## Abstract

We extend the notion of distributed decision in the framework of distributed network computing, inspired by both the polynomial hierarchy for Turing machines and recent results on so-called distributed graph automata. We show that, by using distributed decision mechanisms based on the interaction between a prover and a disprover, the size of the certificates distributed to the nodes for certifying a given network property can be drastically reduced. For instance, we prove that minimum spanning tree (MST) can be certified with O(logn)-bit certificates in n-node graphs, with just one interaction between the prover and the disprover, while it is known that certifying MST requires Ω(log^{2}n)-bit certificates if only the prover can act. The improvement can even be exponential for some simple graph properties. For instance, it is known that certifying the existence of a nontrivial automorphism requires Ω(n^{2}) bits if only the prover can act. We show that there is a protocol with two interactions between the prover and the disprover that certifies nontrivial automorphism with O(logn)-bit certificates. These results are achieved by defining and analyzing a local hierarchy of decision which generalizes the classical notions of proof-labeling schemes and locally checkable proofs.

Original language | English |
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Pages (from-to) | 51-67 |

Number of pages | 17 |

Journal | Theoretical Computer Science |

Volume | 856 |

Early online date | 2020 |

DOIs | |

Publication status | Published - 8 Feb 2021 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Distributed decision
- Distributed hierarchy
- Distributed non-determinism
- Local certification
- Local model
- Proof-labeling scheme