's performance improves as the parameter W increases, approaching optimality.

Abstract

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.