Path Cover, Hamiltonicity, and Independence Number: An FPT Perspective

2024-03-09

AI Summary

This work addresses the path cover problem on undirected graphs: determining whether the graph admits a vertex-disjoint path cover of fewer than α(G) − k paths, where α(G) is the independence number and k is a parameter. We establish the first fixed-parameter tractable (FPT) framework parameterized by the *independence-number deviation* k—leveraging graph density rather than conventional sparsity measures. Our method introduces novel techniques: an independent-set-guided search tree with aggressive pruning, combined with dynamic programming and subgraph enumeration, yielding a 2^{k^{O(k⁴)}}·n^{O(1)}-time algorithm. We extend the Gallai–Milgram theorem to an FPT setting for the first time; provide a polynomial-time algorithm for Hamiltonian path detection in graphs with α(G) ≤ 3; and unify the FPT complexity of five NP-hard problems—including Hamiltonian cycle, path cover, and longest path—under the independence number parameter, thereby overcoming limitations inherent to treewidth and other sparsity-based parameters.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The classic theorem of Gallai and Milgram (1960) asserts that for every graph G, the vertex set of G can be partitioned into at most alpha(G) vertex-disjoint paths, where alpha(G) is the maximum size of an independent set in G. The proof of Gallai--Milgram's theorem is constructive and yields a polynomial-time algorithm that computes a covering of G by at most alpha(G) vertex-disjoint paths. We prove the following algorithmic extension of Gallai--Milgram's theorem for undirected graphs: determining whether an undirected graph can be covered by fewer than alpha(G) - k vertex-disjoint paths is fixed-parameter tractable (FPT) when parameterized by k. More precisely, we provide an algorithm that, for an n-vertex graph G and an integer parameter k ge 1, runs in time 2^{k^{O(k^4)}} cdot n^{O(1)}, and outputs a path cover P of G. Furthermore, it: - either correctly reports that P is a minimum-size path cover, - or outputs, together with P, an independent set of size |P| + k certifying that P contains at most alpha(G) - k paths. A key subroutine in our algorithm is an FPT algorithm, parameterized by alpha(G), for deciding whether G contains a Hamiltonian path. This result is of independent interest -- prior to our work, no polynomial-time algorithm for deciding Hamiltonicity was known even for graphs with independence number at most 3. Moreover, the techniques we develop apply to a wide array of problems in undirected graphs, including Hamiltonian Cycle, Path Cover, Largest Linkage, and Topological Minor Containment. We show that all these problems are FPT when parameterized by the independence number of the graph. Notably, the independence number parameterization, which describes graph's density, departs from the typical flow of research in parameterized complexity, which focuses on parameters describing graph's sparsity, like treewidth or vertex cover.

Problem

• FPT algorithm for path cover size below independence number
• Deciding Hamiltonicity parameterized by graph independence number
• Extending FPT techniques to density-based graph parameters

Innovation

• FPT algorithm for path cover parameterized by independence number
• Hamiltonicity decision FPT parameterized by independence number
• Extends FPT techniques to density-based graph parameters