Planar Disjoint Shortest Paths is Fixed-Parameter Tractable

TL;DR

The paper proves that Planar Disjoint Shortest Paths (PDSP) for undirected planar graphs with positive edge weights is fixed-parameter tractable with runtime 2^{O(k log k)} · n^{O(1)}. It introduces a novel topological framework based on dag-cuts and dag-rings to partition the graph into ring-shaped regions that control winding of solution paths, enabling a bounded enumeration of homology classes. The authors fuse geometric ring decompositions with Schrijver’s Homology Feasibility machinery, via skeletons and a careful enumeration of characteristic words, to reduce PDSP to a polynomial number of tractable subproblems. This approach yields tighter parameter dependence than for the general Planar Disjoint Paths problem and provides a path toward deeper understanding of planarity's impact on shortest-path constraints. The work combines advanced topology, graph minors concepts, and algebraic methods to achieve an FPT algorithm for PDSP with practical implications for planar network design and routing problems.

Abstract

In the Disjoint Shortest Paths problem one is given a graph $G$ and a set $\mathcal{T}=\{(s_1,t_1),\dots,(s_k,t_k)\}$ of $k$ vertex pairs. The question is whether there exist vertex-disjoint paths $P_1,\dots,P_k$ in $G$ so that each $P_i$ is a shortest path between $s_i$ and $t_i$. While the problem is known to be W[1]-hard in general, we show that it is fixed-parameter tractable on planar graphs with positive edge weights. Specifically, we propose an algorithm for Planar Disjoint Shortest Paths with running time $2^{O(k\log k)}\cdot n^{O(1)}$. Notably, our parameter dependency is better than state-of-the-art $2^{O(k^2)}$ for the Planar Disjoint Paths problem, where the sought paths are not required to be shortest paths.

Figures (14)

• Figure 1: An illustration of the rerouting argument from CyganMPP13. Left: linkage $\cal{Q}$ is given as the two black horizontal paths and $\cal{P}$ comprises the purple and the red path. When considered oriented towards the internal face $C_2$, $\cal{P}$ winds four times clockwise around $C_2$, compared to $\cal{Q}$. Right: linkage $\cal{P}'$ has the same endpoints as $\cal{P}$ while its winding number is reduced.
• Figure 2: An illustration of the problematic scenario when the solution path $P$ crosses the geodesic spinal path $Q$ in the tree $T$ many times, because of the winding behavior. The fat segments of $P$ and $Q$ represent subpaths in $G$ of the same length. By analyzing the shortest path structure in this part of the graph, we identify a region (in blue) where these structures coincide for all terminal pairs.
• Figure 3: A refinement operation for a Steiner tree $T$ (in black) with respect to regions $R_1, \dots, R_\ell$ (blue). The purple path $Q$ belongs to a linkage $\cal{Q}$ that connects the two sides of the region $R_1$. We want $T$ to wind within $R_1$ in the same fashion as $Q$ to match the behavior of a certain solution. We remove the subpaths of $T$ that traverse $R_1$ (the dotted segments) and replace them with $Q$. Next, we insert the red arcs to $T$ in order to maintain connectivity. These arcs may not correspond to any paths in the graph so we have to augment it with additional edges that are flagged as forbidden to be used by a solution.
• Figure 4: An example of a dag-cut. The vertices depicted as squares correspond to the set $V_X$ while the vertices depicted as discs correspond to the set $V_Y$.
• Figure 5: An example of a dag-ring. The two dag-cuts $(\gamma,\gamma')$ representing this dag-ring are illustrated with highlighted orange arcs and violet arcs respectively. The dual cycles $\mathrm{Cycle}(\gamma)$ and $\mathrm{Cycle}(\gamma')$ are illustrated as dashed cycles and the cyan region corresponds to $\mathrm{Ring}\xspace(\gamma,\gamma')$.

Theorems & Definitions (109)

• Theorem 1.1
• Corollary 1.2
• Lemma 3.3: Monotonicity of crossings
• proof
• Lemma 3.4
• proof
• Definition 3.5: Shortest paths DAGs
• Lemma 3.6
• proof
• Definition 3.9