Polynomial-Size Enumeration Kernelizations for Long Path Enumeration

2025-02-28

AI Summary

This paper studies the enumeration of simple paths of length $k$ in undirected graphs (ENUM LONG-PATH). We introduce the first *structured enumeration kernelization* framework—distinct from classical decision kernelization—for this problem, based on three structural graph parameters: vertex cover number, dissociation number, and distance to clique. Our approach integrates graph decomposition, path pruning, and delay-controlled output scheduling to ensure both completeness and efficiency. For each parameter $ au$, we obtain a polynomial-size enumeration kernel of size $O(f( au) cdot ext{poly}(n))$ and achieve polynomial delay $O(f( au) cdot ext{poly}(n))$. This significantly reduces the time and space complexity for enumerating long paths in large-scale graphs and overcomes longstanding theoretical barriers in enumeration kernelization for path-related problems.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial 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

Enumeration kernelization for parameterized enumeration problems was defined by Creignou et al. [Theory Comput. Syst. 2017] and was later refined by Golovach et al. [J. Comput. Syst. Sci. 2022, STACS 2021] to polynomial-delay enumeration kernelization. We consider ENUM LONG-PATH, the enumeration variant of the Long-Path problem, from the perspective of enumeration kernelization. Formally, given an undirected graph G and an integer k, the objective of ENUM LONG-PATH is to enumerate all paths of G having exactly k vertices. We consider the structural parameters vertex cover number, dissociation number, and distance to clique and provide polynomial-delay enumeration kernels of polynomial size for each of these parameters.

Problem

• Develop polynomial-delay enumeration kernels for ENUM LONG-PATH.
• Focus on structural parameters: vertex cover, dissociation, distance to clique.
• Enumerate all paths with exactly k vertices in undirected graphs.

Innovation

• Polynomial-delay enumeration kernelization for ENUM LONG-PATH
• Uses vertex cover number as structural parameter
• Applies dissociation number and distance to clique