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Discrete Mathematics and Theoretical Computer Science*. 2021.

doi:10.46298/dmtcs.6280

A distributed graph algorithm is basically an algorithm where every node of a graph can look at its neighborhood at some distance in the graph and chose its output. As distributed environment are subject to faults, an important issue is to be able to check that the output is correct, or in general that the network is in proper configuration with respect to some predicate. One would like this checking to be very local, to avoid using too much resources. Unfortunately most predicates cannot be checked this way, and that is where certification comes into play. Local certification (also known as proof-labeling schemes, locally checkable proofs or distributed verification) consists in assigning labels to the nodes, that certify that the configuration is correct. There are several point of view on this topic: it can be seen as a part of self-stabilizing algorithms, as labeling problem, or as a non-deterministic distributed decision.

This paper is an introduction to the domain of local certification, giving an overview of the history, the techniques and the current research directions.

The first version of this paper was based on the introduction of my PhD thesis. I rewrote the text completely in January 2021. The paper has a list of open questions, ranging from very concrete technical questions to unexplored promising areas.

The journal, DMTCS, is a "gold" open access journal: it is free for authors and readers. It is an arxiv-overlay journal: the submitted and final versions of the paper are on the arxiv.