Abstract
Hybrid bonding terminal (HBT) legalization assigns every terminal of a face-to-face 3D IC to its own pitch-grid site with minimum total Manhattan displacement. Open3DBench reports that a reinforcement-learning (RL) legalizer reduces displacement by 19-22% relative to a sliding-window (SW) heuristic, but none of its legalizers is publicly released. We ask how much displacement any legalizer can still recover. We build an open benchmark of 21 design/pitch instances from Open3DBench placement data, with reconstructed greedy, SW (ViaLegal-style) and BTAssign-style baselines, and GEO-Exact, an exact solver that runs successive shortest augmenting paths over lazily generated spatial candidate edges. An independent dual certificate proves its assignment optimal on all 21 instances (relative primal-dual gap at most $8.2 \times 10^{-16}$). Against these optima, our SW reconstruction with ViaLegal's default configuration is on average 0.011% above optimal (worst 0.119%), while on the 16 contended instances BTAssign-style, a density-guided heuristic (DGHL) and greedy are 3.0%, 6.9% and 64.5% above. Headroom does appear outside this regime: with 2x2 windows SW reaches 34.1%, and above about 58% utilization, on three of six swept designs, it reaches 21.1% on bp, 27.5% on swerv and 8.9% on jpeg (217 further certified instances, and ten bp points checked against an independent exact solver). If the published 19-22% gain were measured on instances like ours, SW would have to be at least 24-28% above optimal on average, far more than any SW variant we tested at nominal pitch; the RL paper's own comparison with the Hungarian optimum puts its SW 3.05% above optimal on average. GEO-Exact is also about an order of magnitude faster than DGHL and SW in our harness (geometric means 8.1x and 40x; a compiled core against Python baselines on a shared server). Code, optimal values and certificates are released.
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URL
https://arxiv.org/abs/2611.14928
