Single-Source Unsplittable Flow Conjecture

• Single-Source Unsplittable Flow Conjecture is a key combinatorial optimization problem that asserts one can obtain an unsplittable flow matching the cost of a fractional flow with an additive maximum-demand violation.
• It leverages techniques from polyhedral combinatorics and face-preserving rounding to bridge the gap between fractional and integral solutions, with applications in network design and scheduling.
• While significant progress has been made for series-parallel and planar graphs, the full conjecture remains open for general acyclic digraphs.

The single-source unsplittable flow conjecture is a central open problem in combinatorial optimization that seeks to tightly link the structure of fractional flows and the existence of efficient, low-cost, unsplittable flows with bounded congestion. This conjecture, attributed to Goemans, posits that in any acyclic network from a single source to multiple sinks, it is always possible to find an unsplittable flow matching the cost of a given fractional solution while incurring at most an additive maximum-demand violation on each arc. The problem and its variants have deep connections to polyhedral combinatorics, network design, rounding theory, and discrepancy minimization.

1. Formal Statement and Definitions

Let $G = (V, A)$ be a directed acyclic graph (DAG) with distinguished source $s \in V$, a set of sinks $$T = \{ t_1, \ldots, t_k \} \subseteq V \setminus \{ s \}$$, and nonnegative demands $d_t \geq 0$ for each $t \in T$. Define $d_{\max} = \max_{t \in T} d_t$.

A fractional flow $x \in \mathbb{R}_+^A$ must route total demand $\sum_{t \in T} d_t$ from $s$ to the sinks, satisfying flow conservation and arc capacities (where relevant). Let $Q_G \subseteq \mathbb{R}_+^A$ be the polytope of such flows.

An unsplittable flow $\mathcal{F}$ consists of a path $P_t$ for each $t \in T$, carrying $d_t$ units along $P_t$, so the arc-load vector is $f^\mathcal{F}(a) = \sum_{t : a \in P_t} d_t$.

Each arc $a$ can have an associated cost $c(a) \geq 0$. The total cost of a flow $y \in \mathbb{R}^A$ is $c^T y = \sum_{a} c(a) y(a)$.

Goemans’ Single-Source Unsplittable Flow Conjecture: Given any $x \in Q_G$, there exists a polynomial-time algorithm to produce an unsplittable flow $\mathcal{F}$ such that:

• (i) $c^T f^\mathcal{F} \leq c^T x$ (no higher cost)
• (ii) $f^\mathcal{F}(a) \leq x(a) + d_{\max}$ for all $a \in A$ (additive $d_{\max}$ violation)

Variants strengthen the conjecture by imposing lower bounds on $f^\mathcal{F}(a) \geq x(a) - d_{\max}$ or by seeking a convex decomposition of the fractional flow into unsplittable flows with the same bounds (Swamy et al., 24 Oct 2025, Almoghrabi et al., 2024, Traub et al., 2023, Liu et al., 23 Nov 2025).

2. Conjectural Frameworks and Strengthenings

Goemans' conjecture is complemented by alternative formulations:

• Morell–Skutella’s Conjecture (weak form): For any $x \in Q_G$, there exists $\mathcal{F}$ such that

$$x(a) - d_{\max} \leq f^{\mathcal{F}}(a) \leq x(a) + d_{\max}, \quad \forall a \in A$$

The strong form adds $c^T f^\mathcal{F} \leq c^T x$.

• This two-sided bound is strictly stronger than the Dinitz–Garg–Goemans upper-bound-only result, and the strong form implies Goemans' original cost version. Furthermore, the weak form enables, via face-preserving rounding and polyhedral arguments, construction of unsplittable flows with cost not exceeding the fractional optimum but with slack $2 d_{\max}$ on arc loads (Swamy et al., 24 Oct 2025).
• The convex decomposition version (Goemans, 1997): Any feasible fractional single-source flow is a convex combination of unsplittable flows, each with per-arc violation at most $d_{\max}$.

The following table organizes several key conjectural statements and resolution progress:

Conjecture	Load Guarantee	Cost Guarantee	Known for ...
Goemans'	$x(a)+d_{\max}$ upper	$c^T f^{\mathcal{F}} \leq c^T x$	Series-parallel, planar$^*$
Morell–Skutella weak	$x(a) \pm d_{\max}$	none	Series-parallel, planar
Morell–Skutella strong	$x(a) \pm d_{\max}$	$c^T f^{\mathcal{F}} \leq c^T x$	Series-parallel

$^*$: Cost guarantee in planar only up to $2 d_{\max}$ violation

3. Algorithmic Methods and Polyhedral Rounding

Progress on the conjecture exploits polyhedral combinatorics, rounding techniques, and face structure of the flow polytope. A pivotal approach is the face-preserving rounding algorithm (FPRA):

• An FPRA takes $x \in Q_G$ and outputs $z \in Z_G$ (unsplittable load vectors) with $z-x \in R$ (entrywise deviation at most $d_{\max}$) and $z$ lying in the same minimal face of $Q_G$. This is crucial to avoid reactivation of zero-flow arcs (Swamy et al., 24 Oct 2025).
• When such a rounding (with error $R$) is combined with an auxiliary linear program minimizing cost over $Q_G \cap (x-R)$, it yields an unsplittable solution with cost at most $c^T x$ and deviation $R-R = [-2 d_{\max}, 2 d_{\max}]^A$.

The approach avoids Lagrangian duality, relying on convexity and face preservation; it generalizes to other settings where additive-error, face-preserving unweighted rounding exists (e.g., weighted ring loading) (Swamy et al., 24 Oct 2025).

4. Structural and Special Cases

Significant advances have been made for certain graph families and problem subclasses:

• Series-Parallel Digraphs: Goemans’ conjecture and the Morell–Skutella two-sided bound hold in full generality, even for multiflows with distinct sources/sinks. The proof constructs a convex combination of unsplittable multiflows where each load deviates from the fractional solution by strictly less than $d_{\max}$ (Almoghrabi et al., 2024). Key ingredients are the sp-tree decomposition, recursive flow adjustment to "almost unsplittable" form, and precise (non-negative) convex decomposition at each node.
• Planar Acyclic Digraphs: For planar SSUF instances, both upper and lower bounds of $d_{\max}$ (without cost) and cost-aware rounding with deviation $2 d_{\max}$ are achievable (Traub et al., 2023). The methodology leverages planar embeddings, non-crossing path decomposition of the fractional flow, and optimal multi-interval discrepancy rounding.
• Prefix-Chain/Chairman Assignment: The conjecture is exactly resolved for DAGs corresponding to parallel prefix chains (matrix assignments): for $m\times n$ fractional assignments and arbitrary nonnegative demands, there exists an integral assignment matching prefix sums within $d_{\max}$ (Liu et al., 23 Nov 2025). The algorithm adapts Tijdeman’s earliest-deadline (matching) strategy to arbitrary weights.

5. Implications, Applications, and Extensions

Confirmations of the conjecture in specialized settings have broad implications:

• Network Design: Guarantees on unsplittable flows with tight congestion and cost bounds underpin robust network routing, resource allocation, and load balancing.
• Scheduling: The prefix-chain rounding result yields $3$-approximation for maximum flow-time minimization in scheduling jobs with release times and machine closing times, and generalizes to $(1+\alpha)$-approximations when an additive discrepancy bound $\alpha$ can be shown (Liu et al., 23 Nov 2025).
• Polyhedral Rounding: The Lagrangian-free FPRA + LP approach is a versatile template, enabling cost-aware rounding for any polyhedral combinatorial structure that admits error-bounded, face-preserving rounding—a principle applicable to spanning trees, ring-loading, and beyond (Swamy et al., 24 Oct 2025).

A plausible implication is that these techniques may extend to other decomposable graph classes (e.g., outerplanar, bounded treewidth) and thus incrementally close the gap in the general DAG case (Almoghrabi et al., 2024).

6. Limitations, Barriers, and Open Problems

Despite progress, the conjecture remains widely open for general acyclic digraphs:

• The full cost variant with $d_{\max}$ deviation is not resolved outside series-parallel and planar classes (planar only up to $2 d_{\max}$). In the general case, only one-sided or high-violation bounds are possible.
• There exist inherent barriers: for cyclic or non-planar graphs, even lower-bound-only versions become NP-hard (Traub et al., 2023). Existing path-decomposition and rounding strategies encounter structural obstructions beyond planar or series-parallel graphs.
• Strengthenings such as requiring support-respecting rounding ("carpooling") or convex decomposition with cost bounds remain open even for small $m$ or special classes (Liu et al., 23 Nov 2025).
• Further, it is unclear whether the two-sided deviation can be sharpened to $d_{\max}$ with cost optimality in all planar DAGs (Traub et al., 2023).

7. Historical Development and Future Directions

Initiated by Goemans in 1997, the conjecture set a benchmark for polyhedral and combinatorial integrality gaps in flows. While early work achieved upper-bounded violation or special demand regimes, the last five years have witnessed sharp structural and algorithmic advances:

• Convex decompositions with tight deviation in series-parallel graphs (Almoghrabi et al., 2024)
• Discrepancy-based methods establishing attainable bounds in planar graphs (Traub et al., 2023)
• Weighted assignment generalizations and applications to scheduling (Liu et al., 23 Nov 2025)
• Polyhedral rounding templates that unify prior disparate approaches (Swamy et al., 24 Oct 2025)

Key open directions include the extension to broader graph classes, refinement of cost and violation parameters, and the resolution of the conjecture in full generality. The underlying combinatorial, geometric, and algorithmic framework continues to inform research in both discrete optimization and related fields such as scheduling and discrepancy theory.