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Low-Step Multi-Commodity Flow Emulators

Bernhard Haeupler, D Ellis Hershkowitz, Jason Li, Antti Roeyskoe, Thatchaphol Saranurak

TL;DR

This work introduces low-step multi-commodity flow emulators for undirected, capacitated graphs, a tool that encodes approximate multi-commodity flows along short paths and avoids the traditional flow-decomposition barrier. By stacking length-constrained expanders into a hierarchical emulator, the authors achieve $(m+k)^{1+ε}$-time, constant-approximate solutions for both concurrent and non-concurrent $k$-commodity flow, with parallelizable algorithms and implicit representations that enable efficient subset queries. The core methodology combines length-constrained expander decompositions, routers, and expansion witnesses to build emulators, then uses flow boosting in the Garg–Konemann framework to reduce congestion to a constant. Extensions to cost-constrained and minimum-cost variants are provided, along with algorithmic and existential results for the emulators and a detailed routing framework on expansion witnesses. This approach advances scalable, near-optimal multi-commodity routing on general graphs and offers practical, parallelizable primitives for large-scale network optimization problems.

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 $O((m+k)^{1+ε})$ 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.

Low-Step Multi-Commodity Flow Emulators

TL;DR

This work introduces low-step multi-commodity flow emulators for undirected, capacitated graphs, a tool that encodes approximate multi-commodity flows along short paths and avoids the traditional flow-decomposition barrier. By stacking length-constrained expanders into a hierarchical emulator, the authors achieve -time, constant-approximate solutions for both concurrent and non-concurrent -commodity flow, with parallelizable algorithms and implicit representations that enable efficient subset queries. The core methodology combines length-constrained expander decompositions, routers, and expansion witnesses to build emulators, then uses flow boosting in the Garg–Konemann framework to reduce congestion to a constant. Extensions to cost-constrained and minimum-cost variants are provided, along with algorithmic and existential results for the emulators and a detailed routing framework on expansion witnesses. This approach advances scalable, near-optimal multi-commodity routing on general graphs and offers practical, parallelizable primitives for large-scale network optimization problems.

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 -commodity flow in time. To bypass the 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 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.
Paper Structure (62 sections, 51 theorems, 46 equations, 1 figure, 8 algorithms)

This paper contains 62 sections, 51 theorems, 46 equations, 1 figure, 8 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Our Techniques
  4. Overview
  5. Preliminaries
  6. Multicommodity Flows.
  7. Edge Representation of Flows.
  8. Demands.
  9. Length-Constrained Expanders
  10. Moving Cuts.
  11. Length-Constrained Expander Decomposition
  12. Routers
  13. Length-Constrained Expansion Witness
  14. Neighborhood Covers.
  15. Embedding.
  16. ...and 47 more sections

Key Result

Theorem 1.1

For every constant $\epsilon \in (0, 1)$, there exists a $(m + k)^{1 + \mathrm{poly}(\epsilon)}$-time $O(2^{-1 / \epsilon})$-approximate algorithm for the concurrent and non-concurrent multicommodity flow value problems, where $k$ is the number of demand pairs. The algorithms work in parallel with d

Figures (1)

  • Figure 1: Dependencies between sections in this paper.

Theorems & Definitions (97)

  • Theorem 1.1: Constant-Approximate Concurrent/Non-Concurrent Flow (informal)
  • Definition 3.3: Cut Characterization of Length-Constrainted Expanders
  • Proposition 3.4
  • Theorem 3.5: Flow Characterization of Length-Constrainted Expanders (Lemma 3.16 of haeupler2022hop)
  • Definition 3.6: Expander Decomposition
  • Definition 3.7: Linked Expander Decomposition
  • Theorem 3.8: implicit in haeupler2022hop, explicit in Theorem 3 of haeupler2023parallel
  • Definition 3.9: Routers
  • Proposition 3.10
  • proof
  • ...and 87 more