Introduction to local certification

Feuilloley, Laurent

doi:10.46298/dmtcs.6280

Cited by 15 publications

(14 citation statements)

References 56 publications

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“…We refer to [15] for an introduction to local certi cation (both in terms of techniques and history), and to [16] for a more complexity-theoretic survey. Most of the relevant related work has been reviewed in the previous sections, but there are two aspects that we would like to detail.…”

Section: Related Work and Discussion Of The Modelmentioning

confidence: 99%

“…Local certi cation is such a mechanism, in the sense that it allows to check any graph property locally. For a given property, a local certi cation is described by a certi cate assignment and a veri cation algorithm: each node receives a certi cate, reads the certi cates of its neighbors and then runs a veri cation algorithm [15]. This algorithm decides whether the node accepts or rejects the certi cation.…”

Section: Local Certificationmentioning

confidence: 99%

“…Let us now sketch some standard tools for local certi cation that we repeatedly use in the rest of the paper. (We refer to [15] for full explanations and proofs.) Before going to certi cation, let us just remind that a vertex accesses its degree locally without any certi cate.…”

Section: Local Certification: Definitions and Basic Techniquesmentioning

confidence: 99%

See 2 more Smart Citations

Local certification of MSO properties for bounded treedepth graphs

Bousquet¹,

Feuilloley²,

Pierron³

2021

Preprint

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The graph model checking problem consists in testing whether an input graph satis es a given logical formula.In this paper, we study this problem in a distributed setting, namely local certi cation. The goal is to assign labels to the nodes of a network to certify that some given property is satis ed, in such a way that the labels can be checked locally.We rst investigate which properties can be locally certi ed with small certi cates. Not surprisingly, this is almost never the case, except for not very expressive logic fragments. Following the steps of Courcelle-Grohe, we then look for meta-theorems explaining what happens when we parameterize the problem by some standard measures of how simple the graph classes are. In that direction, our main result states that any MSO formula can be locally certi ed on graphs with bounded treedepth with a logarithmic number of bits per node, which is the golden standard in certi cation.2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms → Distributed algorithms

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Section: Related Work and Discussion Of The Modelmentioning

confidence: 99%

Section: Local Certificationmentioning

confidence: 99%

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Local certification of MSO properties for bounded treedepth graphs

Bousquet¹,

Feuilloley²,

Pierron³

2021

Preprint

Self Cite

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“…If we say that the complexity is O(f (n)), for some function f , the O(•) notation refers to n → ∞. See [24,31] for more details on proof labelling schemes and local certification in general.…”

Section: Local Verifiersmentioning

confidence: 99%

Testability and local certification of monotone properties in minor-closed classes

Norin¹

2022

Preprint

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The main problem in the area of property testing is to understand which graph properties are testable, which means that with constantly many queries to any input graph G, a tester can decide with good probability whether G satisfies the property, or is far from satisfying the property. Testable properties are well understood in the dense model and in the bounded degree model, but little is known in sparse graph classes when graphs are allowed to have unbounded degree. This is the setting of the sparse model.We prove that for any proper minor-closed class G, any monotone property (i.e., any property that is closed under taking subgraphs) is testable for graphs from G in the sparse model. This extends a result of Czumaj and Sohler (FOCS'19), who proved it for monotone properties with finitely many obstructions. Our result implies for instance that for any integers k and t, k-colorability of K t -minor free graphs is testable in the sparse model.Elek recently proved that monotone properties of bounded degree graphs from minorclosed classes that are closed under disjoint union can be verified by an approximate proof labeling scheme in constant time. We show again that the assumption of bounded degree can be omitted in his result.

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“…As opposed to pre-assigned identifiers, there has been much previous work related to algorithmically solving tasks more efficiently after choosing labels for the nodes of the network, or choosing labels for mobile agents moving in the network (see related surveys [3,16,21,31]). We restrict attention to previous work concerning tasks in radio networks.…”

Section: Related Workmentioning

confidence: 99%

Labeling Schemes for Deterministic Radio Multi-Broadcast

Krisko¹,

Miller²

2021

Preprint

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We consider the multi-broadcast problem in arbitrary connected radio networks consisting of n nodes. There are k designated source nodes for some fixed k ∈ {1, . . . , n}, and each source node has a distinct piece of information that it wants to share with all nodes in the network. When k = 1, this is known as the broadcasting problem, and when k = n, this is known as the gossiping problem. We consider the feasibility of solving multi-broadcast deterministically in radio networks. It is known that multi-broadcast is solvable when the nodes have distinct identifiers (e.g., using round-robin), and, it has been shown by Ellen, Gorain, Miller, and Pelc (2019) that the broadcasting problem is solvable if the nodes have been carefully assigned 2bit labels rather than distinct identifiers. We set out to determine the shortest possible labels so that multi-broadcast can be solved deterministically in the labeled radio network by some universal deterministic distributed algorithm.First, we show that every radio network G with maximum degree ∆ can be labeled using O(min{log k, log ∆})-bit labels in such a way that multi-broadcast with k sources can be accomplished. This bound is tight, in the sense that there are networks such that any labeling scheme sufficient for multi-broadcast with k sources requires Ω(min{log k, log ∆})-bit labels.However, the above result is somewhat unsatisfactory: the bound is tight for certain network topologies (e.g., complete graphs), but there are networks where significantly shorter labels are sufficient. For example, we show how to construct a tree with maximum degree Θ( √ n) in which gossiping (i.e., multi-broadcast with n sources) can actually be solved after labeling the nodes with O(1)-bit labels. So, we set out to find a labeling scheme for multi-broadcast that uses the optimal number of distinct labels in every network. For all trees, we provide a labeling scheme and accompanying algorithm that will solve gossiping, and, we prove an impossibility result that demonstrates that our labeling scheme is optimal for gossiping in each tree. In particular, we prove that Θ(log D(G))-bit labels are necessary and sufficient in every tree G, where D(G) denotes the distinguishing number of G. This result also applies more generally to multi-broadcast in trees with k ∈ {2, . . . , n} sources in the case where the k sources are not known when the labeling scheme is applied.

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Introduction to local certification

Cited by 15 publications

References 56 publications

Local certification of MSO properties for bounded treedepth graphs

Local certification of MSO properties for bounded treedepth graphs

Testability and local certification of monotone properties in minor-closed classes

Labeling Schemes for Deterministic Radio Multi-Broadcast

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