Local treewidth of random and noisy graphs with applications to stopping contagion in networks

TL;DR

This paper investigates local treewidth in sparse random and noisy graphs, providing bounds and applying them to develop fixed parameter tractable algorithms for contagion containment.

Contribution

It offers nearly tight bounds for local treewidth in specific graph models and introduces improved algorithms for contagion control problems.

Findings

Derived tight bounds for local treewidth in sparse random graphs.

Established bounds for local treewidth in noisy trees.

Developed algorithms with subexponential dependence on infected nodes.

Abstract

We study the notion of local treewidth in sparse random graphs: the maximum treewidth over all $k$-vertex subgraphs of an $n$-vertex graph. When $k$ is not too large, we give nearly tight bounds for this local treewidth parameter; we also derive tight bounds for the local treewidth of noisy trees, trees where every non-edge is added independently with small probability. We apply our upper bounds on the local treewidth to obtain fixed parameter tractable algorithms (on random graphs and noisy trees) for edge-removal problems centered around containing a contagious process evolving over a network. In these problems, our main parameter of study is $k$, the number of initially ``infected'' vertices in the network. For the random graph models we consider and a certain range of parameters the running time of our algorithms on $n$-vertex graphs is $2^{o(k)}\textrm{poly}(n)$, improving upon the $2^{\Omega(k)}\textrm{poly}(n)$ performance of the best-known algorithms designed for worst-case instances of these edge deletion problems.