We consider the Minimum-$(k,ρ)$-$\mathrm{Shortcut}$ problem ($\min(k,ρ)\text{-}\mathrm{Shortcut}$), where the goal is to find the smallest set of shortcut edges such that every vertex in a given graph can reach its $ρ$ closest vertices using paths of at most $k$ edges. This is a fundamental graph optimization problem used to accelerate parallel shortest path algorithms. It is well-known that the problem is trivially solvable for the cases $k=1$ and $k\geqρ$. While recent work by Leonhardt, Meyer, and Penschuck (ESA 2024) showed that in undirected graphs $\min(k,ρ)\text{-}\mathrm{Shortcut}$ is NP-hard for $k\geq 3$ if $ρ=Θ(n^ε)$, the boundary where the problem transitions from polynomial-time solvable to NP-hard remained open. In this paper, we narrow this gap significantly. We present a simpler and more direct reduction from the Hitting Set problem which establishes that $\min(k,ρ)\text{-}\mathrm{Shortcut}$ is NP-hard for $k\geq2$ and $ρ\geq k+2$ in both directed and undirected graphs. Complementing this, we use the symmetry of the undirected case to show that $ρ=k+1$ is solvable in polynomial time, a regime where the directed version remains a candidate for NP-hardness. Therefore, we obtain an almost complete characterization of the complexity of $\min(k,ρ)\text{-}\mathrm{Shortcut}$, with the sole remaining open case being $ρ= k+1$ in the directed setting.

翻译：我们考虑最小-(k,ρ)-捷径问题（$\min(k,ρ)\text{-}\mathrm{Shortcut}$），其目标是找到最小的捷径边集，使得给定图中每个顶点都能通过至多k条边的路径到达其最接近的ρ个顶点。这是一个用于加速并行最短路径算法的基础图优化问题。已知该问题在k=1和k≥ρ时平凡可解。尽管Leonhardt、Meyer和Penschuck近期的研究（ESA 2024）表明，在无向图中，当ρ=Θ(n^ε)且k≥3时，$\min(k,ρ)\text{-}\mathrm{Shortcut}$是NP困难的，但该问题从多项式时间可解到NP困难的分界点仍未明确。本文显著缩小了这一差距。我们提出了一种更简单且更直接的从击中集问题的归约，证明了对于有向图和无向图，当k≥2且ρ≥k+2时，$\min(k,ρ)\text{-}\mathrm{Shortcut}$是NP困难的。作为补充，我们利用无向情况下的对称性，证明了当ρ=k+1时问题可在多项式时间内求解，而在该参数下，有向版本仍可能是NP困难的。因此，我们几乎完整刻画了$\min(k,ρ)\text{-}\mathrm{Shortcut}$的复杂度，唯一尚未解决的问题是有向设定中ρ=k+1的情况。