Optimal exchange of data over broadcast networks with adversaries

In the cooperative data exchange problem, a set of clients share a lossless broadcast channel. Each client initially has a subset of packets in the ground set X, and wishes to learn all packets in X. The clients exchange their packets with each other by broadcasting coded or uncoded packets. In this paper, we consider a generalization of this problem for the settings in which an unknown (but of a bounded size) subset of clients are adversarial. The adversarial clients can introduce erasures or errors in their packet transmissions in an arbitrary manner. The problem is to find the minimum total number of transmissions required such that, regardless of the configuration of adversarial clients, all non-adversarial clients can learn the maximally recoverable subset of packets in X. For arbitrary problem instances (i.e., arbitrary sets of packets available at the clients), this problem is NP-hard. Focusing on the settings where the packets are distributed randomly among clients, in this work, we propose a linear-time algorithm which solves (with high probability) the special case of the problem with one adversarial client. This result can also be extended to more general cases with arbitrary number of adversarial clients.