On the complexity of removing Z-cycles from a checkpoints and communication pattern

Communication-induced checkpointing protocols are mechanisms used to produce checkpoints and communication patterns which enjoy desirable properties, such as No-Z-Cycle (NZC). NZC guarantees that each checkpoint can be part of a global consistent checkpoint. It would be nice to define communication-induced checkpointing protocols that enforce NZC, adding a minimum number of checkpoints to remove all the Z-cycles from the distributed computation. In this paper, we prove that this is impossible by formulating the Minimum Z-Cycle Removal (MinZCR) problem and showing that there are no online competitive protocols for it. Moreover, we prove that the problem of enforcing NZC with an optimal number of checkpoints is difficult even if the whole input instance is known because its decision version is NP-complete. Finally, we also prove that MinZCR is difficult to approximate: it is APX-hard and this implies hat no Polynomial Time Approximation Scheme exists for the problem.