Abstract
This work studies 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 W(log n) bits per node. In this work 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
Highlights
This paper studies decision problems related to graph properties from the perspective of distributed graph algorithms
We say that a graph property P is locally checkable if it can be checked by a local algorithm
We catalog graph properties according to their local proof complexities, and we show that the LCP( f ) classes form a natural hierarchy of decision problems
Summary
This paper studies decision problems related to graph properties from the perspective of distributed graph algorithms. In a local algorithm all nodes stop after O(1) communication rounds and announce their outputs. We say that a graph property P is locally checkable if it can be checked by a local algorithm. An easy example of a locally checkable property is determining if a given connected graph is Eulerian: it is sufficient that each node outputs 1 if its degree is even, and 0 otherwise. Another example is checking if a given graph is a line graph. The key insight of Korman et al [15, 16, 18, 19] is to study locally checkable proofs
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