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**Published in:** LIPIcs, Volume 286, 27th International Conference on Principles of Distributed Systems (OPODIS 2023)

Graphical games are a useful framework for modeling the interactions of (selfish) agents who are connected via an underlying topology and whose behaviors influence each other. They have wide applications ranging from computer science to economics and biology. Yet, even though an agent’s payoff only depends on the actions of their direct neighbors in graphical games, computing the Nash equilibria and making statements about the convergence time of "natural" local dynamics in particular can be highly challenging. In this work, we present a novel approach for classifying complexity of Nash equilibria in graphical games by establishing a connection to local graph algorithms, a subfield of distributed computing. In particular, we make the observation that the equilibria of graphical games are equivalent to locally verifiable labelings (LVL) in graphs; vertex labelings which are verifiable with constant-round local algorithms. This connection allows us to derive novel lower bounds on the convergence time to equilibrium of best-response dynamics in graphical games. Since we establish that distributed convergence can sometimes be provably slow, we also introduce and give bounds on an intuitive notion of "time-constrained" inefficiency of best responses. We exemplify how our results can be used in the implementation of mechanisms that ensure convergence of best responses to a Nash equilibrium. Our results thus also give insight into the convergence of strategy-proof algorithms for graphical games, which is still not well understood.

Juho Hirvonen, Laura Schmid, Krishnendu Chatterjee, and Stefan Schmid. On the Convergence Time in Graphical Games: A Locality-Sensitive Approach. In 27th International Conference on Principles of Distributed Systems (OPODIS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 286, pp. 11:1-11:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hirvonen_et_al:LIPIcs.OPODIS.2023.11, author = {Hirvonen, Juho and Schmid, Laura and Chatterjee, Krishnendu and Schmid, Stefan}, title = {{On the Convergence Time in Graphical Games: A Locality-Sensitive Approach}}, booktitle = {27th International Conference on Principles of Distributed Systems (OPODIS 2023)}, pages = {11:1--11:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-308-9}, ISSN = {1868-8969}, year = {2024}, volume = {286}, editor = {Bessani, Alysson and D\'{e}fago, Xavier and Nakamura, Junya and Wada, Koichi and Yamauchi, Yukiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2023.11}, URN = {urn:nbn:de:0030-drops-195015}, doi = {10.4230/LIPIcs.OPODIS.2023.11}, annote = {Keywords: distributed computing, Nash equilibria, mechanism design, best-response dynamics} }

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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

We present a complete classification of the deterministic distributed time complexity for a family of graph problems: binary labeling problems in trees. These are locally checkable problems that can be encoded with an alphabet of size two in the edge labeling formalism. Examples of binary labeling problems include sinkless orientation, sinkless and sourceless orientation, 2-vertex coloring, perfect matching, and the task of coloring edges red and blue such that all nodes are incident to at least one red and at least one blue edge. More generally, we can encode e.g. any cardinality constraints on indegrees and outdegrees.
We study the deterministic time complexity of solving a given binary labeling problem in trees, in the usual LOCAL model of distributed computing. We show that the complexity of any such problem is in one of the following classes: O(1), Θ(log n), Θ(n), or unsolvable. In particular, a problem that can be represented in the binary labeling formalism cannot have time complexity Θ(log^* n), and hence we know that e.g. any encoding of maximal matchings has to use at least three labels (which is tight).
Furthermore, given the description of any binary labeling problem, we can easily determine in which of the four classes it is and what is an asymptotically optimal algorithm for solving it. Hence the distributed time complexity of binary labeling problems is decidable, not only in principle, but also in practice: there is a simple and efficient algorithm that takes the description of a binary labeling problem and outputs its distributed time complexity.

Alkida Balliu, Sebastian Brandt, Yuval Efron, Juho Hirvonen, Yannic Maus, Dennis Olivetti, and Jukka Suomela. Classification of Distributed Binary Labeling Problems. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{balliu_et_al:LIPIcs.DISC.2020.17, author = {Balliu, Alkida and Brandt, Sebastian and Efron, Yuval and Hirvonen, Juho and Maus, Yannic and Olivetti, Dennis and Suomela, Jukka}, title = {{Classification of Distributed Binary Labeling Problems}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {17:1--17:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.17}, URN = {urn:nbn:de:0030-drops-130957}, doi = {10.4230/LIPIcs.DISC.2020.17}, annote = {Keywords: LOCAL model, graph problems, locally checkable labeling problems, distributed computational complexity} }

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Brief Announcement

**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

In order to provide a high resilience and to react quickly to link failures, modern computer networks support fully decentralized flow rerouting, also known as local fast failover. In a nutshell, the task of a local fast failover algorithm is to pre-define fast failover rules for each node using locally available information only. Ideally, such a local fast failover algorithm provides a perfect resilience deterministically: a packet emitted from any source can reach any target, as long as the underlying network remains connected. Feigenbaum et al. showed [Feigenbaum and others, 2012] that it is not always possible to provide perfect resilience; on the positive side, the authors also presented an efficient algorithm which achieves at least 1-resilience, tolerating a single failure in any network.
Interestingly, not much more is known currently about the feasibility of perfect resilience. This brief announcement revisits perfect resilience with local fast failover, both in a model where the source can and cannot be used for forwarding decisions. By establishing a connection between graph minors and resilience, we prove that it is impossible to achieve perfect resilience on any non-planar graph; On the positive side, we can derive perfect resilience for outerplanar and some planar graphs.

Klaus-Tycho Foerster, Juho Hirvonen, Yvonne-Anne Pignolet, Stefan Schmid, and Gilles Tredan. Brief Announcement: What Can(Not) Be Perfectly Rerouted Locally. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 46:1-46:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{foerster_et_al:LIPIcs.DISC.2020.46, author = {Foerster, Klaus-Tycho and Hirvonen, Juho and Pignolet, Yvonne-Anne and Schmid, Stefan and Tredan, Gilles}, title = {{Brief Announcement: What Can(Not) Be Perfectly Rerouted Locally}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {46:1--46:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.46}, URN = {urn:nbn:de:0030-drops-131244}, doi = {10.4230/LIPIcs.DISC.2020.46}, annote = {Keywords: Resilience, Local Failover} }

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**Published in:** LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

Distributed proofs are mechanisms enabling the nodes of a network to collectively and efficiently check the correctness of Boolean predicates on the structure of the network (e.g. having a specific diameter), or on data structures distributed over the nodes (e.g. a spanning tree). We consider well known mechanisms consisting of two components: a prover that assigns a certificate to each node, and a distributed algorithm called verifier that is in charge of verifying the distributed proof formed by the collection of all certificates. We show that many network predicates have distributed proofs offering a high level of redundancy, explicitly or implicitly. We use this remarkable property of distributed proofs to establish perfect tradeoffs between the size of the certificate stored at every node, and the number of rounds of the verification protocol.

Laurent Feuilloley, Pierre Fraigniaud, Juho Hirvonen, Ami Paz, and Mor Perry. Redundancy in Distributed Proofs. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{feuilloley_et_al:LIPIcs.DISC.2018.24, author = {Feuilloley, Laurent and Fraigniaud, Pierre and Hirvonen, Juho and Paz, Ami and Perry, Mor}, title = {{Redundancy in Distributed Proofs}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {24:1--24:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-092-7}, ISSN = {1868-8969}, year = {2018}, volume = {121}, editor = {Schmid, Ulrich and Widder, Josef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.24}, URN = {urn:nbn:de:0030-drops-98139}, doi = {10.4230/LIPIcs.DISC.2018.24}, annote = {Keywords: Distributed verification, Distributed graph algorithms, Proof-labeling schemes, Space-time tradeoffs, Non-determinism} }

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**Published in:** LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

In this work we study the cost of local and global proofs on distributed verification. In this setting the nodes of a distributed system are provided with a nondeterministic proof for the correctness of the state of the system, and the nodes need to verify this proof by looking at only their local neighborhood in the system.
Previous works have studied the model where each node is given its own, possibly unique, part of the proof as input. The cost of a proof is the maximum size of an individual label. We compare this model to a model where each node has access to the same global proof, and the cost is the size of this global proof.
It is easy to see that a global proof can always include all of the local proofs, and every local proof can be a copy of the global proof. We show that there exists properties that exhibit these relative proof sizes, and also properties that are somewhere in between. In addition, we introduce a new lower bound technique and use it to prove a tight lower bound on the complexity of reversing distributed decision and establish a link between communication complexity and distributed proof complexity.

Laurent Feuilloley and Juho Hirvonen. Local Verification of Global Proofs. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{feuilloley_et_al:LIPIcs.DISC.2018.25, author = {Feuilloley, Laurent and Hirvonen, Juho}, title = {{Local Verification of Global Proofs}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {25:1--25:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-092-7}, ISSN = {1868-8969}, year = {2018}, volume = {121}, editor = {Schmid, Ulrich and Widder, Josef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.25}, URN = {urn:nbn:de:0030-drops-98146}, doi = {10.4230/LIPIcs.DISC.2018.25}, annote = {Keywords: Proof-labeling schemes, distributed verification, non-determinism, local proofs} }

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**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

The degree splitting problem requires coloring the edges of a graph red or blue such that each node has almost the same number of edges in each color, up to a small additive discrepancy. The directed variant of the problem requires orienting the edges such that each node has almost the same number of incoming and outgoing edges, again up to a small additive discrepancy.
We present deterministic distributed algorithms for both variants, which improve on their counterparts presented by Ghaffari and Su [SODA'17]: our algorithms are significantly simpler and faster, and have a much smaller discrepancy. This also leads to a faster and simpler deterministic algorithm for (2+o(1))Delta-edge-coloring, improving on that of Ghaffari and Su.

Mohsen Ghaffari, Juho Hirvonen, Fabian Kuhn, Yannic Maus, Jukka Suomela, and Jara Uitto. Improved Distributed Degree Splitting and Edge Coloring. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ghaffari_et_al:LIPIcs.DISC.2017.19, author = {Ghaffari, Mohsen and Hirvonen, Juho and Kuhn, Fabian and Maus, Yannic and Suomela, Jukka and Uitto, Jara}, title = {{Improved Distributed Degree Splitting and Edge Coloring}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {19:1--19:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.19}, URN = {urn:nbn:de:0030-drops-79794}, doi = {10.4230/LIPIcs.DISC.2017.19}, annote = {Keywords: Distributed Graph Algorithms, Degree Splitting, Edge Coloring, Discrepancy} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We extend the notion of distributed decision in the framework of distributed network computing, inspired by recent results on so-called distributed graph automata. We show that, by using distributed decision mechanisms based on the interaction between a prover and a disprover, the size of the certificates distributed to the nodes for certifying a given network property can be drastically reduced. For instance, we prove that minimum spanning tree can be certified with O(log(n))-bit certificates in n-node graphs, with just one interaction between the prover and the disprover, while it is known that certifying MST requires Omega(log^2(n))-bit certificates if only the prover can act. The improvement can even be exponential for some simple graph properties.
For instance, it is known that certifying the existence of a nontrivial automorphism requires Omega(n^2) bits if only the prover can act. We show that there is a protocol with two interactions between the prover and the disprover enabling to certify nontrivial automorphism with O(log(n))- bit certificates. These results are achieved by defining and analysing a local hierarchy of decision which generalizes the classical notions of proof-labelling schemes and locally checkable proofs.

Laurent Feuilloley, Pierre Fraigniaud, and Juho Hirvonen. A Hierarchy of Local Decision. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 118:1-118:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{feuilloley_et_al:LIPIcs.ICALP.2016.118, author = {Feuilloley, Laurent and Fraigniaud, Pierre and Hirvonen, Juho}, title = {{A Hierarchy of Local Decision}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {118:1--118:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.118}, URN = {urn:nbn:de:0030-drops-62536}, doi = {10.4230/LIPIcs.ICALP.2016.118}, annote = {Keywords: Distributed Network Computing, Distributed Algorithm, Distributed Decision, Locality} }

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