*Theory of Computing* · volume 12, article 19, 33 pages, 2016 · doi:10.4086/toc.2016.v012a019

We study *decision problems* related to *graph properties* from the perspective of *nondeterministic distributed algorithms*. For a *yes*-instance there must exist a *proof* that can be verified with a distributed algorithm: all nodes must accept a valid proof, and at least one node must reject an invalid proof. We focus on *locally checkable proofs* that can be verified with a constant-time distributed algorithm. For example, it is easy to prove that a graph is bipartite: the locally checkable proof gives a $2$-coloring of the graph, which only takes $1$ bit per node. However, it is more difficult to prove that a graph is *not* bipartite—it turns out that any locally checkable proof requires $\Omega(\log n)$ bits per node.

In this paper we classify graph properties according to their local proof complexity, i.e., how many bits per node are needed in a locally checkable proof. We establish tight or near-tight results for classical graph properties such as the chromatic number. We show that the local proof complexities form a natural hierarchy of complexity classes: for many classical graph properties, the local proof complexity is either $0$, $\Theta(1)$, $\Theta(\log n)$, or $\operatorname{poly}(n)$ bits per node. Among the most difficult graph properties are proving that a graph is symmetric (has a non-trivial automorphism), which requires $\Omega(n^2)$ bits per node, and proving that a graph is *not* $3$-colorable, which requires $\Omega(n^2/\log n)$ bits per node. Any property of connected graphs admits a trivial proof with $O(n^2)$ bits per node.

- Mika Göös and Jukka Suomela: Locally checkable proofs · PODC 2011