用于分数子空间打包的球心凸优化及其应用
Horospherically convex optimization for fractional subspace packing and its applications
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中文总结 AI 辅助
本文针对向量子空间打包问题的半无限LP松弛，利用球心凸优化理论结合增量Busemann次梯度法得到加性FPTAS，并将其应用于分数线性拟阵匹配和Brascamp-Lieb多面体成员问题的算法设计。
中文摘要 AI 辅助
本文研究向量子空间打包问题的半无限线性规划（LP）松弛，该问题是分数线性拟阵匹配问题的高维推广，且与Brascamp-Lieb多面体密切相关。我们证明该LP的对偶可被建模为“欧氏建筑上的线性规划”，即极小化Busemann函数在水平球交集上的问题，这是Goodwin等人（2026）与Criscitiello和Kim（2025）新近提出的球心凸优化的自然实例。通过应用增量Busemann次梯度法，我们得到该问题的加性完全多项式时间近似方案（FPTAS）。作为应用，我们获得了分数线性拟阵匹配问题的新且更简单的多项式时间算法，以及Brascamp-Lieb多面体成员问题的新算法。
英文摘要
In this paper, we address a semi-infinite LP relaxation of the vector-subspace packing problem. This is a higher-dimensional generalization of the fractional linear matroid parity problem and is closely related to Brascamp-Lieb polytopes. We show that the dual of this LP can be formulated as ``linear programming on a Euclidean building," namely, the problem of minimizing a Busemann function over an intersection of horoballs. This provides a natural example of horospherically convex optimization, recently introduced by Goodwin et al. (2026) and Criscitiello and Kim (2025). By applying the incremental Busemann subgradient method, we obtain an additive FPTAS for the problem. As applications, we obtain a new and simpler polynomial-time algorithm for fractional linear matroid parity, and new algorithms for the membership problem of Brascamp-Lieb polytopes.
