Single-source unsplittable flow discrepancy bound

Establish whether, for every nonnegative flow x on a directed acyclic graph D=(V,A) with a single source s, sink terminals t_1, ..., t_n with demands d_1, ..., d_n > 0, there exists an unsplittable flow y (each demand d_j routed along a single s–t_j path) such that the per-arc discrepancy satisfies |x_a − y_a| ≤ max_{j ∈ [n]} d_j for all arcs a ∈ A.

Background

This conjecture, due to Morell and Skutella, seeks a uniform per-arc L∞ discrepancy bound between any feasible (fractional) single-source multi-sink flow and an unsplittable flow that routes each demand along one path. The weighted chairman assignment problem studied in the paper reduces to a special case of this conjecture, so a resolution would imply the optimal discrepancy bound for that assignment problem.

One-sided bounds are known; Dinitz, Garg, and Goemans previously showed y_a − x_a ≤ max_j d_j for all arcs, and Morell and Skutella showed x_a − y_a ≤ max_j d_j. The full two-sided bound remains open except for special demand structures (e.g., divisibility chains d_1 | d_2 | … | d_n) and certain graph classes (e.g., acyclic planar and series-parallel digraphs).