On the $k$ Shortest Simple Paths Problem in Weighted Directed Graphs

• Authors: Liam Roditty
• Venue: SIAM Journal on Computing
• Year: 2010

We present the first approximation algorithm for finding the $k$ shortest simple paths connecting a pair of vertices in a weighted directed graph that breaks the barrier of $mn$. It is deterministic and has a running time of $O(k(m\sqrt{n}+n^{3/2}\log n))$, where $m$ is the number of edges in the graph and $n$ is the number of vertices. Let $s,t\in V$; the length of the $i$th simple path from $s$ to $t$ computed by our algorithm is at most $\frac{3}{2}$ times the length of the $i$th shortest simple path from $s$ to $t$. The best algorithms for computing the exact $k$ shortest simple paths connecting a pair of vertices in a weighted directed graph are due to Yen [Management Sci., 17 (1970/1971), pp. 712-716] and Lawler [Management Sci., 18 (1971/1972), pp. 401-405]. The running time of their algorithms, using modern data structures, is $O(k(mn+n^2\log n))$. Both algorithms are from the early 70s. Although this problem and other variants of the $k$ shortest path problem has drawn a lot of attention during the last three and a half decades, the $O(k(mn+n^2\log n))$ bound is still unbeaten.