SIAM Journal on Computing, Ahead of Print.
We develop a framework for graph sparsification and sketching, based on a new tool, short cycle decomposition, which is a decomposition of an unweighted graph into an edge-disjoint collection of short cycles, plus a small number of extra edges. A simple observation shows that every graph $G$ on $n$ vertices with $m$ edges can be decomposed in $O(mn)$ time into cycles of length at most $2 \log n,$ and at most $2n$ extra edges. We give an $(m^{1+o(1)})$-time algorithm for constructing a short cycle decomposition, with cycles of length $n^{o(1)},$ and $n^{1+o(1)}$ extra edges. Both the existential and algorithmic variants of this decomposition enable us to make the following progress on several open problems in randomized graph algorithms: (1) We present an algorithm that runs in time $m^{1+o(1)}\varepsilon^{-1.5}$ and returns $(1\pm\varepsilon)$-approximations to effective resistances of all edges, improving over the previous best runtime of $\widetilde{{O}}(\min\{m\varepsilon^{-2}, n^{2} \varepsilon^{-1}\})$. This routine in turn gives an algorithm for approximating the determinant of a graph Laplacian up to a factor of $(1\pm \varepsilon)$ in $m^{1 + o(1)} + n^{\nicefrac{15}{8}+o(1)}\varepsilon^{-\nicefrac{7}{4}}$ time. (2) We show the existence of graphical spectral sketches with about $n\varepsilon^{-1}$ edges, and also give efficient algorithms to construct them. A graphical spectral sketch is a distribution over sparse graphs $H$ such that for a fixed vector ${\mathit{x}}$, we have ${{x}}^{\top} {L}_H {{x}} = (1\pm\varepsilon) {{x}}^{\top} {L}_G {{x}}$ and ${{x}}^{\top} {L}^{+}_H {{x}} = (1\pm\varepsilon) {{x}}^{\top} {L}^{+}_G {{x}}$ with high probability, where ${L}$ is the graph Laplacian and ${L}^{+}$ is its pseudoinverse. This implies the existence of resistance sparsifiers with about $n \varepsilon^{-1}$ edges that preserve the effective resistance between every pair of vertices up to $(1\pm\varepsilon)$. (3) By combining short cycle decompositions with known tools in graph sparsification, we show the existence of nearly linear sized degree-preserving spectral sparsifiers, as well as significantly sparser approximations of Eulerian directed graphs. The latter is critical to recent breakthroughs on faster algorithms for solving linear systems in directed Laplacians. The running time and output qualities of our spectral sketch and degree-preserving (directed) sparsification algorithms are limited by the efficiency of our routines for constructing short cycle decompositions. Improved algorithms for short cycle decompositions will lead to improvement in each of these algorithms.

中文翻译：

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通过短周期分解实现图稀疏化，频谱草图绘制和更快的电阻计算

《 SIAM计算杂志》，预印本。
我们基于新工具短周期分解开发了一种用于图形稀疏化和草图绘制的框架，该工具是将未加权图形分解为短周期的边缘不交集，以及少量额外的边。一个简单的观察表明，在具有$ m $边的$ n $顶点上的每个图形$ G $可以在$ O（mn）$的时间内分解成长度为$ 2 \ log n，$的循环，最多为$ 2n $的额外循环边缘。我们给出了一个$（m ^ {1 + o（1）}）$时间算法来构造一个短周期分解，其循环长度为$ n ^ {o（1）}，$和$ n ^ {1 + o （1）} $个额外的边缘。此分解的存在变量和算法变量都使我们能够在随机图算法中的几个开放问题上取得以下进展：（1）我们提出了一种运行时间为$ m ^ {1 + o（1）} \ varepsilon ^的算法。 {-1。5} $并返回所有边缘有效电阻的$（1 \ pm \ varepsilon）$近似值，比以前的$ \ widetilde {{O}}（\ min \ {m \ varepsilon ^ {-2 }，n ^ {2} \ varepsilon ^ {-1} \}）$。该例程进而给出了一种算法，用于将图拉普拉斯的行列式近似为$ m ^ {1 + o（1）} + n ^ {\ nicefrac {15}中的$（1 \ pm \ varepsilon）$ {8} + o（1）} \ varepsilon ^ {-\ nicefrac {7} {4}} $时间。（2）我们显示了具有约$ n \ varepsilon ^ {-1} $边的图形频谱草图的存在，并给出了构造它们的有效算法。图形频谱草图是稀疏图$ H $的分布，这样对于固定向量$ {\ mathit {x}} $，我们有$ {{x}} ^ {\ top} {L} _H {{x}} =（1 \ pm \ varepsilon）{{x}} ^ {\ top} {L} _G {{x}} $和$ {{x}} ^ {\ top} {L} ^ {+} _ H {{x}} =（1 \ pm \ varepsilon）{{x}} ^ {\ top} {L} ^ {+} _G {{x}} $的概率很高，其中$ {L} $是图拉普拉斯算子，而$ {L} ^ {+} $是其伪逆。这意味着存在具有大约$ n \ varepsilon ^ {-1} $边缘的电阻稀疏器，该边缘保留了每对顶点之间的有效电阻，直至$（1 \ pm \ varepsilon）$。（3）通过结合图稀疏化中的短周期分解与已知工具，我们证明了存在几乎线性大小的保度谱稀疏器，以及欧拉定向图的显着稀疏近似。后者对于有向拉普拉斯算子中解决线性系统更快算法的最新突破至关重要。我们的光谱草图和度保持（定向）稀疏化算法的运行时间和输出质量受到构造短周期分解的例程效率的限制。用于短周期分解的改进算法将导致这些算法中的每一种都有改进。