Accelerated Approximate Optimization of Multi-Commodity Flows on Directed Graphs
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909559895883776 |
|---|---|
| author | Chen, Li Graur, Andrei Sidford, Aaron |
| author_facet | Chen, Li Graur, Andrei Sidford, Aaron |
| contents | We provide $m^{1+o(1)}kε^{-1}$-time algorithms for computing multiplicative $(1 - ε)$-approximate solutions to multi-commodity flow problems with $k$-commodities on $m$-edge directed graphs, including concurrent multi-commodity flow and maximum multi-commodity flow.
To obtain our results, we provide new optimization tools of potential independent interest. First, we provide an improved optimization method for solving $\ell_{q, p}$-regression problems to high accuracy. This method makes $\tilde{O}_{q, p}(k)$ queries to a high accuracy convex minimization oracle for an individual block, where $\tilde{O}_{q, p}(\cdot)$ hides factors depending only on $q$, $p$, or $\mathrm{poly}(\log m)$, improving upon the $\tilde{O}_{q, p}(k^2)$ bound of [Chen-Ye, ICALP 2024]. As a result, we obtain the first almost-linear time algorithm that solves $\ell_{q, p}$ flows on directed graphs to high accuracy. Second, we present optimization tools to reduce approximately solving composite $\ell_{1, \infty}$-regression problems to solving $m^{o(1)}ε^{-1}$ instances of composite $\ell_{q, p}$-regression problem. The method builds upon recent advances in solving box-simplex games [Jambulapati-Tian, NeurIPS 2023] and the area convex regularizer introduced in [Sherman, STOC 2017] to obtain faster rates for constrained versions of the problem. Carefully combining these techniques yields our directed multi-commodity flow algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_24373 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Accelerated Approximate Optimization of Multi-Commodity Flows on Directed Graphs Chen, Li Graur, Andrei Sidford, Aaron Data Structures and Algorithms Optimization and Control We provide $m^{1+o(1)}kε^{-1}$-time algorithms for computing multiplicative $(1 - ε)$-approximate solutions to multi-commodity flow problems with $k$-commodities on $m$-edge directed graphs, including concurrent multi-commodity flow and maximum multi-commodity flow. To obtain our results, we provide new optimization tools of potential independent interest. First, we provide an improved optimization method for solving $\ell_{q, p}$-regression problems to high accuracy. This method makes $\tilde{O}_{q, p}(k)$ queries to a high accuracy convex minimization oracle for an individual block, where $\tilde{O}_{q, p}(\cdot)$ hides factors depending only on $q$, $p$, or $\mathrm{poly}(\log m)$, improving upon the $\tilde{O}_{q, p}(k^2)$ bound of [Chen-Ye, ICALP 2024]. As a result, we obtain the first almost-linear time algorithm that solves $\ell_{q, p}$ flows on directed graphs to high accuracy. Second, we present optimization tools to reduce approximately solving composite $\ell_{1, \infty}$-regression problems to solving $m^{o(1)}ε^{-1}$ instances of composite $\ell_{q, p}$-regression problem. The method builds upon recent advances in solving box-simplex games [Jambulapati-Tian, NeurIPS 2023] and the area convex regularizer introduced in [Sherman, STOC 2017] to obtain faster rates for constrained versions of the problem. Carefully combining these techniques yields our directed multi-commodity flow algorithm. |
| title | Accelerated Approximate Optimization of Multi-Commodity Flows on Directed Graphs |
| topic | Data Structures and Algorithms Optimization and Control |
| url | https://arxiv.org/abs/2503.24373 |