Computing All-Pairs Shortest Paths by Leveraging Low Treewidth
This work provides efficient algorithms for spatial and temporal reasoning, such as in planning and scheduling, though it is incremental as it builds on existing graph width concepts.
The paper tackles the problem of computing all-pairs shortest paths in directed graphs with real weights by introducing two new algorithms that leverage low treewidth, achieving optimal O(n^2) time for constant treewidth graphs and outperforming state-of-the-art methods like Floyd-Warshall and Johnson's algorithm in many cases.
We present two new and efficient algorithms for computing all-pairs shortest paths. The algorithms operate on directed graphs with real (possibly negative) weights. They make use of directed path consistency along a vertex ordering d. Both algorithms run in O(n^2 w_d) time, where w_d is the graph width induced by this vertex ordering. For graphs of constant treewidth, this yields O(n^2) time, which is optimal. On chordal graphs, the algorithms run in O(nm) time. In addition, we present a variant that exploits graph separators to arrive at a run time of O(n w_d^2 + n^2 s_d) on general graphs, where s_d andlt= w_d is the size of the largest minimal separator induced by the vertex ordering d. We show empirically that on both constructed and realistic benchmarks, in many cases the algorithms outperform Floyd-Warshalls as well as Johnsons algorithm, which represent the current state of the art with a run time of O(n^3) and O(nm + n^2 log n), respectively. Our algorithms can be used for spatial and temporal reasoning, such as for the Simple Temporal Problem, which underlines their relevance to the planning and scheduling community.