Polynomial-Time Algorithms for Hamiltonian Path and Cycle in Graphs with Bounded Independence Number

Conceitos Básicos

Hamiltonian path and Hamiltonian cycle problems are solvable in polynomial time for graphs with bounded independence number.

Resumo

The content discusses the computational complexity of Hamiltonian path and Hamiltonian cycle problems in graphs, focusing on graphs with bounded independence number.
Key highlights:

Hamiltonian path and Hamiltonian cycle problems are NP-complete on general graphs, but the authors show that they can be solved in polynomial time for graphs with bounded independence number.
The authors introduce a more general problem called Hamiltonian-ℓ-Linkage, which asks if there exist ℓ disjoint paths that together cover all vertices of the graph. They prove that this problem is also solvable in polynomial time for graphs with bounded independence number.
As an application, the authors provide a complete characterization of the computational complexity of the L(2, 1)-labelling problem on H-free graphs and the related L′(2, 1)-labelling problem on triangle-free H-free graphs.
The authors present a recursive algorithm that solves the Hamiltonian-ℓ-Linkage problem by decomposing the graph into components based on a small vertex cut, and then solving the problem recursively on these components.

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Principais Insights Extraídos De

Hamiltonian path and Hamiltonian cycle are solvable in polynomial time in graphs of bounded independence number

by Niko... às arxiv.org 04-10-2024

https://arxiv.org/pdf/2309.09228.pdf

Perguntas Mais Profundas

What other graph problems beyond Hamiltonian path and cycle can be solved efficiently for graphs with bounded independence number

In addition to the Hamiltonian path and cycle, other graph problems that can be efficiently solved for graphs with bounded independence number include the Path Cover problem and the L(2, 1)-labelling problem. The Path Cover problem, which involves finding the smallest number of vertex-disjoint paths that cover all vertices of the graph, can be determined in polynomial time for graphs with bounded independence number. Similarly, the L(2, 1)-labelling problem, which aims to assign labels to vertices based on distance constraints, can also be efficiently solved for graphs with bounded independence number. These results provide insights into the computational complexity of various graph problems in the context of independence number constraints.

How tight are the running time bounds provided in the algorithm for Hamiltonian-ℓ-Linkage

The running time bounds provided in the algorithm for Hamiltonian-ℓ-Linkage are reasonably tight, considering the complexity of the problem and the constraints of bounded independence number. While the exponents in the running time bounds could potentially be improved with further algorithmic optimizations, the current bounds offer a practical and efficient solution for determining the existence of Hamiltonian linkages in graphs with bounded independence number. By refining the algorithm and exploring advanced data structures or parallel processing techniques, it may be possible to enhance the efficiency of the algorithm and potentially improve the exponents in the running time bounds.

Can the exponents be improved

Beyond the L(2, 1)-labelling problem discussed in the context, the Hamiltonian-ℓ-Linkage problem has various practical applications in network design, routing optimization, and resource allocation. For instance, in telecommunications networks, ensuring the existence of Hamiltonian linkages can help in establishing efficient communication paths between network nodes while minimizing interference and congestion. In transportation systems, Hamiltonian linkages can aid in designing optimal routes for vehicles or pedestrians, considering constraints such as distance and connectivity. Additionally, in supply chain management, Hamiltonian linkages can optimize the flow of goods or information through interconnected nodes, leading to streamlined operations and cost-effective logistics. These practical applications highlight the significance of the Hamiltonian-ℓ-Linkage problem in diverse real-world scenarios beyond theoretical graph theory.