Introduction to local certification

A distributed graph algorithm is basically an algorithm where every node of a graph can look at its neighborhood at some distance in the graph and chose its output. As distributed environment are subject to faults, an important issue is to be able to check that the output is correct, or in general that the network is in proper configuration with respect to some predicate. One would like this checking to be very local, to avoid using too much resources. Unfortunately most predicates cannot be checked this way, and that is where certification comes into play. Local certification (also known as proof-labeling schemes, locally checkable proofs or distributed verification) consists in assigning labels to the nodes, that certify that the configuration is correct. There are several point of view on this topic: it can be seen as a part of self-stabilizing algorithms, as labeling problem, or as a non-deterministic distributed decision. This paper is an introduction to the domain of local certification, giving an overview of the history, the techniques and the current research directions.Comment: Last update: minor editin

Brief Announcement: Local Certification of Graph Decompositions and Applications to Minor-Free Classes

Local certification consists in assigning labels to the nodes of a network to certify that some given property is satisfied, in such a way that the labels can be checked locally. In the last few years, certification of graph classes received a considerable attention. The goal is to certify that a graph G belongs to a given graph class ?. Such certifications with labels of size O(log n) (where n is the size of the network) exist for trees, planar graphs and graphs embedded on surfaces. Feuilloley et al. ask if this can be extended to any class of graphs defined by a finite set of forbidden minors. In this paper, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor H, H-minor-free graphs can indeed be certified with labels of size O(log n). We also show matching lower bounds with a new simple proof technique

Optimal Space Lower Bound for Deterministic Self-Stabilizing Leader Election Algorithms

Given a boolean predicate ? on labeled networks (e.g., proper coloring, leader election, etc.), a self-stabilizing algorithm for ? is a distributed algorithm that can start from any initial configuration of the network (i.e., every node has an arbitrary value assigned to each of its variables), and eventually converge to a configuration satisfying ?. It is known that leader election does not have a deterministic self-stabilizing algorithm using a constant-size register at each node, i.e., for some networks, some of their nodes must have registers whose sizes grow with the size n of the networks. On the other hand, it is also known that leader election can be solved by a deterministic self-stabilizing algorithm using registers of O(log log n) bits per node in any n-node bounded-degree network. We show that this latter space complexity is optimal. Specifically, we prove that every deterministic self-stabilizing algorithm solving leader election must use ?(log log n)-bit per node registers in some n-node networks. In addition, we show that our lower bounds go beyond leader election, and apply to all problems that cannot be solved by anonymous algorithms

Twenty-Two New Approximate Proof Labeling Schemes

Introduced by Korman, Kutten, and Peleg (Distributed Computing 2005), a proof labeling scheme (PLS) is a system dedicated to verifying that a given configuration graph satisfies a certain property. It is composed of a centralized prover, whose role is to generate a proof for yes-instances in the form of an assignment of labels to the nodes, and a distributed verifier, whose role is to verify the validity of the proof by local means and accept it if and only if the property is satisfied. To overcome lower bounds on the label size of PLSs for certain graph properties, Censor-Hillel, Paz, and Perry (SIROCCO 2017) introduced the notion of an approximate proof labeling scheme (APLS) that allows the verifier to accept also some no-instances as long as they are not "too far" from satisfying the property. The goal of the current paper is to advance our understanding of the power and limitations of APLSs. To this end, we formulate the notion of APLSs in terms of distributed graph optimization problems (OptDGPs) and develop two generic methods for the design of APLSs. These methods are then applied to various classic OptDGPs, obtaining twenty-two new APLSs. An appealing characteristic of our APLSs is that they are all sequentially efficient in the sense that both the prover and the verifier are required to run in (sequential) polynomial time. On the negative side, we establish "combinatorial" lower bounds on the label size for some of the aforementioned OptDGPs that demonstrate the optimality of our corresponding APLSs. For other OptDGPs, we establish conditional lower bounds that exploit the sequential efficiency of the verifier alone (under the assumption that NP ? co-NP) or that of both the verifier and the prover (under the assumption that P ? NP, with and without the unique games conjecture)

Local Certification of Graph Decompositions and Applications to Minor-Free Classes

Local certification consists in assigning labels to the nodes of a network to certify that some given property is satisfied, in such a way that the labels can be checked locally. In the last few years, certification of graph classes received a considerable attention. The goal is to certify that a graph G belongs to a given graph class ?. Such certifications with labels of size O(log n) (where n is the size of the network) exist for trees, planar graphs and graphs embedded on surfaces. Feuilloley et al. ask if this can be extended to any class of graphs defined by a finite set of forbidden minors. In this work, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor H, H-minor-free graphs can indeed be certified with labels of size O(log n). We also show matching lower bounds using a new proof technique

Local Certification of Some Geometric Intersection Graph Classes

In the context of distributed certification, the recognition of graph classes has started to be intensively studied. For instance, different results related to the recognition of planar, bounded tree-width and $H$-minor free graphs have been recently obtained. The goal of the present work is to design compact certificates for the local recognition of relevant geometric intersection graph classes, namely interval, chordal, circular arc, trapezoid and permutation. More precisely, we give proof labeling schemes recognizing each of these classes with logarithmic-sized certificates. We also provide tight logarithmic lower bounds on the size of the certificates on the proof labeling schemes for the recognition of any of the aforementioned geometric intersection graph classes

Automated testing and interactive construction of unavoidable sets for graph classes of small path‐width

Let G be a class of graphs with a membership test, k∈N , and let Gk be the class of graphs in G of path-width at most k. We present an interactive framework that finds an unavoidable set for Gk, which is a set of graphs U such that any graph in Gk contains an isomorphic copy of a graph in U. At the core of our framework is an algorithm that verifies whether a set of graphs is, indeed, unavoidable for Gk. While obstruction sets are well-studied, so far there is no general theory or algorithm for finding unavoidable sets. In general, it is undecidable whether a finite set of graphs is unavoidable for a given graph class. However, we give a criterion for termination: our algorithm terminates whenever G is locally checkable of bounded maximum degree and U is a finite set of connected graphs. For example, l-regular graphs, l-colourable graphs, and H-free graphs are locally checkable classes. We put special emphasis on the case that G is the class of cubic graphs and tailor the algorithm to this case. In particular, we introduce the new concept of high-degree-first path-decompositions, which enables highly efficient pruning techniques. We exploit our framework to prove a new lower bound on the path-width of cubic graphs. Moreover, we determine the extremal girth values of cubic graphs of path-width for all and all smallest graphs which take on these extremal girth values. Further, we present a new constructive characterisation of the extremal cubic graphs of path-width 3 and girth 4

Introduction to local certification

A distributed graph algorithm is basically an algorithm where every node of a graph can look at its neighborhood at some distance in the graph and chose its output. As distributed environment are subject to faults, an important issue is to be able to check that the output is correct, or in general that the network is in proper configuration with respect to some predicate. One would like this checking to be very local, to avoid using too much resources. Unfortunately most predicates cannot be checked this way, and that is where certification comes into play. Local certification (also known as proof-labeling schemes, locally checkable proofs or distributed verification) consists in assigning labels to the nodes, that certify that the configuration is correct. There are several point of view on this topic: it can be seen as a part of self-stabilizing algorithms, as labeling problem, or as a non-deterministic distributed decision. This paper is an introduction to the domain of local certification, giving an overview of the history, the techniques and the current research directions