How quickly can you pack short paths? Engineering a search-tree algorithm for disjoint s-t paths of bounded length (2404.10469v1)

Abstract: 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.