Four-Terminal Planar Case of the Dinitz-Garg-Goemans Cost Conjecture
Does the Dinitz-Garg-Goemans cost-preserving unsplittable-flow rounding conjecture survive on acyclic planar instances with only four terminals? An explicit instance answers no: every cost-nonincreasing unsplittable routing has upper overload at least while the maximum demand is .
- Result
- Disproved(see note)
- Status
- Variant only
- AI contribution
- AI co-developed
- Method
- Computation
- Field
- Combinatorial optimization
- Posed by
- Yefim Dinitz, Naveen Garg & Michel Goemans
- Year posed
- 1999
- Years open
- 27y
- Solved
- 2026-07-23
- Model
- GPT-5.6 Pro
- Vendor
- OpenAI
- Collaborators
- Matthew Protti
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 10 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
restricted planar four-terminal case
What the AI did
GPT-5.6 Pro carried out much of the construction search, symbolic derivation, proof development, exact-verifier development, adversarial critique and manuscript preparation. The human author selected and framed the problem, directed the investigation, caught a cost-normalization error, required exact and adversarial checks, set the claim scope and approved the release. A later Codex session independently re-encoded the key graph, finite and symbolic checks and ran deterministic stress and release checks.
Verification
The immutable v0.1.0 public disclosure ships a manuscript, exact data, an exhaustive verifier over all 16 routings and all 13 arcs, mutation tests, deterministic hashes and a separate AI-assisted computational cross-check. The attained certificate is ; the package also proves the limiting lower bound , with sharpness only in the stated fixed-topology, equal-full-cost, two-cheap-choice model. Released 2026-07-23, one day after the first public disproof of the general conjecture, and developed independently of it.
Sources
Submitted by LuckyHawk816