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Locally Checkable Proofs
"... This work studies decision problems from the perspective of nondeterministic distributed algorithms. For a yesinstance 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 loc ..."
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This work studies decision problems from the perspective of nondeterministic distributed algorithms. For a yesinstance 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 constanttime distributed algorithm. For example, it is easy to prove that a graph is bipartite: the locally checkable proof gives a 2colouring 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 Ω(log n) bits per node. In this work we classify graph problems according to their local proof complexity, i.e., how many bits per node are needed in a locally checkable proof. We establish tight or neartight results for classical graph properties such as the chromatic number. We show that the proof complexities form a natural hierarchy of complexity classes: for many classical graph problems, the proof complexity is either 0, Θ(1), Θ(log n), or poly(n) bits per node. Among the most difficult graph properties are symmetric graphs, which require Ω(n 2) bits per node, and non3colourable graphs, which require Ω(n 2 / log n) bits per node—any pure graph property admits a trivial proof of size O(n 2).
Hierarchies in Local Distributed Decision∗
"... We study the complexity theory for the local distributed setting introduced by Korman, Peleg and Fraigniaud in their seminal paper [3]. They have defined three complexity classes LD (Local Decision), NLD (Nondeterministic Local Decision) and NLD#n. The class LD consists of all languages which can be ..."
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We study the complexity theory for the local distributed setting introduced by Korman, Peleg and Fraigniaud in their seminal paper [3]. They have defined three complexity classes LD (Local Decision), NLD (Nondeterministic Local Decision) and NLD#n. The class LD consists of all languages which can be decided with a constant number of communication rounds. The class NLD consists of all languages which can be verified by a nondeterministic algorithm with a constant number of communication rounds. In order to define the nondeterministic classes, they have transferred the notation of nondeterminism into the distributed setting by the use of certificates and verifiers. The classNLD#n consists of all languages which can be verified by a nondeterministic algorithm where each node has access to an oracle for the number of nodes. They have shown the hierarchy LD ( NLD ( NLD#n. Our main contributions are strict hierarchies within the classes defined by Korman, Peleg and Fraigniaud. We define additional complexity classes: the class LD(t) consists of all languages which can be decided with at most t communication rounds. The class NLD(O(f)) consists of all languages which can be verified by a local verifier such that the size of the certificates that are needed to verify the language are bounded by a function from O(f). Our main result is the following hierarchy within the nondeterministic classes: LD ( NLD(O(1)) ( NLD(O(log n)) ( NLD(O(n)) ( NLD(O(n2)) ⊆ NLD(O(n2 + w)) = NLD. In order to prove this hierarchy, we give several lower bounds on the sizes of certificates that are needed to verify some languages from NLD. For the deterministic classes we prove the following hierarchy: