Deleting Edges to Restrict the Size of an Epidemic: A New Application for Treewidth

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 $$2^{O(|\mathcal {F}|w^r)}n$$ 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.