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 $n/10$, 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.

翻译：参数化算法的一个优势在于，每个问题都可以通过本质上无穷无尽的参数集进行参数化。通常，所选参数的确定受到参数间理论关系的指导，其总体目标是针对越来越小的参数实现FPT算法。然而，针对较小参数的FPT算法通常具有更高的运行时间，且尚不清楚在真实世界数据中，参数值的下降与运行时间界的上升何者占据主导地位。这一问题无法仅通过纯理论考量得到解答，任何答案都需要了解典型参数值。为提供图问题参数化复杂性研究的数据驱动指导，我们首次对源自真实世界应用的一组基准图进行了参数值的全面比较。我们的研究涵盖度相关参数（如最大度或退化度）、基于邻域的参数（如邻域多样性和模宽度）、基于调制器的参数（如顶点覆盖数和反馈顶点集数）以及图的树宽。我们的结果可能有助于评估FPT运行时间界对真实世界实例可解性的重要性。例如，$n$顶点图的顶点覆盖数$vc$通常仅略低于$n/2$。因此，$O(2^{vc})$的运行时间界仅比$O(1.4^{n})$稍好。相比之下，树宽$tw$几乎始终低于$n/3$，且常接近$n/10$，这使得$O(2^{tw})$的运行时间在真实世界实例上更为实用。我们公开提供实现代码及完整实验数据。特别地，这为多个图参数（如4-路径顶点覆盖数和顶点完整性）提供了首个实现方案。