We study the Short Path Packing problem which asks, given a graph $G$, integers $k$ and $\ell$, and vertices $s$ and $t$, whether there exist $k$ pairwise internally vertex-disjoint $s$-$t$ paths of length at most $\ell$. The problem has been proven to be NP-hard and fixed-parameter tractable parameterized by $k$ and $\ell$. Most previous research on this problem has been theoretical with limited practical implemetations. We present an exact FPT-algorithm based on a search-tree approach in combination with greedy localization. While its worst case runtime complexity of $(k\cdot \ell^2)^{k\cdot \ell}\cdot n^{O(1)}$ is larger than the state of the art, the nature of search-tree algorithms allows for a broad range of potential optimizations. We exploit this potential by presenting techniques for input preprocessing, early detection of trivial and infeasible instances, and strategic selection of promising subproblems. Those approaches were implemented and heavily tested on a large dataset of diverse graphs. The results show that our heuristic improvements are very effective and that for the majority of instances, we can achieve fast runtimes.

翻译：我们研究短路径打包问题：给定图$G$、整数$k$和$\ell$以及顶点$s$和$t$，判断是否存在$k$条内部顶点不相交的$s$-$t$路径，且每条路径长度不超过$\ell$。该问题已被证明是NP难的，且关于参数$k$和$\ell$是固定参数可解的。此前该问题的研究多为理论性质，实用化实现较少。我们提出一种基于搜索树方法并融合贪婪定位的精确FPT算法。虽然其最坏情况运行时间复杂度$(k\cdot \ell^2)^{k\cdot \ell}\cdot n^{O(1)}$高于现有最优算法，但搜索树算法的特性允许进行广泛潜在优化。我们通过以下技术利用这一特性：输入预处理、平凡与不可行实例的早期检测、以及有前景子问题的策略性选择。这些方法已在包含多样图结构的大规模数据集上实现并充分测试。结果表明，我们的启发式改进效果显著，对大多数实例都能实现快速运行时间。