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On the Complexity of Network Synchronization

We show that if a minimal-time solution to a fundamental distributed computation primitive, synchronizing a network path of finite-state processors, exists on the three-dimensional, undirected grid, then we can conclude the purely complexity-theoretic result P=NP. 
 
Every previous result on network synchronization for various network topologies either demonstrates the existence of fast synchronization solutions or proves that a synchronization solution cannot exist at all To date, it is unknown whether there is a network topology for which there exists a synchronization solution but for which no minimal-time synchronization solution exists Under the assumption that P ≠ NP, this paper solves this longstanding open problem in the affirmative.