超越割平衡：非线性有向拉普拉斯算子的谱稀疏化

中文摘要 AI 辅助

具有常数割平衡的有向图允许近线性的有向割稀疏化。该条件要求每个割的两个方向上的总弧权重彼此在常数因子范围内。我们询问该条件是否也允许关于非线性有向拉普拉斯算子能量的近线性谱稀疏化。对于加权有向图 $G=(V,E,w)$，设 \\[ Q_G^+(x)=\sum_{(u,v)\in E}w_{uv}(x_u-x_v)_+^2, \qquad (t)_+:=\max\{t,0\}. \\] 该能量在二元向量上与出割函数一致。谱稀疏化器是一个非负重新加权的子图，它同时对所有 $x\in\mathbb R^V$ 在 $1\pm\varepsilon$ 因子内保持 $Q_G^+(x)$。我们证明仅割平衡本身并不能产生近线性谱稀疏化：对于常数误差，简单无权重欧拉有向图的最坏情况支撑大小为 $\widetilde\Theta(n^{3/2})$，尽管欧拉有向图是完美割平衡的并且允许近线性有向割稀疏化。相比之下，我们证明每个 $n$ 顶点锦标赛都有一个具有 $\widetilde O(n/\varepsilon^3)$ 条弧的谱稀疏化器，而不对其割平衡做任何假设。这包括割平衡无界的传递锦标赛。因此，完美平衡并不保证近线性谱稀疏化，而无界不平衡也不排除它。最后，我们使用凸对偶性证明，保持 $Q_G^+$ 也保持每个可行需求向量的非负流的最优二次成本。因此，该保证包含超出有向割值的信息。

英文摘要

Digraphs with constant cut balance admit nearly linear directed cut sparsifiers. This condition requires the total arc weights in the two directions of every cut to be within a constant factor of each other. We ask whether this condition also permits nearly linear spectral sparsification with respect to the energy of the nonlinear directed Laplacian. For a weighted digraph $G=(V,E,w)$, let \[ Q_G^+(x)=\sum_{(u,v)\in E}w_{uv}(x_u-x_v)_+^2, \qquad (t)_+:=\max\{t,0\}. \] This energy agrees with the outgoing-cut function on binary vectors. A spectral sparsifier is a nonnegatively reweighted subgraph that preserves $Q_G^+(x)$ within a factor of $1\pm\varepsilon$ simultaneously for all $x\in\mathbb R^V$. We show that cut balance alone does not yield nearly linear spectral sparsifiers: for constant error, the worst-case support size for simple unweighted Eulerian digraphs is $\widetildeΘ(n^{3/2})$, although Eulerian digraphs are perfectly cut-balanced and admit nearly linear directed cut sparsifiers. In contrast, we prove that every $n$-vertex tournament has a spectral sparsifier with $\widetilde O(n/\varepsilon^3)$ arcs, without any assumption on its cut balance. This includes the transitive tournament, whose cut balance is unbounded. Thus perfect balance does not guarantee nearly linear spectral sparsification, while unbounded imbalance does not preclude it. Finally, we use convex duality to show that preserving $Q_G^+$ also preserves, for every feasible demand vector, the optimum quadratic cost of a nonnegative flow. Hence the guarantee contains information beyond directed cut values.

发表机构

• National Institute of Informatics（国立情报学研究所）

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