Parameterized Shortest Path Reconfiguration

TL;DR

This work analyzes the parameterized complexity of Shortest Path Reconfiguration (SPR) and Shortest Shortest Path Reconfiguration (SSPR) under token jumping and token sliding. It establishes W[1]-hardness for SPR parameterized by the path length $k$ and for SSPR parameterized by $k+\ell$, via a reduction from Regular Multicolored Clique, and strengthens this to graphs of constant degeneracy; on the positive side, it proves FPT when parameterized by $\ell$ on nowhere-dense graphs and by structural parameters treedepth, cluster deletion number, and modular width. The paper also discusses a foundational, constant-structure approach: leveraging gadgets, buffers, and collapses to encode clique instances and to control token movement, ensuring the reductions preserve parameter bounds. These results delineate the tractability frontier for SPR/SSPR, highlighting the potential for FO-model checking and modular-decomposition-based kernels in restricted graph classes while leaving open the status for feedback vertex set parameterization. Overall, the work clarifies when reconfiguring shortest paths remains feasible and when it becomes intractable, guiding future algorithm design for network design and related reconfiguration tasks.

Abstract

An st-shortest path, or st-path for short, in a graph G is a shortest (induced) path from s to t in G. Two st-paths are said to be adjacent if they differ on exactly one vertex. A reconfiguration sequence between two st-paths P and Q is a sequence of adjacent st-paths starting from P and ending at Q. Deciding whether there exists a reconfiguration sequence between two given $st$-paths is known to be PSPACE-complete, even on restricted classes of graphs such as graphs of bounded bandwidth (hence pathwidth). On the positive side, and rather surprisingly, the problem is polynomial-time solvable on planar graphs. In this paper, we study the parameterized complexity of the Shortest Path Reconfiguration (SPR) problem. We show that SPR is W[1]-hard parameterized by k + \ell, even when restricted to graphs of bounded (constant) degeneracy; here k denotes the number of edges on an st-path, and \ell denotes the length of a reconfiguration sequence from P to Q. We complement our hardness result by establishing the fixed-parameter tractability of SPR parameterized by \ell and restricted to nowhere-dense classes of graphs. Additionally, we establish fixed-parameter tractability of SPR when parameterized by the treedepth, by the cluster-deletion number, or by the modular-width of the input graph.

Figures (6)

• Figure 1: The graph parameters studied in this paper. A connection between two parameters indicates the existence of a function in the one above that lower-bounds the one below.
• Figure 2: Example of a hard instance. Only edges of starting and ending paths are shown. Rectangles represent vertices of the feedback vertex set who neighborhood into the adjacent paths can be arbitrary.
• Figure 3: Example of a graph $\Gamma(2, H, 2) = \mathcal{I}^2 \oplus H \oplus \mathcal{J}^{2}$, where $\mu(R_1) = \mu(H_1) = \mu(H_2) = \mu(H_3) = V_1$, $\mu(R_2) = V_2$, $\mu(R_3) = V_3$, and $\mu(R_\beta) = V_1$. Edges inside $H$ are omitted.
• Figure 4: Example of a graph $\Gamma(2, H(h^1_1), 2)$ obtained after collapsing $H$ on $h^1_1$.
• Figure 5: An example of our reduction in the case of token jumping.

Theorems & Definitions (49)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• proof
• Lemma 7
• proof
• Lemma 8
• proof
• Lemma 9