曲面嵌入图的最小圈与同调基
Minimum cycle and homology bases of surface embedded graphs
摘要：
我们研究的问题，找到一个最小的循环基础（一个最小的重量集的循环，形成一个基础的循环空间）和最小的同源基础（一个最小的重量集的循环，产生$1$ -维（$\mathbb{Z}_2$）-同源类）的无向图嵌入在一个表面上。这两个问题是密切相关的，因为图的最小圈基包含它的最小同调基，而任何图的$1$ -骨架的最小同调基恰是它的最小圈基。对于最小圈基问题，我们给出了嵌入在亏格为$g$的可定向曲面上的图的确定性$O(n^ω+2^{2g}n^2+m)$ -时间算法。最著名的表面嵌入图的现有算法是那些一般的图：$O(m^ω)$时间蒙特卡罗算法和确定性$O(nm^2/\log n + n^2 m)$时间算法。对于最小同调基问题，我们给出了一个确定性的$O((g+b)^3 n \log n + m)$ -时间算法嵌入到一个可定向或不可定向表面上的属$g$与$b$边界组件的图，假设最短路径是唯一的，改进了现有的算法的许多值的$g$和$n$。唯一最短路径的假设可以使用随机化以高概率避免，或者通过将同源基算法的运行时间增加因子$O(\log n)$而确定性地避免。
英文摘要：
We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the $1$-dimensional ($\mathbb{Z}_2$)-homology classes) of an undirected graph embedded on a surface. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the $1$-skeleton of any graph is exactly its minimum cycle basis. For the minimum cycle basis problem, we give a deterministic $O(n^ω+2^{2g}n^2+m)$-time algorithm for graphs embedded on an orientable surface of genus $g$. The best known existing algorithms for surface embedded graphs are those for general graphs: an $O(m^ω)$ time Monte Carlo algorithm and a deterministic $O(nm^2/\log n + n^2 m)$ time algorithm. For the minimum homology basis problem, we give a deterministic $O((g+b)^3 n \log n + m)$-time algorithm for graphs embedded on an orientable or non-orientable surface of genus $g$ with $b$ boundary components, assuming shortest paths are unique, improving on existing algorithms for many values of $g$ and $n$. The assumption of unique shortest paths can be avoided with high probability using randomization or deterministically by increasing the running time of the homology basis algorithm by a factor of $O(\log n)$.
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