Deleting edges to restrict the size of an epidemic

TL;DR

This paper presents algorithms for deleting edges in small-treewidth graphs to prevent large connected components, thereby limiting epidemic spread, with practical testing on cattle movement data.

Contribution

It introduces fixed-parameter algorithms for edge deletion problems in small-treewidth graphs to control epidemic sizes, including improvements for specific cases.

Findings

Algorithms run efficiently on small-treewidth graphs

Implemented methods tested on real cattle movement data

Effective in preventing large epidemic components

Abstract

Motivated by applications in network epidemiology, we consider the problem of determining whether it is possible to delete at most $k$ edges from a given input graph (of small treewidth) so that the resulting graph avoids a set $\mathcal{F}$ of forbidden subgraphs; of particular interest is the problem of determining whether it is possible to delete at most $k$ edges so that the resulting graph has no connected component of more than $h$ vertices, as this bounds the worst-case size of an epidemic. While even this special case of the problem is NP-complete in general (even when $h=3$), we provide evidence that many of the real-world networks of interest are likely to have small treewidth, and we describe an algorithm which solves the general problem in time \genruntime ~on an input graph having $n$ vertices and whose treewidth is bounded by a fixed constant $w$, if each of the subgraphs we wish to avoid has at most $r$ vertices. For the special case in which we wish only to ensure that no component has more than $h$ vertices, we improve on this to give an algorithm running in time $O((wh)^{2w}n)$, which we have implemented and tested on real datasets based on cattle movements.