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arXiv · 2406.14384

Low-Step Multi-Commodity Flow Emulators

Bernhard Haeupler·D Ellis Hershkowitz·Jason Li·Antti Roeyskoe·Thatchaphol Saranurak
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Abstract

We introduce the concept of low-step multi-commodity flow emulators for any undirected, capacitated graph. At a high level, these emulators contain approximate multi-commodity flows whose paths contain a small number of edges, shattering the infamous flow decomposition barrier for multi-commodity flow. We prove the existence of low-step multi-commodity flow emulators and develop efficient algorithms to compute them. We then apply them to solve constant-approximate $k$-commodity flow in $O((m+k)^{1+\epsilon})$ time. To bypass the $O(mk)$ flow decomposition barrier, we represent our output multi-commodity flow implicitly; prior to our work, even the existence of implicit constant-approximate multi-commodity flows of size $o(mk)$ was unknown. Our results generalize to the minimum cost setting, where each edge has an associated cost and the multi-commodity flow must satisfy a cost budget. Our algorithms are also parallel.

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Bernhard Haeupler, D Ellis Hershkowitz, Jason Li, Antti Roeyskoe, Thatchaphol Saranurak. 2024-06-20. Low-Step Multi-Commodity Flow Emulators. https://doi.org/10.1145/3618260.3649689

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