Computing Tree Width: From Theory to Practice and Back

While the theoretical aspects concerning the computation of tree width – one of the most important graph parameters – are well understood, it is not clear how it can be computed practically. As tree width has a wide range of applications, e. g. in bioinformatics or artificial intelligence, this lack of understanding hinders the applicability of many important algorithms in the real world. The Parameterized Algorithms and Computational Experiments (PACE) challenge therefore chose the computation of tree width as one of its challenge problems in 2016 and again in 2017. In 2016, Hisao Tamaki (Meiji University) presented a new algorithm that outperformed the other approaches (including SAT solvers and branch-and-bound) by far. An implementation of Tamaki’s algorithm allowed Larisch (King-Abdullah University of Science and Engineering) and Salfelder (University of Leeds) to solve over 50% of the test suite of PACE 2017 (containing graphs with over 3500 nodes) in under six seconds (and the remaining 50% in under 30 min). Before PACE 2016, no algorithm was known to compute tree width on graphs with about 100 nodes. As a wide range of parameterized algorithms require the computation of a tree decomposition as a first step, this breakthrough result allows practical implementations of these algorithms for the first time. This work starts with a gentle introduction to tree width and its use in parameterized complexity, followed by an algorithmic approach for the exact computation of the tree width of a graph, based on a variant of the well-studied cops-and-robber game. Finally, we present a streamlined version of Tamaki’s algorithms due to Bannach and Berndt based on this game.