Kadison--Singer 划分与 Bilu--Linial 图符号的多项式时间算法
Kadison--Singer partitions and Bilu--Linial graph signings in polynomial time
- Massachusetts Institute of Technology（麻省理工学院）
- Stanford University（斯坦福大学）
- Technical University of Munich（慕尼黑工业大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出谱偏差的两个多项式时间算法：确定性舍入定理实现Kadison--Singer半划分与图符号，以及Las Vegas算法求解Bilu--Linial符号问题，达到接近最优的谱界。
AI中文摘要：
我们在谱偏差领域证明了两个主要的算法结果。首先，我们给出一个针对任意秩的有理正半定矩阵的确定性多项式时间舍入定理。该算法从任意有理分数符号开始，并为每个原始矩阵分配一个符号。其偏差小于 $3.37\\,\\|\sum_i \mathrm{Tr}(A_i)A_i\\|^{1/2}$。这产生了误差低于 $1.69\sqrt{\varepsilon}$ 的 Kadison--Singer 半划分，以及同时控制带符号邻接矩阵和带符号度数的确定性图符号。证明基于 Ezeunala 和 Jiang (2026) 的谱势方法，并引入了一种选择舍入方向的新方法。我们证明了舍入过程的多项式比特复杂度。其次，我们给出了一个针对任意给定图的 Bilu--Linial 符号问题的 Las Vegas 算法。如果 $G$ 有 $n$ 个顶点且最大度 $\Delta\ge3$，该算法几乎必然终止。它在期望上使用少于 $100n^3$ 次插入尝试，并返回一个满足 $\\|A_s\\|<2\sqrt{2(\Delta-1)}$ 的符号。对于二分图，其单边形式给出了尖锐的普遍界 $\\|A_s\\|<2\sqrt{\Delta-1}$。该算法通过插入顶点并在拒绝插入后递归删除和恢复邻居来构建符号。在分析中，与 Bilu--Linial 猜想的 $\sqrt2$ 差距来自于双边情形中顶点删除界的一个因子二。在 $d$-正则二分 Ramanujan 基上，相同的符号产生该给定基的 Ramanujan $2$-提升。
英文摘要：
We prove two main algorithmic results in spectral discrepancy. First, we give a deterministic polynomial-time rounding theorem for rational positive semidefinite matrices of arbitrary rank. The algorithm starts from any rational fractional signing and assigns one sign per original matrix. Its discrepancy is less than $3.37\,\|\sum_i \mathrm{Tr}(A_i)A_i\|^{1/2}$. This yields Kadison--Singer half-partitions with error below $1.69\sqrt{\varepsilon}$, as well as deterministic graph signings that control signed adjacency and signed degrees simultaneously. The proof builds on the spectral-potential method of Ezeunala and Jiang (2026) and introduces a new way to choose rounding directions. We prove polynomial bit complexity for the rounding procedure. Second, we give a Las Vegas algorithm for the Bilu--Linial signing problem on an arbitrary prescribed graph. If $G$ has $n$ vertices and maximum degree $Δ\ge3$, the algorithm terminates almost surely. It uses fewer than $100n^3$ insertion attempts in expectation and returns a signing with $\|A_s\|<2\sqrt{2(Δ-1)}$. For bipartite graphs its one-sided form gives the sharp universal bound $\|A_s\|<2\sqrt{Δ-1}$. The algorithm builds the signing by inserting vertices and recursively deleting and restoring neighbors after rejected insertions. In the analysis, the $\sqrt2$ gap to the Bilu--Linial conjecture comes from a factor of two in the bound for vertex deletions in the two-sided case. On a $d$-regular bipartite Ramanujan base the same signing produces a Ramanujan $2$-lift of that prescribed base.
