Positive-Instance Driven Dynamic Programming for Graph Searching

Abstract

Research on the similarity of a graph to being a tree -called the treewidth of the graphhas seen an enormous rise within the last decade, but a practically fast algorithm for this task has been discovered only recently by Tamaki (ESA 2017). It is based on dynamic programming and makes use of the fact that the number of positive subinstances is typically substantially smaller than the number of all subinstances. Algorithms producing only such subinstances are called positive-instance driven (PID). We give an alternative and intuitive view on this algorithm from the perspective of the corresponding configuration graphs in certain two-player games. This allows us to develop PID-algorithms for a wide range of important graph parameters such as treewidth, pathwidth, and treedepth. We analyse the worst case behaviour of the approach on some well-known graph classes and perform an experimental evaluation on real world and random graphs.

A Machine Learning Approach to Algorithm Selection for Exact Computation of Treewidth

“…Our analysis ignores nodes that offer under 50% accuracy and nodes that contain very few (less than four) samples, as those are deemed not to bring a significant amount of insight to the analysis. q1 ≤ 4.5 gini = 0.594 samples = 213 salue= [24,93,96] class = tdlib entropy ≤ 4.73 gini = 0.452 samples = 116 salue= [11,23,82] class = tdlib minimum ≤ 3.5 gini = 0.321 samples = 90 salue= [5,12,73] class= tdlib gini = 0.281 samples = 87 salue= [4,10,73] class = tdlib variation ≤ 0.603 gini = 0.44 samples = 97 salue= [13,70,14] class The first line in each internal node indicates the condition according to which that node splits instances.…”

“…Given its relevance as a 'gateway to tractability', treewidth has attracted enormous interest not just from the combinatorial optimisation and theoretical computer science communities but also from many other domains such as artificial intelligence (particularly Bayesian inference and constraint satisfaction: see Reference [5] for a highly comprehensive list), computational biology [6] and operations research [7]. Unfortunately, computing the treewidth (equivalently, a minimum-width tree decomposition) is NP-hard [8].…”