Horospherically convex optimization for fractional subspace packing and its applications

Abstract

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.