Deletion-Star Ceilings and a 1.28249 Lower Bound for Cost-Preserving Single-Source Unsplittable Flows

For a single-source unsplittable flow we ask for a rounding of a feasible fractional flow $\textbf{x}$ to one path per terminal that satisfies both the cost constraint $\textbf{c}^\top\textbf{y}\le\textbf{c}^\top\textbf{x}$ and the overload constraint $y_a\le x_a+Cd_{\max}$ on every arc. Goemans conjectured $C=1$; this was disproved in July 2026 by a seven-vertex instance with critical constant $16/15$, and our previous work raised the bound to $(299-41\sqrt{41})/32=1.1397\ldots$ with a four-terminal planar family. We continue that program on three fronts.

Records: we present a seventeen-terminal common-point interval instance that certifies

$$ C\ \ge\ \frac{1282494797984843521}{10^{18}}=1.282494797984843521\ldots, $$

verified independently by enumerating all $2^{17}$ routings from the raw $67$-arc list, together with an $18$-atom exact convex-hull certificate. 

Upper bounds: we prove the first unconditional upper bound below 2 for a nontrivial family of the extremal cells. Representing the class as a weighted two-permutation prefix system, we introduce the deletion-star residual $B$, show that a bound $B\le\beta$ yields congestion at most $1+\beta$, and prove that on the codimension two cost face defined by the equality $\sum_if_i=k-2$ one always has $B\le\tfrac12$, for arbitrary orders, arbitrary demands (zero demands allowed) and arbitrary shares. This gives congestion at most $\tfrac32$ on every common-point cell whose first cost-good routing rank is $k-1$. The constant $\tfrac12$ is asymptotically sharp at every fixed codimension $q$, with the exact value $B_{mq,q}=\tfrac12-\tfrac1{2m}$, and the sharpness persists through a non-divisible two-scale demand perturbation. Throughout the paper, theorems are named by the mechanism that proves them (proportional credits, a single deletion star, the multi-star threshold), and subsequent negative results (of which this program has produced several) make the scope of these methods more narrow rather than retract theorems.

Structure: we show that the naive continuum limit of the class has value zero, so the class supremum is a genuinely microscopic quantity; we give a cost-preserving transfer through Knuth's two-way rounding network with exact radius $B(\textbf{d})=\min_{\lambda\ge0}(\lambda+\sum_i|d_i-\lambda|)$; we prove the lower-box statement (LB) for all $2^{k-1}$ deque orders and for a maximal switched-interval class of signed tree networks; and we record exact minimal obstructions showing that every scalar two-order induction fails, leaving a convex Pareto transfer as the surviving mechanism. Finally, literal three-order overlays are dominated at $6/5$.

Every numerical claim is certified in exact rational or symbolic arithmetic by programs that rebuild each instance from its raw arc list; the artifacts are published at https://github.com/snikolenko/unsplittable-flows.