最优Max-Cut SDP解的几何结构
The Geometry of Optimal Max-Cut SDP Solutions
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- Indian Institute of Technology Bombay（印度理工学院孟买分校）
- The Ohio State University（俄亥俄州立大学）
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中文总结 AI 辅助
本文研究Max-Cut半定松弛精确时最优割符号向量张成对偶松弛核的条件，证明双非负矩阵经偶数复制可作证书，并刻画有理核障碍与张成亏缺。
中文摘要 AI 辅助
当Max-Cut的半定松弛是精确的，其最优割符号向量位于最优对偶松弛变量的核中。我们研究这些向量何时张成该核，以及哪些矩阵可以作为具有此性质的证书出现。我们的主要结果表明，在足够大的偶数复制后，每个具有正对角元和有理核的双非负矩阵都可以作为这样的证书。复制保持秩、完全正性和有限时的cp-秩。特别地，在秩至少为三的每个秩上，都存在连通的精确实例，其符号张成的松弛变量不是完全正的，包括一个显式的十四顶点例子。我们还刻画了算术障碍：某个正列复制具有符号张成的核当且仅当原始核是有理的。对于任意实核，我们确定了最终的张成亏缺。一个重数公式分离了重复列块及其可行和的贡献。进一步的结果区分了割张成与最优面几何，通过均匀完全图和Hadamard矩阵刻画了正交极值性，并给出了支持精确有限分类的行列式界。
英文摘要
When the semidefinite relaxation of Max-Cut is exact, its optimal cut sign vectors lie in the kernel of the optimal dual slack. We study when they span this kernel and which matrices can occur as certificates with this property. Our main result realizes every doubly nonnegative matrix with positive diagonal and rational kernel, after sufficiently large even replication, as such a certificate. Replication preserves rank, complete positivity, and cp-rank when finite. In particular, connected exact instances with non-completely-positive sign-spanned slacks exist at every rank at least three, including an explicit fourteen-vertex example. We also characterize the arithmetic obstruction: some positive column replication has a sign-spanned kernel if and only if the original kernel is rational. For arbitrary real kernels, we determine the eventual spanning deficit. A multiplicity formula separates the contributions of repeated-column blocks and their feasible sums. Further results distinguish cut spans from optimal-face geometry, characterize orthogonal extremality through uniform complete graphs and Hadamard matrices, and give determinant bounds supporting exact finite classifications.
