Provably Good Prim-Dijkstra Revisited: New Theory and a Practical Algorithm for a Classical VLSI Routing Problem with LLMs

Large language models may make precise but dormant algorithmic problems practical to revisit, and may expose new paths toward fundamental ones. We demonstrate this possibility through Prim-Dijkstra routing, a classic VLSI problem whose terminal-only Manhattan complexity remained open despite decades of practical work. We prove weak NP-completeness, derive a continuous cost-radius tradeoff with a balanced (2,2) guarantee, and build HP-RCRST, a height-partition-based multi-mode solver. On 28 development instances, its stronger modes Pareto-dominate the published-method union on 23 and tie on five. The case shows how conflicting conjectures, counterexamples, formal checks, and implementation can reopen neglected questions. Code and reproducibility materials are available at https://github.com/CODA-Team/hp-rcrst.