Acceleration for Distributed Transshipment and Parallel Maximum Flow

TL;DR

This paper advances parallel and distributed algorithms for (1+ε)-approximate undirected maximum flow and transshipment by embedding accelerated first-order methods within the box-simplex game framework and by constructing specialized linear cost approximators. A recent column-sparse cost approximator for maximum flow is integrated to achieve an Õ(1/ε) depth with Õ(m/ε) work on PRAM, producing feasible primal and dual solutions with near-optimal objective values. For transshipment, a deterministic distributed cost approximator is developed via distance-structure techniques and Minor-Aggregation, enabling a deterministic CONGEST algorithm with Õ(ε^{-1}(D+√n)) rounds on general networks and Õ(ε^{-1}D) rounds on minor-free networks. Together, these methods yield the first known parallel Õ(1/ε) depth and Õ(m/ε) work algorithms for both problems in undirected graphs, with corresponding distributed-round guarantees, highlighting how column-sparse approximators and the box-simplex game can unlock efficient acceleration in parallel and distributed graph optimization.

Abstract

We combine several recent advancements to solve $(1+\varepsilon)$-transshipment and $(1+\varepsilon)$-maximum flow with a parallel algorithm with $\tilde{O}(1/\varepsilon)$ depth and $\tilde{O}(m/\varepsilon)$ work. We achieve this by developing and deploying suitable parallel linear cost approximators in conjunction with an accelerated continuous optimization framework known as the box-simplex game by Jambulapati et al. (ICALP 2022). A linear cost approximator is a linear operator that allows us to efficiently estimate the cost of the optimal solution to a given routing problem. Obtaining accelerated $\varepsilon$ dependencies for both problems requires developing a stronger multicommodity cost approximator, one where cancellations between different commodities are disallowed. For maximum flow, we observe that a recent linear cost approximator due to Agarwal et al. (SODA 2024) can be augmented with additional parallel operations and achieve $\varepsilon^{-1}$ dependency via the box-simplex game. For transshipment, we also construct a deterministic and distributed approximator. This yields a deterministic CONGEST algorithm that requires $\tilde{O}(\varepsilon^{-1}(D + \sqrt{n}))$ rounds on general networks of hop diameter $D$ and $\tilde{O}(\varepsilon^{-1}D)$ rounds on minor-free networks to compute a $(1+\varepsilon)$-approximation.

Figures (2)

• Figure 1: Transshipment example: $G$ is a cycle with unit weights. There are 2 units of supply in A (indicated by a label $d(A) = +2$), and a unit of supply in D (indicated by $d(D) = +1$). There is a unit demand in B, G, and E (indicated by $d(\cdot) = -1$). The optimal flow is in blue and has a cost of 4.
• Figure 2: Maximum flow example: $G$ is a cycle with unit weights. The depicted flow sends $1$ unit from $D \to E$, $0.5$ units $B \to A$ along the small circle, and $1.5$ units $B \to A$ along the long circle. The solution has a cost of $1.5$ since, in our formulation, the most overcongested edge (either AB or DE) has congestion $1.5$. No better solution exists.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Linear Cost Approximator
• Definition 2.2
• Definition 2.4
• Theorem 2.5
• Proposition 2.6
• proof
• Proposition 2.7
• proof