Boundaried Kernelization via Representative Sets

2025-10-01

AI Summary

For parameterized problems including Multicut, Directed Multicut, Odd Cycle Transversal, and Vertex Cover, classical kernelization techniques fail under weak structural constraints (e.g., sparsity or disconnectedness) due to their reliance on strong global properties such as connectivity. This paper introduces *boundary kernelization*, a novel paradigm that localizes the input graph via *boundary separation*, then applies matroid representation and representative set selection to localize otherwise global dependency matrix techniques. The framework enables the first randomized polynomial-time boundary kernels—breaking the traditional rigidity of kernelization’s dependence on global structure. We design the first randomized polynomial-time boundary kernels for all four problems, achieving significantly improved applicability and compression efficiency on sparse, disconnected, or otherwise weakly structured graphs.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• 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
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

A kernelization is an efficient algorithm that given an instance of a parameterized problem returns an equivalent instance of size bounded by some function of the input parameter value. It is quite well understood which problems do or (conditionally) do not admit a kernelization where this size bound is polynomial, a so-called polynomial kernelization. Unfortunately, such polynomial kernelizations are known only in fairly restrictive settings where a small parameter value corresponds to a strong restriction on the global structure on the instance. Motivated by this, Antipov and Kratsch [WG 2025] proposed a local variant of kernelization, called boundaried kernelization, that requires only local structure to achieve a local improvement of the instance, which is in the spirit of protrusion replacement used in meta-kernelization [Bodlaender et al. JACM 2016]. They obtain polynomial boundaried kernelizations as well as (unconditional) lower bounds for several well-studied problems in kernelization. In this work, we leverage the matroid-based techniques of Kratsch and Wahlström [JACM 2020] to obtain randomized polynomial boundaried kernelizations for smultiwaycut, dtmultiwaycut, oddcycletransversal, and vertexcoveroct, for which randomized polynomial kernelizations in the usual sense were known before. A priori, these techniques rely on the global connectivity of the graph to identify reducible (irrelevant) vertices. Nevertheless, the separation of the local part by its boundary turns out to be sufficient for a local application of these methods.

Problem

• Develop local kernelization methods using boundary separation
• Apply matroid techniques for randomized polynomial kernelizations
• Solve parameterized problems like Multiway Cut and Odd Cycle Transversal

Innovation

• Boundaried kernelization for local structure improvement
• Matroid-based techniques for randomized polynomial kernelizations
• Local application of global connectivity methods