Local Sherman's Algorithm for Multi-commodity Flow

TL;DR

This work delivers the first local algorithm for multi-commodity flow, applicable to unit-capacity expanders where it attains a $(1+\epsilon)$-approximate solution with running time roughly $(m+\epsilon^{-3}k^3D)n^{o(1)}$, thereby breaking the km barrier in this restricted setting. The core idea is to localize Sherman's approximate maximum flow algorithm by embedding it in a Multiplicative Weights Update framework and to replace per-round full-edge updates with rounding of small vertex potentials to zero, enabling sublinear-time oracle calls via a refined MWU analysis with approximate weights. The multi-commodity extension mirrors the single-commodity approach but uses a MWU over commodity-vertex-pair constraints, yielding a feasible $k$-commodity flow with residual per-vertex demands bounded by $\epsilon\deg(v)$ and a running time of $O(\epsilon^{-2}\log n\, (\|b\|_0+\epsilon^{-1}\|b\|_1))$; an expander-routing step then completes the residual demands. On $\phi$-expanders, combining these mechanisms yields a deterministic algorithm that either certifies infeasibility or computes a $k$-commodity flow with congestion $1+\epsilon$ in time $(m+\epsilon^{-3}k^3D)\cdot\mathrm{poly}(1/\phi)\cdot 2^{O(\sqrt{\log n\log\log n})}$, highlighting a scalable pathway for near-optimal multi-commodity routing in highly connected graphs.

Abstract

We give the first local algorithm for computing multi-commodity flow and apply it to obtain a $(1+ε)$-approximate algorithm for computing a $k$-commodity flow on an expander with $m$ edges in $(m+ε^{-3}k^3D)n^{o(1)}$ time, where $D$ is the total demand. This is the first $(1+ε)$-approximate algorithm that breaks the $km$ multi-commodity flow barrier, albeit only on expanders. All previous algorithms either require $Ω(km)$ time or a big constant approximation. Our approach is by localizing Sherman's flow algorithm when put into the Multiplicative Weight Update (MWU) framework. We show that, on each round of MWU, the oracle could instead work with the *rounded weights* where all polynomially small weights are rounded to zero. Since there are only few large weights, one can implement the oracle call with respect to the rounded weights in sublinear time. This insight is generic and may be of independent interest.

Theorems & Definitions (32)

• Theorem 1.1: Informal version of \ref{['thm:main-multi']})
• Theorem 1.2
• proof
• Theorem 2.1: Local Sherman: Single-commodity
• Lemma 2.2: Infeasibility certificate
• proof
• Theorem 2.3: MWU with Approximate Weights
• proof
• Lemma 2.4
• proof