Scenario-dependent deviations in a fixed unsplittable-flow gadget

For the fixed four-terminal planar acyclic unsplittable-flow gadget and coordinatewise positive cost-difference scenarios, determine the normalized additive upper arc-deviation supremum as the number of scenarios varies. The targets are (299−41√41)/32, 17/8, 3 and 4 for one, two, three and at least four scenarios.

Why it matters and what counts as progress

Why it matters: It isolates exactly how scenario count changes the extremal deviation for this gadget, including nonattainment of the two-scenario value under cost non-increase.

What counts as progress: Determine and attain the scenario-dependent extrema, distinguishing a supremum from an attained maximum under cost non-increase.

How the problem developed

23 Jul 2026 · Partial result [1]
Matthew Protti Released a one-scenario lower-bound construction and a fixed four-terminal planar gadget with equal-cost alternatives.
28 Jul 2026 · Partial result [2]
Matthew Protti Established the exact 17/8 supremum for two scenarios on the fixed gadget.
4 Aug 2026 · Theorem [3]
Matthew Protti Reported the fixed-gadget scenario ladder: (299−41√41)/32 for one scenario, 17/8 for two, 3 for three, and 4 for at least four.

AI contribution

Scenario-Cover Geometry for a Four-Terminal Planar Unsplittable-Flow Gadget

Matthew Protti

For the released fixed four-terminal planar acyclic gadget with coordinatewise positive cost-difference scenarios, the normalized additive upper arc-deviation supremum is (299−41√41)/32 with one scenario, 17/8 with two, 3 with three, and 4 with at least four. The two-scenario value is not attained under scenario-wise cost non-increase.

AI involvementAI-centered

ImportanceModest

Technical verificationPartial

Human understandingPartial