Efficient Algorithms for Listing k Disjoint st-Paths in Graphs

Abstract

Given a connected graph G of m edges and n vertices, we consider the basic problem of listing all the choices of k vertex-disjoint st-paths, for any two input vertices s, t of G and a positive integer k. Our algorithm takes O(m) time per solution, using O(m) space and requiring O(F k (G)) setup time, where F k (G) = O(m min{k, n 2/3 log n, √ m log n}) is the cost of running a max-flow algorithm on G to compute a flow of size k. The proposed techniques are simple and apply to other related listing problems discussed in the paper.

Towards bridging theory and practice

“…(1) BC-DFS. A common practice to develop polynomial delay algorithm for enumeration problem is to ensure that each search branch has at least one output, which is used by T-DFS and T-DFS2 algorithms in [24,54]. However, we observe that this strategy may lead to high overhead in our problem, which results in the poor practical performance.…”

“…al studied the problem of s-t simple path enumeration, but their solution only supports the undirected graphs. Two polynomial delay algorithms are proposed in [24,54], which take O(km) time per output. The counting of s-t simple paths is a well-known #P hard problem, which has been studied with different approaches such as recursive expressions of the adjacency matrix (e.g., [8,22,35]) and immanantal equations [5].…”

“…Thus, the time complexity of T-DFS is O(kmδ) where δ is the number of hc-s-t paths. Recently, Grossi et al study the problem of listing k disjoint s-t paths in [24], and a new algorithm, namely T-DFS2 in this paper, is proposed for hc-s-t path enumeration following the same aggressive checking strategy. It can reduce shortest path distance computation by skipping some vertices associated with only one output in the following search.…”

“…Nevertheless, authors stress that it is difficult to apply the above technique to directed graph, thus T-DFS takes O(km) time per output on directed graph as shown in Section 3.2. Recently, Grossi et al study the problem of listing k disjoint s-t paths in [24]. As the enumeration of hc-s-t paths is a special case of k disjoint s-t path listing problem, they propose a polynomial delay technique following the aggressive checking strategy.…”

“…State-of-the-art. To the best of our knowledge, two polynomial delay algorithms for the problem of hc-s-t path enumeration are proposed in two theoretical papers [24,54], namely T-DFS and T-DFS2 in this paper. Their theoretical results are quite attractive, with O(km) time per output where m is the number of edges.…”