Unsplittable Transshipments

TL;DR

We address unsplittable transshipments, a multi-source, multi-sink generalization of SSUF, and show how to efficiently transform any feasible fractional transshipment into an unsplittable one with per-arc increase bounded by the maximum sink/source demand $d_{\max}$. Building on the DGG framework, we develop a Modified DGG algorithm that preserves source supplies by using a super-source and by splitting sinks within singular digraphs, ensuring confluence and a tree-like source–sink bipartite structure. The analysis proves structural invariants, confluence of routing, and a bound $y_a < x_a + d_{\max}$; it also derives bounds on the number of paths and identifies the cost of enforcing confluence, plus results on routing in rounds and maximum routable demand under additional assumptions. These results extend the reach of unsplittable flow techniques to more realistic, multi-source networks, with implications for logistics and transport where splitting a commodity is impractical, and establish a foundation for further exploration of bounds, hardness, and k-splittable variants.

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.

Figures (4)

• Figure 1: Backward path discovery in the Modified DGG Algorithm starting at junction vertex $w$ with singular arcs depicted in thick: Left: a non-funnel vertex $\neq s^*$ is found along some backward path, allowing to continue the construction of a nice alternating cycle; Right: the only non-funnel vertex found on backward paths is $s^*$, yielding a singular digraph rooted at junction vertex $w$.
• Figure 2: Family of instances (depicted for $q=4$) with unit arc capacities and $d_{\max}=1$, admitting a feasible non-confluent unsplittable transshipment, while any confluent unsplittable transshipment has congestion $1-1/q$, tending to $d_{\max}$ as $q \to \infty$.
• Figure 3: A non-integral unsplittable transshipment, formed by the $s^1$-$t^1$-path with flow value $3/2$ highlighted in yellow, the $s^1$-$t^2$-path with flow value $7/2$ highlighted in magenta, the $s^2$-$t^1$-path with flow value $13/2$ highlighted in blue, and the $s^2$-$t^2$-path with flow value $17/2$ highlighted in green.
• Figure 4: An instance to show $d_{\max}$ is a tight lower bound for the required capacity violation.

Theorems & Definitions (18)

• Lemma 1
• proof
• proof : Proof of Invariant \ref{['inv:flow_satisfies_demands']}
• proof : Proof of Invariant \ref{['inv:vertex_w_sinks_2_inc']}
• proof : Proof of Invariant \ref{['inv:moving_removes_sing']}
• Lemma 2
• proof
• Theorem 3
• proof
• Corollary 4