# Positive-Instance Driven Dynamic Programming for Graph Searching

## TL;DR

This paper introduces a positive-instance driven dynamic programming approach for efficiently computing graph parameters like treewidth, pathwidth, and treedepth, with theoretical analysis and experimental validation on various graph types.

## Contribution

It provides an alternative, intuitive perspective on Tamaki's treewidth algorithm and extends the PID approach to multiple graph parameters.

## Key findings

- Efficient algorithms for treewidth, pathwidth, and treedepth based on positive-instance driven dynamic programming.
- Analysis of worst-case behavior on well-known graph classes.
- Experimental results demonstrating effectiveness on real-world and random graphs.

## Abstract

Research on the similarity of a graph to being a tree - called the treewidth of the graph - has 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.