Distributed graph problems through an automata-theoretic lens

Chang, Yi‐Jun; Studený, Jan; Suomela, Jukka

doi:10.1016/j.tcs.2023.113710

Cited by 2 publications

(10 citation statements)

References 31 publications

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“…However, for all other cases the type directly determines both solvability, and all these cases are also decidable in polynomial time. The correctness of this classification is proved in the full version [14].…”

Section: Introductionmentioning

confidence: 87%

“…Moreover, Π can be specified in the node-edge-checkable formalism. We will give the details in the full version [14]. From now on, all LCL problems considered are by default problems defined using the node-edge-checkable formalism.…”

Section: Definition 1 (Lcl Problem) An Lcl Problem π In the Node-edge...mentioning

confidence: 99%

“…Throughout this work, our main focus will be on paths and cycles. This may at first seem highly restrictive, but as we will show in the full version [14], once we understand LCL problems in paths and cycles, through reductions we will also gain understanding on so-called edge-checkable problems in rooted trees.…”

Section: Introductionmentioning

confidence: 99%

“…Both of the canonical problems can be solved in the message-passing setting with small messages and with little local memory. Furthermore, when we look at rooted trees (see the full version [14]), our algorithms are "one-sided": each node only needs to receive information from its parent. It follows that our algorithms work also e.g.…”

Section: Introductionmentioning

confidence: 99%

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Distributed Graph Problems Through an Automata-Theoretic Lens

Chang

Studený

Suomela

2021

Structural Information and Communication Complexity

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The locality of a graph problem is the smallest distance T such that each node can choose its own part of the solution based on its radius-T neighborhood. In many settings, a graph problem can be solved efficiently with a distributed or parallel algorithm if and only if it has a small locality. In this work we seek to automate the study of solvability and locality: given the description of a graph problem Π, we would like to determine if Π is solvable and what is the asymptotic locality of Π as a function of the size of the graph. Put otherwise, we seek to automatically synthesize efficient distributed and parallel algorithms for solving Π. We focus on locally checkable graph problems; these are problems in which a solution is globally feasible if it looks feasible in all constant-radius neighborhoods. Prior work on such problems has brought primarily bad news: questions related to locality are undecidable in general, and even if we focus on the case of labeled paths and cycles, determining locality is PSPACE-hard (Balliu et al., PODC 2019). We complement prior negative results with efficient algorithms for the cases of unlabeled paths and cycles and, as an extension, for rooted trees. We study locally checkable graph problems from an automata-theoretic perspective by representing a locally checkable problem Π as a nondeterministic finite automaton M over a unary alphabet. We identify polynomial-time-computable properties of the automaton M that near-completely capture the solvability and locality of Π in cycles and paths, with the exception of one specific case that is co-NP-complete.

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Section: Introductionmentioning

confidence: 87%

Section: Definition 1 (Lcl Problem) An Lcl Problem π In the Node-edge...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Distributed Graph Problems Through an Automata-Theoretic Lens

Chang

Studený

Suomela

2021

Structural Information and Communication Complexity

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“…Finally, Büchi-Elgot-Trakhtenbrot theorem states that MSO properties are exactly the ones that are recognized by a regular automaton, thus we get Theorem 2.2 in the case of directed paths. The automata point of view (without the relation to logics) has been used before to understand the complexity of locally checkable labelings on cycles and paths, see in particular in [11]. Now, a tree automaton is the analogue of a regular automaton for rooted trees.…”

Section: Mso Certification On Trees Via Tree Automatamentioning

confidence: 99%

What Can Be Certified Compactly? Compact local certification of MSO properties in tree-like graphs

Feuilloley

Bousquet

Pierron

2022

Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing

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Local certification consists in assigning labels (called certificates) to the nodes of a network to certify a property of the network or the correctness of a data structure distributed on the network. The verification of this certification must be local: a node typically sees only its neighbors in the network. The main measure of performance of a certification is the size of its certificates.In 2011, Göös and Suomela identified Θ(log ) as a special certificate size: below this threshold little is possible, and several key properties do have certifications of this type. A certification with such small certificates is now called a compact local certification, and it has become the gold standard of the area, similarly to polynomial time for centralized computing. A major question is then to understand which properties have (log ) certificates, or in other words: what is the power of compact local certification?Recently, a series of papers have proved that several well-known network properties have compact local certifications: planarity, bounded-genus, etc. But one would like to have more general results, i.e. meta-theorems. In the analogue setting of polynomial-time centralized algorithms, a very fruitful approach has been to prove that restricted types of problems can be solved in polynomial time in graphs with restricted structures. These problems are typically those that can be expressed in some logic, and the graph structures are whose with bounded width or depth parameters. We take a similar approach and prove the first meta-theorems for local certification.More precisely, the logic we use is MSO, the most classic fragment for logics on graphs, where one can quantify on vertices and sets of vertices, and consider adjacency between vertices. We prove the relevance of this choice in the context of local certification by first considering properties of trees. On trees, we prove that MSO properties can be certified with labels of constant size, whereas the typical non-MSO property of isomorphism requires ˜ ( ) size certificates (where ˜ hides polylogarithmic factors). We then move on to graphs of bounded treedepth, a well-known parameter that basically measures how far a graph is from a star. We first prove that an optimal certification for bounded treedepth uses certificates of size Θ(log ), and then prove that in bounded treedepth graphs, every MSO property has a compact certification.To establish our results, we use a variety of techniques, originating from model checking, tree automata theory, communication complexity, and combinatorics.A preliminary version of this paper appeared on the arxiv under the name "Local certification of MSO properties for bounded treedepth graphs" [9].

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Distributed graph problems through an automata-theoretic lens

Cited by 2 publications

References 31 publications

Distributed Graph Problems Through an Automata-Theoretic Lens

Distributed Graph Problems Through an Automata-Theoretic Lens

What Can Be Certified Compactly? Compact local certification of MSO properties in tree-like graphs

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