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Dinitz-Garg-Goemans conjecture is false. This graph theory problem was open for ~30 years. The graph below has fractional flow cost 58. Any unsplittable flow (with capacity violation <=15) has cost at least 60. Chat with GPT 5.6 Pro where this was found: chatgpt.com/share/6a60b2eb…
I know counterexamples to old conjectures are becoming a meme at this point. But I really cared about this problem and spent many weeks thinking about it a while ago (in both directions, proof and disproof). I think almost all graph flows experts thought about this problem.
The conjecture was based on absolutely stunning result of Dinitz, Garg, and Goemans: any fractional flow can be routed to unsplittable flow by violating graph capacities by at most max(demand). The chat with gpt pro here is an absolute meme
"But I really cared about this problem and spent many weeks thinking about it a while ago" No, you didn't, lol.