[Paper Review] The Minimum Shared Edges Problem on Planar Graphs

Till Fluschnik, Manuel Sorge|arXiv (Cornell University)|Feb 3, 2016

Advanced Graph Theory Research3 references3 citations

TL;DR

This paper proves that the Minimum Shared Edges problem on planar graphs is NP-hard, even when restricted to graphs of maximum degree four. The authors reduce from Vertex Cover using a grid-based construction with specialized gadgets (bundles, feathers, rainbows) to model vertex and edge selection, demonstrating that shared edge constraints correspond precisely to vertex cover solutions in the reduction.

ABSTRACT

We study the Minimum Shared Edges problem introduced by Omran et al. [Journal of Combinatorial Optimization, 2015] on planar graphs: Planar MSE asks, given a planar graph G = (V,E), two distinct vertices s,t in V , and two integers p, k, whether there are p s-t paths in G that share at most k edges, where an edges is called shared if it appears in at least two of the p s-t paths. We show that Planar MSE is NP-hard by reduction from Vertex Cover. We make use of a grid-like structure, where the alignment (horizontal/vertical) of the edges in the grid correspond to selection and validation gadgets respectively.

Motivation & Objective

• To establish the computational complexity of the Minimum Shared Edges problem on planar graphs.
• To determine whether the problem remains NP-hard under planarity and bounded degree constraints.
• To investigate the parameterized complexity of the problem with respect to the number of shared edges.
• To extend the hardness result to directed planar graphs with bounded out- and in-degree.

Proposed method

• Construct a planar grid graph where rows represent vertices and columns represent edges from a Vertex Cover instance.
• Use $(m^3, M, k'+1)$-feathers and $(1,M,k'+1)$-feathers to encode vertex and edge selection constraints.
• Replace vertical connections with directed gadget components to simulate undirected edges in a directed setting.
• Apply edge subdivision and binary tree replacements to reduce vertex degrees to at most four while preserving planarity and solution equivalence.
• Use rainbows to replace bundles and adjust the shared edge budget to maintain equivalence after structural modifications.
• Direct all edges except vertical connections, and replace each undirected vertical connection with a directed gadget ensuring at least $k'+1$ shared arcs per gadget.

Experimental results

Research questions

• RQ1 Is the Minimum Shared Edges problem NP-hard on planar graphs?
• RQ2 Does the NP-hardness of the problem persist when the graph has maximum degree four?
• RQ3 Can the reduction be adapted to directed planar graphs with bounded out- and in-degree?
• RQ4 Is the problem fixed-parameter tractable when parameterized by the number of shared edges?
• RQ5 Does the hardness result extend to planar graphs of maximum degree three?

Key findings

• The Minimum Shared Edges problem on planar graphs is NP-hard, even when restricted to graphs of maximum degree four.
• The reduction from Vertex Cover uses a grid-like structure with specialized gadgets to simulate vertex and edge selection.
• Each $s$-$t$ route in the constructed instance corresponds to a valid vertex cover in the original instance, and vice versa.
• The shared edge budget $k'$ is set to $k imes (m^3 + m + 1)$, ensuring that only solutions corresponding to valid vertex covers achieve the required shared edge count.
• The construction maintains planarity throughout all transformations, including degree reduction and directed adaptation.
• The problem remains NP-hard on directed planar graphs with maximum out-degree and in-degree three, as shown by the directed gadget replacement.