Distributed Black Virus Decontamination and Rooted Acyclic Orientations

In a network supporting mobile agents, a particular threat is that posed by the presence of a black virus (BV), a harmful entity capable of destroying any agent arriving at the site where it resides, and of then moving to all the neighbouring sites. A moving BV can only be destroyed if it arrives at a site where an anti-viral agent is located. The objective for a team of mobile antiviral system agents, called cleaners, is to locate and permanently eliminate the BV, whose initial location is unknown. The goal is to perform this task with the minimum number of network infections and agent casualties. The problem of optimal black virus decontamination (BVD) has been investigated for special classes of highly regular network topologies; a (centralized) solution exists for networks of known arbitrary topology. In this paper, we consider the BVD problem in networks of arbitrary and unknown topology; we prove that it can be solved optimally in a purely decentralized way by asynchronous agents provided with 2-hop visibility. In fact, we prove that our proposed protocols always correctly decontaminate the network with the minimum number of system agents' casualties and network infections. Furthermore, we show that the total number of system agents is also optimal. Finally, we prove an interesting correspondence between the BVD problem and the problem of computing a rooted acyclic orientation of a given graph with minimum outdegrees. As a consequence, our protocols provide a distributed optimal solution to this graph optimization problem.