Global certification via perfect hashing

TL;DR

This work addresses global certification for the existence of a graph homomorphism to a fixed graph $H$, a natural generalization of graph coloring, in distributed settings with a single global certificate. It introduces a novel use of perfect hashing to compress node identifiers and constructs a certificate of size $O(n \log n' + \log \log M)$ that can be verified locally by each node. The results show that the previously conjectured $\Theta(n \log n)$ bound for bipartiteness is tight only in general, and they extend the approach to CSPs, yielding near-optimal certificate sizes and broad applicability to space-conscious distributed verification. This technique has potential implications for space-bounded distributed computation and may influence future labeling schemes and CSP certification methods.

Abstract

In this work, we provide an upper bound for global certification of graph homomorphism, a generalization of graph coloring. In certification, the nodes of a network should decide if the network satisfies a given property, thanks to small pieces of information called certificates. Here, there is only one global certificate which is shared by all the nodes, and the property we want to certify is the existence of a graph homomorphism to a given graph. For bipartiteness, a special case of graph homomorphism, Feuilloley and Hirvonen proved in~\cite{FeuilloleyH18} some upper and lower bounds on the size of the optimal certificate, and made the conjecture that their lower bound could be improved to match their upper bound. We prove that this conjecture is false: their lower bound was in fact optimal, and we prove it by providing the matching upper bound using a known result of perfect hashing.

Theorems & Definitions (9)

• Conjecture 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 5
• Definition 6
• Theorem 6
• Theorem 7
• proof : Proof of Theorem \ref{['thm:new_bound']}.