Computational Complexity and Exact Algorithms for Pressure-Driven Hydrogen Charging

Abstract

We study the problem of deciding the feasibility of pressure-driven hydrogen charging from supply cylinders to demand cylinders without replenishment, asking whether an order of transfers allows every demand cylinder to reach its target pressure. Hydrogen can be transferred from a cylinder to another only when the supply cylinder has higher pressure than the demand cylinder. The charging to each demand cylinder must be completed before starting another, and each transfer continues until the connected cylinders reach equal pressures or the demand cylinder reaches its target. We consider equal-capacity supply cylinders and arbitrary positive rational demand cylinder capacities. Let $n$ be the number of demand cylinders and $L$ the binary input length. With one supply cylinder, feasibility is decidable in polynomial time; with two, it is NP-complete even with cylinder capacities equal to $1$ and nonnegative integer pressures. We prove that feasibility is preserved even if the supply cylinders are always used in non-decreasing order of their current pressures when filling each demand cylinder. We also show that majorization of residual supply cylinder pressure states justifies discarding majorized states in dynamic programming over sets of completed demand cylinders without losing any feasible remaining demand cylinder order. With two supply cylinders, one state per completed demand cylinder set suffices, yielding $2^n\operatorname{poly}(L)$ time and space. With three, the exponential time hypothesis (ETH) excludes deterministic $2^{o(n\log n)}\operatorname{poly}(L)$-time algorithms. This lower bound is tight in the exponent because the rule for order of supply cylinders allows feasibility to be decided in $n!\operatorname{poly}(L)$ time by enumerating only demand cylinder orders.