Edge Deletion to Restrict the Size of an Epidemic

TL;DR

This paper investigates the computational complexity of edge deletion problems to prevent the formation of certain subgraphs, proving hardness results and identifying fixed-parameter tractability for specific cases related to epidemic control.

Contribution

It resolves an open question by showing the problem is W[1]-hard for treewidth, and provides FPT algorithms and kernelization results for special cases involving the deletion of edges to limit component sizes.

Findings

The problem is W[1]-hard when parameterized by treewidth.

The problem is W[2]-hard when parameterized by solution size, feedback vertex set, or pathwidth.

The problem is FPT when parameterized by vertex cover number for the case of deleting edges to limit component size.

Abstract

Given a graph $G=(V,E)$, a set $\mathcal{F}$ of forbidden subgraphs, we study $\mathcal{F}$-Free Edge Deletion, where the goal is to remove minimum number of edges such that the resulting graph does not contain any $F\in \mathcal{F}$ as a subgraph. For the parameter treewidth, the question of whether the problem is FPT has remained open. Here we give a negative answer by showing that the problem is W[1]-hard when parameterized by the treewidth, which rules out FPT algorithms under common assumption. Thus we give a solution to the conjecture posted by Jessica Enright and Kitty Meeks in [Algorithmica 80 (2018) 1857-1889]. We also prove that the $\mathcal{F}$-Free Edge Deletion problem is W[2]-hard when parameterized by the solution size $k$, feedback vertex set number or pathwidth of the input graph. A special case of particular interest is the situation in which $\mathcal{F}$ is the set $\mathcal{T}_{h+1}$ of all trees on $h+1$ vertices, so that we delete edges in order to obtain a graph in which every component contains at most $h$ vertices. This is desirable from the point of view of restricting the spread of disease in transmission network. We prove that the $\mathcal{T}_{h+1}$-Free Edge Deletion problem is fixed-parameter tractable (FPT) when parameterized by the vertex cover number. We also prove that it admits a kernel with $O(hk)$ vertices and $O(h^2k)$ edges, when parameterized by combined parameters $h$ and the solution size $k$.