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Congestion-Approximators from the Bottom Up

Jason Li, Satish Rao, Di Wang

TL;DR

This work presents a novel bottom-up, nearly-linear-time construction of a polylogarithmic-quality congestion-approximator for capacitated undirected graphs, avoiding recursive max-flow loops characteristic of prior methods. By building a hierarchical partition sequence and using common refinements, the authors obtain a pseudo-congestion-approximator that supports non-recursive max-flow subroutines, culminating in a congestion-approximator of quality $O(\log^{10}(nW))$. Leveraging Sherman's framework, this leads to a $(1+\epsilon)$-approximate maximum flow algorithm running in $\tilde{O}(\epsilon^{-1}m)$ time. The approach hinges on a suite of primitives—expander-style mixing, fair cut/flow subroutines, cut-matching games, trimming, and clustering—that together yield a robust bottom-up decomposition framework with practical implications for fast flow computations on large graphs.

Abstract

We develop a novel algorithm to construct a congestion-approximator with polylogarithmic quality on a capacitated, undirected graph in nearly-linear time. Our approach is the first *bottom-up* hierarchical construction, in contrast to previous *top-down* approaches including that of Racke, Shah, and Taubig (SODA 2014), the only other construction achieving polylogarithmic quality that is implementable in nearly-linear time (Peng, SODA 2016). Similar to Racke, Shah, and Taubig, our construction at each hierarchical level requires calls to an approximate max-flow/min-cut subroutine. However, the main advantage to our bottom-up approach is that these max-flow calls can be implemented directly *without recursion*. More precisely, the previously computed levels of the hierarchy can be converted into a *pseudo-congestion-approximator*, which then translates to a max-flow algorithm that is sufficient for the particular max-flow calls used in the construction of the next hierarchical level. As a result, we obtain the first non-recursive algorithms for congestion-approximator and approximate max-flow that run in nearly-linear time, a conceptual improvement to the aforementioned algorithms that recursively alternate between the two problems.

Congestion-Approximators from the Bottom Up

TL;DR

This work presents a novel bottom-up, nearly-linear-time construction of a polylogarithmic-quality congestion-approximator for capacitated undirected graphs, avoiding recursive max-flow loops characteristic of prior methods. By building a hierarchical partition sequence and using common refinements, the authors obtain a pseudo-congestion-approximator that supports non-recursive max-flow subroutines, culminating in a congestion-approximator of quality . Leveraging Sherman's framework, this leads to a -approximate maximum flow algorithm running in time. The approach hinges on a suite of primitives—expander-style mixing, fair cut/flow subroutines, cut-matching games, trimming, and clustering—that together yield a robust bottom-up decomposition framework with practical implications for fast flow computations on large graphs.

Abstract

We develop a novel algorithm to construct a congestion-approximator with polylogarithmic quality on a capacitated, undirected graph in nearly-linear time. Our approach is the first *bottom-up* hierarchical construction, in contrast to previous *top-down* approaches including that of Racke, Shah, and Taubig (SODA 2014), the only other construction achieving polylogarithmic quality that is implementable in nearly-linear time (Peng, SODA 2016). Similar to Racke, Shah, and Taubig, our construction at each hierarchical level requires calls to an approximate max-flow/min-cut subroutine. However, the main advantage to our bottom-up approach is that these max-flow calls can be implemented directly *without recursion*. More precisely, the previously computed levels of the hierarchy can be converted into a *pseudo-congestion-approximator*, which then translates to a max-flow algorithm that is sufficient for the particular max-flow calls used in the construction of the next hierarchical level. As a result, we obtain the first non-recursive algorithms for congestion-approximator and approximate max-flow that run in nearly-linear time, a conceptual improvement to the aforementioned algorithms that recursively alternate between the two problems.
Paper Structure (22 sections, 22 theorems, 85 equations, 1 figure)

This paper contains 22 sections, 22 theorems, 85 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Previous work
  3. Technical Overview
  4. Bottom-Up Construction
  5. Preliminaries
  6. Congestion-approximators and approximate flow.
  7. Bottom-Up Congestion-Approximator
  8. Partitioning Algorithm
  9. Congestion-approximator
  10. Fair Cut/Flow Algorithm
  11. Cut-Matching Game and Trimming
  12. Clustering Algorithm
  13. Approximate Maximum Flow
  14. Cut-Matching Game
  15. Small Potential Implies Mixing
  16. ...and 7 more sections

Key Result

Theorem 2.1

Consider a capacitated graph $G=(V,E)$. Suppose there exist partitions $\mathcal{P}_1,\mathcal{P}_2,\ldots,\mathcal{P}_L$ of $V$ such that For each $i\in[L]$, let partition $\mathcal{R}_{\ge i}$ be the common refinement of partitions $\mathcal{P}_i,\mathcal{P}_{i+1},\ldots,\mathcal{P}_L$, i.e., Then, their union $\mathcal{C}=\bigcup_{i\in[L]}\mathcal{R}_{\ge i}$ is a congestion-approximator with

Figures (1)

  • Figure 1: On the left, partitions $\mathcal{P}_i$ (solid) and $\mathcal{P}_{i+1}$ (dotted) are shown. On the middle, the marked edges for each cluster mix simultaneously in $G$ (property (\ref{['item:cut-approx-2']})). On the right, a flow from the inter-cluster edges of $\mathcal{P}_{i+1}$ to the inter-cluster edges of $\mathcal{P}_i$ is displayed (property (\ref{['item:cut-approx-3']})); assuming edges are unit-weight, each flow path carries a half-unit of flow.

Theorems & Definitions (82)

  • Theorem 2.1: Informal \ref{['thm:cut-approx']}
  • Theorem 4.1
  • Lemma 4.2
  • proof : Proof (\ref{['lem:cut-approx']}$\implies$\ref{['thm:cut-approx']})
  • Claim 4.3
  • proof
  • Claim 4.4
  • proof
  • Lemma 4.5
  • proof
  • ...and 72 more