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

2025-09-08

AI Summary

This study addresses the practical applicability of parameterized algorithms by investigating how theoretical graph parameters correlate with real-world instances. Method: We conduct a large-scale empirical analysis—first of its kind across diverse real-world graph benchmarks—measuring distributions of over ten structural parameters, including degeneracy, neighborhood diversity, modular width, vertex cover number, feedback vertex set size, and treewidth. Contribution/Results: We reveal critical discrepancies between theoretical assumptions and empirical behavior: treewidth is typically ≈ *n*/9 (well below the worst-case *n*/3), rendering *O*(2^tw) algorithms practically viable; in contrast, vertex cover number often approaches *n*/2, substantially eroding its fixed-parameter tractability advantage. We publicly release the first unified computational framework supporting emerging parameters (e.g., 4-path vertex cover number) and a comprehensive experimental dataset, establishing a data-driven foundation for the design, selection, and optimization of parameterized algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Algorithm Configuration
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Comparing parameter values in real-world graphs for complexity analysis
• Assessing practical significance of FPT algorithms on benchmark instances
• Providing data-driven guidelines for parameterized algorithm selection

Innovation

• Comprehensive comparison of graph parameter values
• Implementation of several graph parameter algorithms
• Data-driven guideline for parameterized complexity studies