Minimum Cuts in Surface-Embedded Graphs
📝 Original Paper Info
- Title: Minimum Cuts in Surface Graphs- ArXiv ID: 1910.04278
- Date: 2019-10-11
- Authors: Erin W. Chambers and Jeff Erickson and Kyle Fox and Amir Nayyeri
📝 Abstract
We describe algorithms to efficiently compute minimum $(s,t)$-cuts and global minimum cuts of undirected surface-embedded graphs. Given an edge-weighted undirected graph $G$ with $n$ vertices embedded on an orientable surface of genus $g$, our algorithms can solve either problem in $g^{O(g)} n \log \log n$ or $2^{O(g)} n \log n$ time, whichever is better. When $g$ is a constant, our $g^{O(g)} n \log \log n$ time algorithms match the best running times known for computing minimum cuts in planar graphs. Our algorithms for minimum cuts rely on reductions to the problem of finding a minimum-weight subgraph in a given $\mathbb{Z}_2$-homology class, and we give efficient algorithms for this latter problem as well. If $G$ is embedded on a surface with $b$ boundary components, these algorithms run in $(g + b)^{O(g + b)} n \log \log n$ and $2^{O(g + b)} n \log n$ time. We also prove that finding a minimum-weight subgraph homologous to a single input cycle is NP-hard, showing it is likely impossible to improve upon the exponential dependencies on $g$ for this latter problem.💡 Summary & Analysis
This research paper focuses on finding efficient algorithms to compute minimum cuts in surface-embedded graphs, which are undirected graphs that can be embedded onto a surface with certain characteristics. The authors aim to solve the problem of calculating minimum $(s,t)$-cuts and global minimum cuts within these types of graphs by utilizing their unique properties.The primary contribution of this paper is the development of algorithms that compute minimum cuts in $g^{O(g)} n \log \log n$ or $2^{O(g)} n \log n$ time, where $g$ represents the genus (a measure of complexity) of the surface and $n$ is the number of vertices. When the genus $g$ is a constant, these algorithms match the efficiency of those used for planar graphs.
The authors achieve this by reducing the problem to finding minimum-weight subgraphs within a given $\mathbb{Z}_2$-homology class. This reduction leverages the topological properties of surface-embedded graphs, enabling faster computation compared to general graph algorithms.
This work is significant as it provides a more efficient way to handle problems involving complex network structures such as communication networks or transportation systems where minimizing costs (represented by weights) between different parts of the system is crucial. By reducing the computational complexity, this research can lead to quicker and more effective solutions for real-world applications.
📄 Full Paper Content (ArXiv Source)
📊 논문 시각자료 (Figures)
