Near-minimizing hop count in Regime II

Determine the order and fluctuations of the hop-count profile \(H_{\lceil n x\rceil}((1+\varepsilon)T_n)\) for fixed \(\varepsilon>0\) in Regime II.

Background

The paper establishes a lower bound of order $\log\log n$ for the hop count of bounded-cost paths in Regime II and constructs hub-chain paths with the same order of magnitude. These results do not determine the hop count of paths whose passage time is within a fixed multiplicative factor of the optimum.

The unresolved problem is to identify both the asymptotic order and the fluctuations of the near-minimizing hop-count profile.