有限阿贝尔凯莱图上Goemans-Linial半定规划的整性间隙边界
Integrality Gap Bounds for the Goemans-Linial SDP on Finite Abelian Cayley Graphs
- Maastricht University（马斯特里赫特大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究有限阿贝尔凯莱图上Goemans-Linial半定规划的整性间隙，证明指数不超过4时该松弛精确，给出生成元阶不超过R时的R/2近似，还构造了整性间隙为16/15的无限族图。
AI中文摘要：
在均匀最稀疏割问题中，我们需要找到一个顶点集，使得其切割的边数相对于它所分离的顶点对数量尽可能少。Goemans-Linial半定规划（SDP）结合Arora-Rao-Vazirani舍入方法，可对n个顶点的任意图给出O(√log n)的近似比。本文研究该松弛在有限阿贝尔凯莱图上的情况：首先，当图G=Cayley(Γ,S)的第二个归一化拉普拉斯特征值由一个像大小不超过4的傅里叶特征实现时，λ₂(G)=SDP_GL(G)=ψ(G)。从几何上看，该特征将顶点映射到正多边形，当多边形顶点数不超过4时，平方弦距离满足三角不等式；对相等的特征纤维进行分组可得到一个循环商，其中最优割可被精确找到，因此该松弛在指数不超过4的群上的有限阿贝尔凯莱图上是精确的。其次，我们将S的每个生成元s替换为其循环子群（包含单位元）中的均匀随机元素，设r_s为s的阶，α(r_s)为模拟此类移动所需的±s步的平均数量，ρ(S)=max_{s∈S}α(r_s)为最坏情况值。完全循环平均消除了特征相位，选择非平凡特征χ*最小化辅助特征值并取K=kerχ*，可得ψ(G)≤ψ_G(K)≤(q*/(q*-1))·ρ(S)·SDP_GL(G)≤2ρ(S)·SDP_GL(G)，其中q*=|χ*(Γ)|。若所有生成元的阶不超过R，则这是一个R/2的近似。最后，我们构造了一个无限族的有限阿贝尔凯莱图，其Goemans-Linial整性间隙恰好为16/15。
英文摘要：
In the uniform sparsest cut problem we are asked to find a vertex set that cuts few edges relative to the number of vertex pairs it separates. The Goemans-Linial SDP coupled with the Arora-Rao-Vazirani rounding gives an $\mathcal{O}(\sqrt{\log n})$ approximation on arbitrary graphs on $n$ vertices. We study this relaxation on finite Abelian Cayley graphs. First we show that when the second normalized Laplacian eigenvalue of $G= \mathrm{Cayley}(Γ, S)$ is realized by a Fourier character with image size at most four then $λ_2(G)=\mathrm{SDP}_{\mathrm{GL}}(G)=ψ(G)$. Geometrically, a character maps the vertices onto a regular polygon where the squared chord distance satisfies the triangle inequalities exactly when the polygon has at most four vertices. Grouping equal character fibers gives a cyclic quotient where the optimal cut can be found exactly and so the relaxation is exact on finite Abelian Cayley graphs on groups of exponent at most four. Second, we replace each generator $s$ of $S$ by a uniformly random element of its cyclic subgroup (including identity). If $r_s$ is the order of $s$, we let $α(r_s)$ to be the average number of $\pm s$ steps needed to simulate such a move, and let $ρ(S)=\max_{s\in S}α(r_s)$ be its worst case. Full cyclic averaging eliminates character phases and choosing a nontrivial character $χ^*$ minimizing the auxiliary eigenvalue and taking $K=\mathrm{ker}χ^*$ gives \[ ψ(G)\leqψ_G(K)\leq\frac{q^*}{q^*-1} \cdotρ(S)\cdot\mathrm{SDP}_{\mathrm{GL}}(G)\leq 2ρ(S)\cdot\mathrm{SDP}_{\mathrm{GL}}(G), \] where $q^*=|χ^*(Γ)|$. If all generator orders are at most $R$, this is an $R/2$ approximation. Finally, we construct an infinite family of finite Abelian Cayley graphs with Goemans-Linial integrality gap exactly $16/15$.
