# The Minimum Shared Edges Problem on Grid-like Graphs

## TL;DR

This paper investigates the computational complexity of the Minimum Shared Edges problem on grid-like graphs, providing efficient solutions for certain grid sizes and establishing hardness results and kernelization limits in a parameterized complexity framework.

## Contribution

It offers linear-time algorithms for MSE on bounded grids with specific size relations and proves NP-hardness on subgraphs, also analyzing kernelization bounds in parameterized complexity.

## Key findings

- Linear-time solvability on certain bounded grids
- NP-hardness on subgraphs of bounded grids
- No polynomial kernel for combined parameters under standard assumptions

## Abstract

We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route $p$ paths from a start vertex to a target vertex in a given graph while using at most $k$ edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number $p$ of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number $p$ of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter $k$, $p$, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs.