Unsplittable Transshipments

TL;DR

This paper introduces the Unsplittable Transshipment Problem, extending classical flow models with new algorithmic techniques to efficiently convert fractional solutions into integral ones while respecting demands and capacities.

Contribution

It generalizes a key result for transshipments, providing an efficient method to convert fractional solutions into near-integral solutions with bounded deviations.

Findings

Efficient algorithm to convert transshipment solutions into unsplittable transshipments.

Bounds on the number of rounds needed to satisfy all demands.

Extension of classical flow results to more complex transshipment scenarios.

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.