A graph is called a $(k,\rho)$-graph iff every node can reach $\rho$ of its nearest neighbors in at most k hops. This property proved useful in the analysis and design of parallel shortest-path algorithms. Any graph can be transformed into a $(k,\rho)$-graph by adding shortcuts. Formally, the $(k,\rho)$-Minimum-Shortcut problem asks to find an appropriate shortcut set of minimal cardinality. We show that the $(k,\rho)$-Minimum-Shortcut problem is NP-complete in the practical regime of $k \ge 3$ and $\rho = \Theta(n^\epsilon)$ for $\epsilon > 0$. With a related construction, we bound the approximation factor of known $(k,\rho)$-Minimum-Shortcut problem heuristics from below and propose algorithmic countermeasures improving the approximation quality. Further, we describe an integer linear problem (ILP) solving the $(k,\rho)$-Minimum-Shortcut problem optimally. Finally, we compare the practical performance and quality of all algorithms in an empirical campaign.

翻译：如果一个图中的每个节点在至多 k 跳内都能到达其 $\rho$ 个最近邻，则该图被称为 $(k,\rho)$-图。该性质在并行最短路径算法的分析与设计中已被证明具有实用价值。通过添加捷径，任意图都能被转换为 $(k,\rho)$-图。形式化地，$(k,\rho)$-最小捷径问题要求寻找一个基数最小的适当捷径集合。我们证明，在 $k \ge 3$ 且 $\rho = \Theta(n^\epsilon)$（其中 $\epsilon > 0$）的实际参数范围内，$(k,\rho)$-最小捷径问题是 NP-完全的。通过一个相关构造，我们从下界限定了已知 $(k,\rho)$-最小捷径问题启发式算法的近似比，并提出了改进近似质量的算法对策。此外，我们描述了一个最优求解 $(k,\rho)$-最小捷径问题的整数线性规划（ILP）模型。最后，我们通过一系列实验对比了所有算法的实际性能和质量。