基于节点特征的Max-$k$-Cut问题
Max-$k$-Cut via Node Features
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中文总结 AI 辅助
该研究从节点特征视角分析Max-$k$-Cut问题，推导Frieze-Jerrum松弛精确条件，关联其与向量平衡，证明贪心算法的近似保证，还给出后验最优性间隙证书。
中文摘要 AI 辅助
我们从节点特征的视角研究Max-$k$-Cut问题，其中每个顶点关联一个特征向量，边权重由两两内积给出。我们首先从该视角考察Max-$k$-Cut的半定松弛，利用法锥论证推导Frieze-Jerrum松弛精确性的一般充分条件，并证明该条件在两种特征结构场景下成立：完美特征平衡（各部分的聚合特征向量相等）和特征主导（少量大的非负特征向量决定最优划分结构）。随后我们证明Max-$k$-Cut目标等价于最小化分配给$k$个部分的聚合特征向量的平方范数之和，从而将该问题与向量平衡关联起来。受此观察启发，我们证明贪心特征平衡算法保留经典的$1-1/k$最坏情况近似保证，且在特征主导下能恢复最优划分。对于具有非负特征的秩1特征图，Chandra和Wong的贪心负载平衡经典界给出了可计算的后验最优性间隙证书，该证书仅依赖返回的划分，无需最优值的先验知识。
英文摘要
We study the Max-$k$-Cut problem from a node-feature perspective, where each vertex is associated with a feature vector and edge weights are given by pairwise inner products. We first examine the semidefinite relaxation of Max-$k$-Cut from this perspective. Using a normal-cone argument, we derive a general sufficient condition for exactness of the Frieze--Jerrum relaxation and show that it is satisfied in two feature-structural regimes: perfect feature balance, where the aggregate feature vectors of the parts are equal, and feature dominance, where a small set of large nonnegative feature vectors determines the structure of an optimal partition. We then show that the Max-$k$-Cut objective is equivalent to minimizing the sum of squared norms of the aggregate feature vectors assigned to the $k$ parts, thereby connecting the problem to vector balancing. Motivated by this observation, we show that a greedy feature-balancing algorithm retains the classical $1-1/k$ worst-case approximation guarantee and recovers an optimal partition under feature dominance. For rank-$1$ feature graphs with nonnegative features, classical bounds of Chandra and Wong for greedy load balancing yield a computable \emph{a posteriori} optimality-gap certificate that depends only on the returned partition and requires no knowledge of the optimum.
