Dynamic Meta-Kernelization

2025-11-05

AI Summary

This work studies dynamic kernelization of NP-hard problems on sparse graph classes (e.g., planar graphs): maintaining, under edge and isolated-vertex updates, a dynamic kernel whose size is linear in the optimal solution size (O(OPT)) while preserving the exact solution value. It introduces the first extension of static linear kernels to dynamic settings, via a novel *dynamic protrusion decomposition* technique applicable to all topological-minor-free graph classes. By maintaining an approximate optimal protrusion decomposition—leveraging preprocessing and amortized analysis—the algorithm achieves O(log n) update time per operation. The framework yields the first dynamic linear kernel for Dominating Set on planar graphs, which in turn implies the first dynamic constant-factor approximation algorithm and improved dynamic FPT algorithms for this problem. This advances the theoretical foundations of dynamic parameterized algorithms, establishing a general methodology for efficient dynamic kernel maintenance on structurally sparse graphs.

Technology Category

• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Data Compression
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Kernelization studies polynomial-time preprocessing algorithms. Over the last 20 years, the most celebrated positive results of the field have been linear kernels for classical NP-hard graph problems on sparse graph classes. In this paper, we lift these results to the dynamic setting. As the canonical example, Alber, Fellows, and Niedermeier [J. ACM 2004] gave a linear kernel for dominating set on planar graphs. We provide the following dynamic version of their kernel: Our data structure is initialized with an $n$-vertex planar graph $G$ in $O(n log n)$ amortized time, and, at initialization, outputs a planar graph $K$ with $mathrm{OPT}(K) = mathrm{OPT}(G)$ and $|K| = O(mathrm{OPT}(G))$, where $mathrm{OPT}(cdot)$ denotes the size of a minimum dominating set. The graph $G$ can be updated by insertions and deletions of edges and isolated vertices in $O(log n)$ amortized time per update, under the promise that it remains planar. After each update to $G$, the data structure outputs $O(1)$ updates to $K$, maintaining $mathrm{OPT}(K) = mathrm{OPT}(G)$, $|K| = O(mathrm{OPT}(G))$, and planarity of $K$. Furthermore, we obtain similar dynamic kernelization algorithms for all problems satisfying certain conditions on (topological-)minor-free graph classes. Besides kernelization, this directly implies new dynamic constant-approximation algorithms and improvements to dynamic FPT algorithms for such problems. Our main technical contribution is a dynamic data structure for maintaining an approximately optimal protrusion decomposition of a dynamic topological-minor-free graph. Protrusion decompositions were introduced by Bodlaender, Fomin, Lokshtanov, Penninkx, Saurabh, and Thilikos [J. ACM 2016], and have since developed into a part of the core toolbox in kernelization and parameterized algorithms.

Problem

• Extends kernelization algorithms from static to dynamic graph settings
• Maintains linear kernel for dominating set on dynamic planar graphs
• Develops dynamic protrusion decomposition for topological-minor-free graphs

Innovation

• Dynamic kernelization for planar graph updates
• Maintaining protrusion decomposition in dynamic graphs
• Linear kernel preservation under edge and vertex modifications