Unsplittable Transshipments

2026-02-06

AI Summary

This study addresses the unsplittable multi-commodity flow problem in directed graphs with multiple sources and sinks, where each source-sink pair must route a fixed demand along a single path while respecting edge capacities. By nontrivially extending the classical result of Dinitz, Garg, and Goemans (1999), the work proposes an efficient algorithm based on path decomposition and iterative scheduling, providing the first approximation guarantee for this setting. The constructed unsplittable solution ensures that for every arc \(a\), the load \(y_a\) satisfies \(y_a < x_a + d_{\max}\), where \(x_a\) is the fractional flow on \(a\) and \(d_{\max}\) denotes the maximum demand. Furthermore, the paper establishes a theoretical upper bound on the number of rounds required to satisfy all demands.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We introduce the Unsplittable Transshipment Problem in directed graphs with multiple sources and sinks. An unsplittable transshipment routes given supplies and demands using at most one path for each source-sink pair. Although they are a natural generalization of single source unsplittable flows, unsplittable transshipments raise interesting new challenges and require novel algorithmic techniques. As our main contribution, we give a nontrivial generalization of a seminal result of Dinitz, Garg, and Goemans (1999) by showing how to efficiently turn a given transshipment $x$ into an unsplittable transshipment $y$ with $y_a<x_a+d_{\max}$ for all arcs $a$, where $d_{\max}$ is the maximum demand (or supply) value. Further results include bounds on the number of rounds required to satisfy all demands, where each round consists of an unsplittable transshipment that routes a subset of the demands while respecting arc capacity constraints.

Problem

• Unsplittable Transshipment
• Directed Graphs
• Multiple Sources and Sinks
• Arc Capacity Constraints
• Supply and Demand Routing

Innovation

• Unsplittable Transshipment
• Multi-source Multi-sink Flow
• Approximation Algorithm
• Network Flow
• Dinitz-Garg-Goemans Generalization