The Parameter Report: An Orientation Guide for Data-Driven Parameterization

Abstract: A strength of parameterized algorithmics is that each problem can be parameterized by an essentially inexhaustible set of parameters. Usually, the choice of the considered parameter is informed by the theoretical relations between parameters with the general goal of achieving FPT-algorithms for smaller and smaller parameters. However, the FPT-algorithms for smaller parameters usually have higher running times and it is unclear whether the decrease in the parameter value or the increase in the running time bound dominates in real-world data. This question cannot be answered from purely theoretical considerations and any answer requires knowledge on typical parameter values. To provide a data-driven guideline for parameterized complexity studies of graph problems, we present the first comprehensive comparison of parameter values for a set of benchmark graphs originating from real-world applications. Our study covers degree-related parameters, such as maximum degree or degeneracy, neighborhood-based parameters such as neighborhood diversity and modular-width, modulator-based parameters such as vertex cover number and feedback vertex set number, and the treewidth of the graphs. Our results may help assess the significance of FPT-running time bounds on the solvability of real-world instances. For example, the vertex cover number $vc$ of $n$-vertex graphs is often only slightly below $n/2$. Thus, a running time bound of $O(2^{vc})$ is only slightly better than a running time bound of $O(1.4^{n})$. In contrast, the treewidth $tw$ is almost always below $n/3$ and often close to $n/9$, making a running time of $O(2^{tw})$ much more practical on real-world instances. We make our implementation and full experimental data openly available. In particular, this provides the first implementations for several graph parameters such as 4-path vertex cover number and vertex integrity.