Congestion-Approximators from the Bottom Up

TL;DR

This paper introduces a novel bottom-up hierarchical algorithm for constructing congestion-approximators in capacitated graphs, achieving nearly-linear time complexity and avoiding recursion, unlike previous top-down methods.

Contribution

It presents the first non-recursive, nearly-linear time algorithm for congestion-approximators using a bottom-up approach, improving upon prior recursive methods.

Findings

First non-recursive congestion-approximator algorithm

Achieves polylogarithmic quality in nearly-linear time

Demonstrates efficiency without recursive max-flow calls

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.