图的弱嵌入的识别
Recognizing Weak Embeddings of Graphs
摘要：
我们提出了一个有效的算法，在聚类和图嵌入之间的接口的问题。图$G$到2-流形$M$的嵌入$φ:G\rightarrow M$将$V(G)$中的顶点映射到不同的点，将$E(G)$中的边映射到相应顶点之间的相邻不相交的Jordan弧。在聚类、制图和可视化应用中，由于数据压缩或低分辨率，附近的顶点和边经常被捆绑到同一点或重叠弧。这引起了决定给定映射$φ:G\rightarrow M$是否来自嵌入的计算问题。一个映射$φ:G\rightarrow M$是一个\textbf{weak embedding}，如果它可以被扰动到一个嵌入$ψ_\varepsilon:G\rightarrow M$中，其中对每个$\varepsilon>0$都有$\|φ-ψ_\varepsilon\|<\varepsilon$，其中$\|.\|$是unform范数。最近Fulek和Kynčl发现了一个识别弱嵌入的多项式时间算法。它将问题简化为求解$\mathbb{Z}_2$上的线性方程组。它以$O(n^{2ω})\leq O(n^{4.75})$时间运行，其中$ω\in [2,2.373)$是矩阵乘法指数，$n$是$G$的顶点和边的数量。我们将运行时间提高到$O(n\log n)$。我们的算法在概念上也更简单：我们执行一系列的\emph{local operations}，逐渐“解开”图像$φ(G)$到嵌入$ψ(G)$，或报告$φ$不是一个弱嵌入。它直接结合了子图方向的局部约束，从而消除了求解大型线性方程组的需要。
英文摘要：
We present an efficient algorithm for a problem in the interface between clustering and graph embeddings. An embedding $φ:G\rightarrow M$ of a graph $G$ into a 2-manifold $M$ maps the vertices in $V(G)$ to distinct points and the edges in $E(G)$ to interior-disjoint Jordan arcs between the corresponding vertices. In applications in clustering, cartography, and visualization, nearby vertices and edges are often bundled to the same point or overlapping arcs, due to data compression or low resolution. This raises the computational problem of deciding whether a given map $φ:G\rightarrow M$ comes from an embedding. A map $φ:G\rightarrow M$ is a \textbf{weak embedding} if it can be perturbed into an embedding $ψ_\varepsilon:G\rightarrow M$ with $\|φ-ψ_\varepsilon\|<\varepsilon$ for every $\varepsilon>0$, where $\|.\|$ is the unform norm. A polynomial-time algorithm for recognizing weak embeddings has recently been found by Fulek and Kynčl. It reduces the problem to solving a system of linear equations over $\mathbb{Z}_2$. It runs in $O(n^{2ω})\leq O(n^{4.75})$ time, where $ω\in [2,2.373)$ is the matrix multiplication exponent and $n$ is the number of vertices and edges of $G$. We improve the running time to $O(n\log n)$. Our algorithm is also conceptually simpler: We perform a sequence of \emph{local operations} that gradually "untangles" the image $φ(G)$ into an embedding $ψ(G)$, or reports that $φ$ is not a weak embedding. It combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
