A Heuristic for Dijkstra's Algorithm With Many Targets and its Use in Weighted Matching Algorithms

We consider the single-source many-targets shortest-path (SSMTSP) problem in directed graphs with non-negative edge weights. A source node $s$ and a target set $T$ is specified and the goal is to compute a shortest path from $s$ to a node in $T$. Our interest in the shortest path problem with many targets stems from its use in weighted bipartite matching algorithms. A weighted bipartite matching in a graph with $n$ nodes on each side reduces to $n$ SSMTSP problems, where the number of targets varies between $n$ and $1$. The SSMTSP problem can be solved by Dijkstra's algorithm. We describe a heuristic that leads to a significant improvement in running time for the weighted matching problem; in our experiments a speed-up by up to a factor of 10 was achieved. We also present a partial analysis that gives some theoretical support for our experimental findings