发表机构
University of Rochester（罗切斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究加权图边复杂度的计算难度，通过添加通用顶点建立与对合对称性的联系，证明其近似计算是NP困难的，并给出NP完全性及非实傅里叶系数的示例。
AI 中文摘要
加权图的边复杂度是其邻接矩阵在所有顶点标号下的傅里叶$\ell^1$范数与$\ell^2$范数之比的最小值。我们通过将其与图的对称性联系起来，研究寻找该最小值的难度。添加一个具有足够大关联权重的通用顶点，可产生一个显式的傅里叶$\ell^1$下界。我们证明，当且仅当源图具有无不动点的对合自同构时，等号成立。双边估计将该下界之上的超出量与到具有此类对称性的最近加权图的平方Frobenius距离进行比较。对于常数加权度的源图，当添加的权重趋于无穷时，这些估计确定了精确的首项。对于简单源图，一个更强的分离结果证明了加权边复杂度的加法$\frac{1}{256N^{\frac{7}{2}}}$近似是NP困难的，即使在具有奇数阶$N$且至多两个不同的正整数权重（每个权重至多为$N^2$）的连通图上也是如此。我们还证明了识别具有实傅里叶标号的简单图是NP完全的。一个七顶点示例表明，即使存在实傅里叶标号，每个最小化标号也可能具有非实傅里叶系数。该示例的精确有理数证书包含在附录中。
英文摘要
The edge complexity of a weighted graph is the smallest ratio of the Fourier $\ell^1$ and $\ell^2$ norms of its adjacency matrix over all vertex labelings. We study the difficulty of finding this minimum by relating it to a graph symmetry. Adding a universal vertex with sufficiently large incident weight produces an explicit Fourier $\ell^1$ lower bound. We show that equality holds exactly when the source graph has a fixed-point-free involutory automorphism. Two-sided estimates compare the excess above this bound with the squared Frobenius distance to the nearest weighted graph having such a symmetry. For sources of constant weighted degree, these estimates determine the exact leading term as the added weight tends to infinity. A stronger separation for simple source graphs proves that additive $\frac{1}{256N^{\frac{7}{2}}}$ approximation of weighted edge complexity is NP-hard, even on connected graphs of odd order $N$ with at most two distinct positive integer weights, each at most $N^2$. We also prove that recognizing a simple graph with a real Fourier labeling is NP-complete. A seven-vertex example shows that every minimizing labeling can have nonreal Fourier coefficients even when real Fourier labelings exist. An exact rational certificate for this example is included in the appendix.
Comments21 pages
