The Algorithmic Geometry of Decompression Schedules

Abstract

We cast decompression planning as hybrid optimal control with ppO${}_2$ and narcotic depth feasibility, affine tissue ceilings, and convex oversaturation penalties. A rearrangement theorem gives monotone ascent for terminal objectives. Admitting excursions below the start depth breaks this, a one compartment instance with integrated oversaturation makes re-descent strictly beat every monotone profile, so monotone ascent is only an operational constraint. Without cumulative oxygen exposure the minimum inert feasible gas dominates every pure or relaxed policy along each fixed depth path, giving exact pure attainment with finitely many switches. A CNS/OTU oxygen budget is the path coupling state that restores gas choice and prevents zero inert holds, we derive its oxygen shadow price and gas switching inequality, and a risk price threshold makes no-stop ascent globally optimal. For fixed stops an exact dwell switching identity gives a forward state sweep and a backward adjoint sweep, on-gassing holds are dominated, and the one compartment problem has closed form scalarised and capped solutions. For saturation decompression rectangular uncertainty collapses to one worst case endpoint, while finite bounce exposures admit a unique strictly interior worst rate with closed form location and finitely many critical rates. Clipped oversaturation is not Markov, but full tissue vectors support safe dominance and one sided enclosures certify a hard cap. The finite menu problem is NP-hard with one tissue and has $2^m$ nondominated labels at bounded horizon, yet for fixed tissue dimension and polynomial conditioning, state compression gives an FPTAS for scalarisation and risk repair its exact cap counterpart. A frontier conjugacy theorem characterises which capped points scalarisation recovers, and worked examples match the dwell law, saturation endpoint equality, and time/risk frontier.