Robust Cuts Over Time: Combatting the Spread of Invasive Species with Unreliable Biological Control

Widespread accounts of the harmful effects of invasive species have stimulated both practical and theoretical studies on how the spread of these destructive agents can be contained. In practice, a widely used method is the deployment of biological control agents, that is, the release of an additional species (which may also spread) that creates a hostile environment for the invader. Seeding colonies of these protective biological control agents can be used to build a kind of living barrier against the spread of the harmful invader, but the ecological literature documents that attempts to establish colonies of biological control agents often fail (opening gaps in the barrier). Further, the supply of the protective species is limited, and the full supply may not be available immediately. This problem has a natural temporal component: biological control is deployed as the extent of the harmful invasion grows. How can a limited supply of unreliable biological control agents best be deployed over time to protect the landscape against the spread of a harmful invasive species? To explore this question we introduce a new family of stochastic graph vaccination problems that generalizes ideas from social networks and multistage graph vaccination. We point out a deterministic (1− 1/e)-approximation algorithm for a deterministic base case studied in the social networks literature (matching the previous best randomized (1− 1/e) guarantee for that problem). Next, we show that the randomized (1− 1/e) guarantee (and a deterministic 1/2 guarantee) can be extended to our much more general family of stochastic graph vaccination problems in which vaccinations (a.k.a. biological control colonies) spread but may be unreliable. For the non-spreading vaccination case with unreliable vaccines, we give matching results in trees. Qualitatively, our extension is from computing “cuts over time” to computing “robust cuts