Multi-sources simultaneous communication in the wireless mobility model is NP-complete

In this article we consider a mobility model M=(S, D, U, L, R, V, C, O), where S is a set of sources, D a set of directions, U a set of users, L a set of user movements, R a set of source movements, V a set of velocities of sources, C a set of source coverages, and O a set of obstacles. Particularly, we study a problem called MULTI-SOURCES SIMULTANEOUS COMMUNICATION PROBLEM in this model. This problem is stated as follows: given a mobility model M=(S, D, U, L, R, V, C, O), k pairs of distinct sources {s₁, s´₁}, {s₂, s´₂},...,{s_k, s´_k}, and a time t, can all k pairs of sources simultaneously communicate throughout the duration t of the model without sharing a source? We show that the complexity of this problem is at least as hard as the ONE-IN-THREE 3-SATISFIABILITY unless P=NP.