Prophylactic Vaccination of Networks

Motivated by preventative vaccination in a graph against the worst-case outbreak of an infectious disease, we propose new important graph cut problems. In the most basic problem Min-Max Component Size, we are given a capacity bound, and the goal is to remove a set of nodes (or edges) whose capacity does not exceed the bound, and minimizes the size of the largest resulting connected component. In generalizations of this problem, we consider the objectives of minimizing the size of the k largest components for a fixed k, and the maximum number of special “terminal” nodes inside any component. Under the assumption that each edge of a network will deterministically transmit a disease from either endpoint to the other, these problems naturally model the goal of targeting limited vaccinations of nodes (or edges) in such a way that the number of infected nodes in a worst-case outbreak is minimized. We present (O(1), O(log n)) bicriteria approximation algorithms for the Min-Max Component Size problem and for the generalization to k infected components. Our algorithms are based on LP rounding and region growing techniques. If instead, a bound k on the number of terminals inside any component is given, and the goal is to minimize the total capacity of nodes (or edges) removed, we improve the approximation guarantee to an (O(1), O(log k)) bicriteria result. Finally, if k is a constant, and the edge-cut version is considered, we show how to obtain a combinatorial single-criterion approximation not violating the bound on the component size, and approximating the capacity of edges removed to within O(log k). ∗Supported in part by NSF CAREER award 0545855, and NSF grant DDDAS-TMRP 0540420