Search

Local certification is a topic originating from distributed computing, where a prover tries to convince the vertices of a graph $G$ that $G$ satisfies some property $\mathcal{P}$. To convince the vertices, the prover gives a small piece of information, called certificate, to each vertex, and the vertices then decide whether the property $\mathcal{P}$ is satisfied by just looking at their certificate and the certificates of their neighbors. When studying a property $\mathcal{P}$ in the perspective of local certification, the aim is to find the optimal size of the certificates needed to certify $\mathcal{P}$, which can be viewed a measure of the local complexity of $\mathcal{P}$. A certification scheme is considered to be efficient if the size of the certificates is polylogarithmic in the number of vertices. While there have been a number of meta-theorems providing efficient certification schemes for general graph classes, the proofs of the lower bounds on the size of the certificates are usually very problem-dependent. In this work, we introduce a notion of hardness reduction in local certification, and show that we can transfer a lower bound on the certificates for a property $\mathcal{P}$ to a lower bound for another property $\mathcal{P}'$, via a (local) hardness reduction from $\mathcal{P}$ to $\mathcal{P}'$. We then give a number of applications in which we obtain polynomial lower bounds for many classical properties using such reductions., Comment: 35 pages

In local certification, vertices of a $n$-vertex graph perform a local verification to check if a given property is satisfied by the graph. This verification is performed thanks to certificates, which are pieces of information that are given to the vertices. In this work, we focus on the local certification of $P_5$-freeness, and we prove a $O(n^{3/2})$ upper bound on the size of the certificates, which is (to our knowledge) the first subquadratic upper bound for this property.

Local certification is the area of distributed network computing asking the following question: How to certify to the nodes of a network that a global property holds, if they are limited to a local verification? In this area, it is often essential to have identifiers, that is, unique integers assigned to the nodes. In this short paper, we show how to reduce the range of the identifiers, in three different settings. More precisely, we show how to rename identifiers in the classical local certification setting, when we can (resp. cannot) choose the new identifiers, and we show how a global certificate can help to encode very compactly a new identifier assignment that is not injective in general, but still useful. We conclude with a number of applications of these three results.

Detecting specific structures in a network has been a very active theme of research in distributed computing for at least a decade. In this paper, we start the study of subgraph detection from the perspective of local certification. Remember that a local certification is a distributed mechanism enabling the nodes of a network to check the correctness of the current configuration, thanks to small pieces of information called certificates. Our main question is: For a given graph $H$, what is the minimum certificate size that allows checking that the network does not contain $H$ as a (possibly induced) subgraph? We show a variety of lower and upper bounds, uncovering an interesting interplay between the optimal certificate size, the size of the forbidden subgraph, and the locality of the verification. Along the way we introduce several new technical tools, in particular what we call the \emph{layered map}, which is not specific to forbidden subgraphs and that we expect to be useful for certifying many other properties.

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.

Local certification is a distributed mechanism enabling the nodes of a network to check the correctness of the current configuration, thanks to small pieces of information called certificates. For many classic global properties, like checking the acyclicity of the network, the optimal size of the certificates depends on the size of the network, $n$. In this paper, we focus on properties for which the size of the certificates does not depend on $n$ but on other parameters. We focus on three such important properties and prove tight bounds for all of them. Namely, we prove that the optimal certification size is: $\Theta(\log k)$ for $k$-colorability (and even exactly $\lceil \log k \rceil$ bits in the anonymous model while previous works had only proved a $2$-bit lower bound); $(1/2)\log t+o(\log t)$ for dominating sets at distance $t$ (an unexpected and tighter-than-usual bound) ; and $\Theta(\log \Delta)$ for perfect matching in graphs of maximum degree $\Delta$ (the first non-trivial bound parameterized by $\Delta$). We also prove some surprising upper bounds, for example, certifying the existence of a perfect matching in a planar graph can be done with only two bits. In addition, we explore various specific cases for these properties, in particular improving our understanding of the trade-off between locality of the verification and certificate size., Comment: Accepted at STACS 2024