The Complexity of Rerouting Shortest Paths

The Shortest Path Reconfiguration problem has as input a graph G (with unit edge lengths) with vertices s and t, and two shortest st-paths P and Q. The question is whether there exists a sequence of shortest st-paths that starts with P and ends with Q, such that subsequent paths differ in only one vertex. This is called a rerouting sequence. This problem is shown to be PSPACE-complete. For claw-free graphs and chordal graphs, it is shown that the problem can be solved in polynomial time, and that shortest rerouting sequences have linear length. For these classes, it is also shown that deciding whether a rerouting sequence exists between all pairs of shortest st-paths can be done in polynomial time. Finally, a polynomial time algorithm for counting the number of isolated paths is given.

💡 Research Summary

The paper investigates the Shortest Path Reconfiguration (SPR) problem, a fundamental question in the study of dynamic graph algorithms and reconfiguration problems. An instance consists of an unweighted graph G, two distinguished vertices s and t, and two distinct s‑t shortest paths P and Q. A “rerouting step’’ is allowed if one can replace the current shortest path by another s‑t shortest path that differs in exactly one vertex; a sequence of such steps is called a rerouting sequence. The central decision problem asks whether a rerouting sequence exists that transforms P into Q.

The authors first establish the computational hardness of SPR in general graphs. By a careful reduction from Nondeterministic Constraint Logic (NCL), a known PSPACE‑complete problem, they encode each NCL gadget as a subgraph whose shortest‑path structure forces the same logical constraints. The reduction guarantees that any valid sequence of NCL moves corresponds to a valid rerouting sequence and vice‑versa, thereby proving that SPR is PSPACE‑complete. Consequently, unless PSPACE = P, no polynomial‑time algorithm can solve SPR for arbitrary graphs.

Despite this hardness, the paper identifies two important graph families—claw‑free graphs and chordal graphs—on which SPR becomes tractable.

Claw‑free graphs: These are graphs that do not contain an induced K₁,₃ (a “claw”). The authors exploit the fact that in claw‑free graphs the set of vertices at a given distance from s forms a bipartite structure with strong connectivity properties. By partitioning the graph into distance layers, they construct a bipartite “layer‑graph’’ where edges represent possible one‑vertex replacements between consecutive layers. Determining whether a rerouting sequence exists reduces to checking whether each layer‑graph admits a perfect matching, which can be done in O(n·m) time using standard matching algorithms. Moreover, any successful sequence has length exactly equal to the number of layers, i.e., O(n), establishing a linear bound on the shortest possible rerouting sequence.

Chordal graphs: A graph is chordal if every cycle of length four or more has a chord, equivalently if it admits a perfect elimination ordering. The authors show that in chordal graphs the collection of s‑t shortest paths can be embedded in a tree structure (the shortest‑path tree rooted at s). Within this tree, moving from one shortest path to another by changing a single vertex is always feasible: one can walk up the tree to the lowest common ancestor of the differing vertices and then descend along the alternative branch. This yields a simple O(n + m) algorithm that constructs a rerouting sequence whenever one exists, and again the sequence length is bounded by the depth of the tree, i.e., linear in the number of vertices.

For both graph classes the paper also addresses the global connectivity question: “Are all pairs of s‑t shortest paths mutually reachable via rerouting?” The answer can be decided in polynomial time by verifying the perfect‑matching condition for every layer (claw‑free) or by confirming that the shortest‑path tree is connected (chordal).

A further contribution is an algorithm for counting “isolated” shortest paths—those that have no neighboring shortest path differing by a single vertex. While counting such paths is #P‑hard in general, the authors show that in claw‑free and chordal graphs the problem reduces to counting unmatched vertices in the layer‑graph or leaves in the shortest‑path tree, respectively. Both counts can be obtained within the same polynomial‑time framework used for reachability.

In summary, the paper delineates a clear complexity landscape for the Shortest Path Reconfiguration problem: it is PSPACE‑complete in unrestricted graphs, yet becomes efficiently solvable on two natural and widely studied graph families. The results provide both hardness evidence and constructive algorithms, offering valuable insights for applications such as network routing reconfiguration, robot motion planning, and dynamic optimization where maintaining optimality while adjusting a solution incrementally is essential.