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

2026-07-18

AI Summary

This study addresses the long-standing open problem of computational complexity for the Prim-Dijkstra problem in VLSI routing under Manhattan-distance-constrained terminals. The authors establish, for the first time, that this problem is weakly NP-complete and derive a trade-off relationship between continuous cost and radius, leading to a novel theoretical framework with a (2,2)-balanced guarantee. Building upon this foundation, they propose HP-RCRST, a multi-mode solver integrating hierarchical partitioning, formal verification, and counterexample generation techniques. Evaluated on 28 benchmark instances, the strong mode of HP-RCRST achieves Pareto dominance over the union of all existing methods on 23 instances and matches their performance on the remaining five.

Abstract

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.

Problem

• VLSI routing
• Prim-Dijkstra
• Manhattan complexity
• NP-completeness
• algorithmic problem

Innovation

• Prim-Dijkstra routing
• weak NP-completeness
• cost-radius tradeoff
• HP-RCRST
• VLSI routing