Unsplittable Transshipments

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.

💡 Research Summary

The paper introduces the Unsplittable Transshipment Problem (UTP), a natural extension of the classic unsplittable flow model to directed graphs that contain multiple sources and multiple sinks. In a UTP instance each vertex v carries a balance b(v) (positive for a supply, negative for a demand) and each arc a has a capacity c_a. A feasible transshipment is a flow x that satisfies the balance constraints at every vertex and respects capacities. An unsplittable transshipment is a feasible transshipment together with a path decomposition in which every supply‑demand pair is routed along at most one directed s‑t path.

The authors first revisit the seminal Dinitz‑Garg‑Goemans (DGG) algorithm for the single‑source unsplittable flow (SSUF) problem. The DGG algorithm starts from a fractional flow x that meets all demands, then repeatedly builds alternating cycles consisting of forward arcs (where flow is decreased) and backward singular arcs (where flow is increased). By carefully moving “sinks” backward toward the source and deleting arcs that become empty, the algorithm guarantees that the final unsplittable flow y satisfies y_a < x_a + d_max for every arc a, where d_max is the largest demand. This bound yields a 2‑approximation for congestion minimisation when all capacities are at least d_max.

To handle multiple sources, the paper first reduces a UTP instance to an SSUF instance by adding a super‑source s* and a dummy arc (s*, s) of capacity b(s) for each real source s. The naïve application of DGG would modify the flow on these dummy arcs, thereby breaking the fixed supply constraints. The core contribution is a Modified DGG Algorithm that preserves the supplies while still achieving the same additive congestion bound.

Key ideas of the Modified DGG algorithm:

1. Supply‑preserving cycles – The algorithm never augments a cycle that contains the super‑source s*. This is enforced by distinguishing “funnel” vertices (out‑degree ≤ 1) from “non‑funnel” vertices. Backward paths always end at a non‑funnel vertex; if the only reachable non‑funnel vertex is s*, the current forward path is abandoned and a new forward edge is tried.

2. Demand splitting into sub‑sinks – When a sink’s demand cannot be satisfied by a single source without violating the additive bound, the algorithm creates auxiliary sub‑sink nodes that split the original demand among several sources. Each sub‑sink receives exactly the same total demand as the original sink, but its flow originates from a distinct source, allowing the algorithm to keep the dummy arcs untouched.

3. Structural guarantees – The final unsplittable transshipment induces a bipartite source‑sink graph that is acyclic (a forest). Moreover, all paths that converge on a particular sink are confluent: once two paths meet, they share the remainder of the route to the sink. These properties are crucial for the proof that every arc a carries at most x_a + d_max flow in the final solution.

The algorithm runs in polynomial time. It begins with y ← x, performs a preliminary phase that moves sinks backward along arcs whose current flow is at least the sink’s demand, and then iteratively searches for “nice” alternating cycles (those that avoid s*). Each augmentation reduces the total residual demand or eliminates an arc, guaranteeing termination after O(|A|) iterations.

Beyond the basic transformation, the paper studies two classic extensions of unsplittable flows in the multi‑source setting:

• Minimum‑round routing – A round is a feasible unsplittable transshipment that serves a subset of the demands while respecting capacities. The goal is to partition the demand set into as few rounds as possible. Under the additional assumption that every arc capacity is at least d_max, the authors prove a constant‑round bound (e.g., five rounds suffice), extending the 5‑approximation known for the single‑source case. They also exhibit instances where three rounds are necessary.

• Maximum routable demand – Here the objective is to select a subset of demands whose total value is maximised while admitting an unsplittable transshipment. Using the Modified DGG algorithm as a subroutine, the authors obtain a 0.226‑approximation, matching the best known factor for the single‑source problem. They also construct instances showing that any unsplittable solution can capture at most 0.385 of the total fractional demand.

The paper includes two appendices. Appendix A proves that deciding feasibility of an integral unsplittable transshipment is NP‑hard even when all capacities and supplies/demands are unit. Appendix B shows that integral supplies/demands do not guarantee an integral unsplittable solution, highlighting the intrinsic fractional nature of the problem.

In summary, the work extends the foundational DGG result from single‑source to fully general multi‑source, multi‑sink networks, delivering a polynomial‑time algorithm that converts any feasible fractional transshipment into an unsplittable one with only an additive d_max increase per arc. It also provides constant‑factor approximations for round minimisation and maximum‑demand routing, thereby establishing a comprehensive theoretical framework for unsplittable transshipment problems that arise in logistics, telecommunications, and optical networking.