Beyond Cut Balance: Spectral Sparsification of the Nonlinear Directed Laplacian

2026-09-08

AI Summary

研究探讨了非线性有向拉普拉斯算子的谱稀疏化问题，通过特定能量函数和凸对偶方法证明了仅切平衡不足以实现近线性谱稀疏化。

Abstract

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 $\widetilde\Theta(n^{3/2})$, 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.

Problem

• Spectral Sparsification
• Nonlinear Directed Laplacian
• Cut Balance

Innovation

• Spectral Sparsification
• Nonlinear Directed Laplacian
• Cut Balance
• Tournaments