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

Abstract: LLMs 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.

• The paper proves that the rooted terminal-only Manhattan cost–radius spanning-tree problem is weakly NP-complete through a reduction from positive-integer Partition, resolving a long-standing complexity question.
• The paper develops a height-partition algorithm that runs in linear time after MST construction and guarantees both total length and source radius within a factor of 2 of their respective optima.
• The paper presents HP-RCRST, whose search-enhanced modes dominate published-method baselines on 23 of 28 development instances, while the certified mode scales to 2,048 terminals with median runtime of 27 milliseconds.

The problem and its history

The paper revisits the rooted, terminal-only Manhattan cost–radius spanning-tree problem (RCRST), a formulation that traces to Cong et al.'s 1992 "provably good" bounded-radius routing work and Alpert et al.'s 1995 Prim–Dijkstra (PD) construction. Given terminals $V \subset \mathbb{Z}^2$, a source $r$, and a spanning tree $T$ of the complete terminal graph, the objectives are total length $L(T)$ and source radius $R(T)$; the decision problem asks whether a tree exists with $L(T)\le B$ and $R(T)\le D$. The minimum spanning tree minimizes $L$ but can inflate $R$; the shortest-path tree does the reverse. The complexity of this exact decision question was left open in 1992, and no direct global-radius approximation theory had been joined to a modern implementation. The paper closes both gaps: it proves weak NP-completeness, derives a continuous $(1+\Delta/H,\,1+H/\Delta)$ tradeoff whose balanced point is $r$0, and builds HP-RCRST, a deterministic multi-mode solver.

A distinctive methodological feature is the use of five deliberately conflicting LLM-driven investigations—hardness, blind exact, informed exact, approximation, and synthesis—under controlled information boundaries, with claim ledgers, counterexample ledgers, and an evaluator frozen before the final solver was written. The author retained responsibility for proof reconstruction, literature due diligence, and final claims.

Weak NP-completeness

The paper proves that RCRST is NP-complete under many-one reductions, via a reduction from positive-integer Partition. The gadget places axis terminals $r$1 with $r$2, upper terminals $r$3, and a far terminal $r$4 where $r$5, $r$6. Budgets are $r$7 and $r$8. A subset summing to $r$9 yields a tree meeting both budgets exactly; conversely, cut-counting arguments force every interblock prefix cut to have exactly one tree crossing lying on the root path, forcing all internal edges $T$0, and a telescoping bridge inequality $T$1 pins the selected-item sum $T$2 to exactly $T$3.

The classification is explicitly qualified as weak NP-hardness: coordinates scale linearly with the Partition instance, so strong hardness and pseudopolynomial exact algorithms remain open. Membership in NP follows from exact binary arithmetic on an $T$4-edge certificate. This answers the question left open by Cong et al., though only at the level of ordinary NP-completeness.

The height-partition theorem

The constructive result is a bottom-up partition of a rooted MST. Processing vertices in postorder, the algorithm retains each parent edge iff doing so keeps residual component height at most $T$5; otherwise it cuts the edge and reconnects the child directly to the root. The theorem states, for every real $T$6:

$T$7

so $T$8, and $T$9 gives the balanced $L(T)$0 guarantee in linear time after rooting. The proof charges each shortcut's net addition to the deleted parent edge plus the closed residual's edges; MST cut optimality bounds the net addition, and disjointness of charged residuals bounds total charge by $L(T)$1.

The paper also establishes a factor-two boundary: for every $L(T)$2, an explicit finite metric admits no single tree with both $L(T)$3 and $L(T)$4. This boundary is deliberately narrow—it uses arbitrary metrics not embeddable in the Manhattan plane, concerns one tree rather than a portfolio, and leaves open a below-two planar-$L(T)$5 guarantee or a portfolio-level approximate-Pareto theorem. Compared with KRY/LAST's per-terminal stretch envelope $L(T)$6, the height partition traces the tighter global-radius envelope $L(T)$7, but does not replace the stronger per-terminal statement.

Notably, the proof was redirected by a counterexample: a conjectured per-terminal stretch invariant survived over half a million tests before failing on a 58-terminal subdivided broom with common factor $L(T)$8. Charging whole residual components instead of local edges fixed the accounting unit—a change of amortization object, not of constant.

HP-RCRST: certified core plus empirical search

The solver separates a theorem-bearing core from empirical frontier improvement. Every candidate is a parent array validated by full reconstruction and exact integer remeasurement; incremental move scores never certify reported objectives. Four modes share the same construction pool:

Mode	Guarantee	Character
certified	universal $L(T)$9 witness preserved	MST, star, five height thresholds
fast	empirical	shallow bounded repair
balanced	empirical	larger primary budget for small $R(T)$0
quality	empirical	intensive branch, companion branch, exact enumeration

Exactness is confined to bounded domains: complete Prüfer-code enumeration for $R(T)$1, and an $R(T)$2 dynamic program that is exact only for centered partitions of one fixed MST. The two rational cutting-plane engines from the exact investigations were deliberately excluded because their uncontrolled tails conflicted with production use. A scalarization witness—Pareto points unsupported by any weighted sum, found independently by both exact conditions—explains why neither exact route could prove membership in P despite converging on identical cutting-plane architectures.

Experimental results

On the frozen 28-instance development comparison against the nondominated union of published methods (classical constructions, PD-II swept over 19 parameters, MSPD/MSS over its prescribed settings), balanced and quality never lose: they strictly dominate the union on 23 instances and tie on five, with aggregate normalized hypervolume difference $R(T)$3. Each dominates PD-II alone on 25 instances and MSPD/MSS alone on 23.

The certified mode tells a different story. It is covered by the published union on 26 of 28 instances (aggregate hypervolume difference $R(T)$4), yet its best balanced representative achieves mean empirical common factor $R(T)$5 on development data and $R(T)$6 held-out—well inside the worst-case ceiling of two, versus $R(T)$7 for the much denser reference. The resolution is that a one-dimensional height sweep finds good central points but cannot populate the shoulders between MST and SPT endpoints; on grid_5x5, alternative MST tie structures dominate the entire certified portfolio. Search therefore changes available topologies, not merely sample density: constructions alone fail to cover the reference on 18 instances, while quality mode has zero losses.

Held-out ordering is monotone: fast strictly covers certified on 44/50 without losses, balanced covers fast on 21 with 29 ties, quality covers balanced on nine with 41 ties. Scaling is strong: all 360 invocations completed, with certified median 27 ms at $R(T)$8 versus 141–182 ms for empirical modes, peak memory under 4.1 MiB, single-threaded portable -O3 builds. Quality reproduces the complete exact frontier on all four exhaustively enumerable instances. Ablations show non-additive contributions—the scalable tier makes the largest jump, deep search removes remaining losses—and a separately budgeted companion branch recovers eight frontier points hidden by strong centered seeds.

Limitations and open questions

The paper is explicit about boundaries. The reduction proves only weak hardness; strong NP-hardness and pseudopolynomial algorithms remain unresolved. The factor-two lower bound holds for arbitrary metrics, leaving planar-$R(T)$9 improvements and portfolio-level guarantees open. Only the certified mode carries a universal guarantee; fast, balanced, and quality make empirical claims tied to released benchmark records. The 28-instance set was visible during development and supports regression, not out-of-sample generalization claims. Baseline artifacts lack same-machine timings, so no cross-paper runtime ratio is asserted. The Lean formalization covers only elementary modules (tree model, Manhattan parity, gcd length lattice); the main theorems are ordinary mathematical proofs, not machine-checked ones. The public repository is a reproducibility subset, not a transcript of model interactions.

Conclusion

The paper settles the terminal-only Manhattan cost–radius decision problem as weakly NP-complete, supplies a linear-time $L(T)\le B$0 bicriteria construction optimal within its metric-only proof technique, and demonstrates empirically that a certified core plus bounded search Pareto-covers the published-method union on 23 of 28 instances while running in tens of milliseconds at two thousand terminals. Its broader contribution is procedural: conflicting LLM investigations with information boundaries, preserved counterexamples, and a frozen evaluator moved a dormant classical question through conjecture, falsification, proof, and implementation—with proofs, witnesses, and exact remeasurement, rather than model output, serving as the standards of evidence.