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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

In recent work, Gourv{è}s, Lesca, and Wilczynski (IJCAI 17) propose a variant of the classic housing markets model in which the matching between agents and objects evolves through Pareto-improving swaps between pairs of agents who are adjacent in a social network. To explore the swap dynamics of their model, they pose several basic questions concerning the set of reachable matchings, and investigate the computational complexity of these questions when the graph structure of the social network is a star, path, or tree, or is unrestricted.
We are interested in how to direct the agents to swap objects with each other in order to arrive at a reachable matching that is both efficient and most agreeable. In particular, we study the computational complexity of reaching a Pareto-efficient matching that maximizes the number of agents who prefer their match to their initial endowments. We consider various graph structures of the social network: path, star, tree, or being unrestricted. Additionally, we consider two assumptions regarding preference relations of agents: strict (ties among objects not allowed) or weak (ties among objects allowed). By designing two polynomial-time algorithms and two NP-hardness reductions, we resolve the complexity of all cases not yet known. Our main contributions include a polynomial-time algorithm for path networks with strict preferences and an NP-hardness result in a star network with weak preferences.

Fu Li and Xiong Zheng. Maximum Votes Pareto-Efficient Allocations via Swaps on a Social Network. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 71:1-71:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{li_et_al:LIPIcs.MFCS.2021.71, author = {Li, Fu and Zheng, Xiong}, title = {{Maximum Votes Pareto-Efficient Allocations via Swaps on a Social Network}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {71:1--71:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.71}, URN = {urn:nbn:de:0030-drops-145112}, doi = {10.4230/LIPIcs.MFCS.2021.71}, annote = {Keywords: Housing markets, Distributed process, Algorithms, Complexity} }

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**Published in:** LIPIcs, Volume 184, 24th International Conference on Principles of Distributed Systems (OPODIS 2020)

We study the Byzantine lattice agreement (BLA) problem in asynchronous distributed message passing systems. In the BLA problem, each process proposes a value from a join semi-lattice and needs to output a value also in the lattice such that all output values of correct processes lie on a chain despite the presence of Byzantine processes. We present an algorithm for this problem with round complexity of O(log f) which tolerates f < n/5 Byzantine failures in the asynchronous setting without digital signatures, where n is the number of processes. This is the first algorithm which has logarithmic round complexity for this problem in asynchronous setting. Before our work, Di Luna et al give an algorithm for this problem which takes O(f) rounds and tolerates f < n/3 Byzantine failures. We also show how this algorithm can be modified to work in the authenticated setting (i.e., with digital signatures) to tolerate f < n/3 Byzantine failures.

Xiong Zheng and Vijay Garg. Byzantine Lattice Agreement in Asynchronous Systems. In 24th International Conference on Principles of Distributed Systems (OPODIS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 184, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{zheng_et_al:LIPIcs.OPODIS.2020.4, author = {Zheng, Xiong and Garg, Vijay}, title = {{Byzantine Lattice Agreement in Asynchronous Systems}}, booktitle = {24th International Conference on Principles of Distributed Systems (OPODIS 2020)}, pages = {4:1--4:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-176-4}, ISSN = {1868-8969}, year = {2021}, volume = {184}, editor = {Bramas, Quentin and Oshman, Rotem and Romano, Paolo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2020.4}, URN = {urn:nbn:de:0030-drops-134894}, doi = {10.4230/LIPIcs.OPODIS.2020.4}, annote = {Keywords: Byzantine Lattice Agreement, Asynchronous} }

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

We propose three algorithms for the Byzantine lattice agreement problem in synchronous systems. The first algorithm runs in min {3h(X) + 6,6√{f_a} + 6}) rounds and takes O(n² min{h(X), √{f_a}}) messages, where h(X) is the height of the input lattice X, n is the total number of processes in the system, f is the maximum number of Byzantine processes such that n ≥ 3f + 1 and f_a ≤ f is the actual number of Byzantine processes in an execution. The second algorithm takes 3log n + 3 rounds and O(n² log n) messages. The third algorithm takes 4 log f + 3 rounds and O(n² log f) messages. All algorithms can tolerate f < n/3 Byzantine failures. This is the first work for the Byzantine lattice agreement problem in synchronous systems which achieves logarithmic rounds. In our algorithms, we apply a slightly modified version of the Gradecast algorithm given by Feldman et al [Feldman and Micali, 1988] as a building block. If we use the Gradecast algorithm for authenticated setting given by Katz et al [Katz and Koo, 2006], we obtain algorithms for the Byzantine lattice agreement problem in authenticated settings and tolerate f < n/2 failures.

Xiong Zheng and Vijay Garg. Byzantine Lattice Agreement in Synchronous Message Passing Systems. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 32:1-32:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{zheng_et_al:LIPIcs.DISC.2020.32, author = {Zheng, Xiong and Garg, Vijay}, title = {{Byzantine Lattice Agreement in Synchronous Message Passing Systems}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {32:1--32:16}, 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.2020.32}, URN = {urn:nbn:de:0030-drops-131106}, doi = {10.4230/LIPIcs.DISC.2020.32}, annote = {Keywords: Lattice agreement, Byzantine Failure, Gradecast} }

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**Published in:** LIPIcs, Volume 153, 23rd International Conference on Principles of Distributed Systems (OPODIS 2019)

We propose parallel and distributed algorithms for the housing allocation problem. In this problem, there is a set of agents and a set of houses. Each agent has a strict preference list for a subset of houses. We need to find a matching for agents to houses such that some criterion is optimized. One such criterion which has attracted much attention is Pareto optimality. A matching is Pareto optimal if no coalition of agents can be strictly better off by exchanging houses among themselves. We also study the housing market problem, a variant of the housing allocation problem, where each agent initially owns a house. In addition to Pareto optimality, we are also interested in finding the a matching in the core of a housing market. A matching is in the core if there is no coalition of agents that can be better off by breaking away from other agents and switching houses only among themselves in the initial allocation.
In the first part of this work, we show that computing a Pareto optimal matching of a house allocation is in CC and computing a matching in the core of a housing market is CC-hard, where CC is the class of problems logspace reducible to the comparator circuit value problem. Given a matching of agents to houses, we show that verifying whether it is Pareto optimal is in NC. We also show that verifying whether it is in the core can be done in NC. We then give an algorithm to show that computing a maximum cardinality Pareto optimal matching for the housing allocation problem is in RNC² and quasi-NC².
In the second part of this work, we present a distributed version of the top trading cycle algorithm for finding a matching in the core of a housing market. To that end, we first present two algorithms for finding all the disjoint cycles in a functional graph. The first algorithm is a Las Vegas algorithm which terminates in O(log l) rounds with high probability, where l is the length of the longest cycle. The second algorithm is a deterministic algorithm which terminates in O(log* n log l) rounds, where n is the number of nodes in the graph. Both algorithms work in the synchronous distributed model and use messages of size O(log n). By applying these two algorithms for finding cycles in a functional graph, we give the distributed top trading cycle algorithm which terminates in O(n) rounds and requires O(n²) messages.

Xiong Zheng and Vijay K. Garg. Parallel and Distributed Algorithms for the Housing Allocation Problem. In 23rd International Conference on Principles of Distributed Systems (OPODIS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 153, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{zheng_et_al:LIPIcs.OPODIS.2019.23, author = {Zheng, Xiong and Garg, Vijay K.}, title = {{Parallel and Distributed Algorithms for the Housing Allocation Problem}}, booktitle = {23rd International Conference on Principles of Distributed Systems (OPODIS 2019)}, pages = {23:1--23:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-133-7}, ISSN = {1868-8969}, year = {2020}, volume = {153}, editor = {Felber, Pascal and Friedman, Roy and Gilbert, Seth and Miller, Avery}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2019.23}, URN = {urn:nbn:de:0030-drops-118090}, doi = {10.4230/LIPIcs.OPODIS.2019.23}, annote = {Keywords: Parallel Algorithm, Distributed Algorithm, Housing Allocation, Housing Markets, Pareto optimality} }

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**Published in:** LIPIcs, Volume 153, 23rd International Conference on Principles of Distributed Systems (OPODIS 2019)

This paper studies the lattice agreement problem in asynchronous systems and explores its application to building a linearizable replicated state machine (RSM). First, we propose an algorithm to solve the lattice agreement problem in O(log f) asynchronous rounds, where f is the number of crash failures that the system can tolerate. This is an exponential improvement over the previous best upper bound of O(f). Second, Faleiro et al have shown in [Faleiro et al. PODC, 2012] that combination of conflict-free data types and lattice agreement protocols can be applied to implement a linearizable RSM. They give a Paxos style lattice agreement protocol, which can be adapted to implement a linearizable RSM and guarantee that a command by a client can be learned in at most O(n) message delays, where n is the number of proposers. Later, Xiong et al in [Xiong et al. DISC, 2018] gave a lattice agreement protocol which improves the O(n) message delay guarantee to O(f). However, neither of the protocols is practical for building a linearizable RSM. Thus, in the second part of the paper, we first give an improved protocol based on the one proposed by Xiong et al. Then, we implement a simple linearizable RSM using our improved protocol and compare our implementation with an open source Java implementation of Paxos. Results show that better performance can be obtained by using lattice agreement based protocols to implement a linearizable RSM compared to traditional consensus based protocols.

Xiong Zheng, Vijay K. Garg, and John Kaippallimalil. Linearizable Replicated State Machines With Lattice Agreement. In 23rd International Conference on Principles of Distributed Systems (OPODIS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 153, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{zheng_et_al:LIPIcs.OPODIS.2019.29, author = {Zheng, Xiong and Garg, Vijay K. and Kaippallimalil, John}, title = {{Linearizable Replicated State Machines With Lattice Agreement}}, booktitle = {23rd International Conference on Principles of Distributed Systems (OPODIS 2019)}, pages = {29:1--29:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-133-7}, ISSN = {1868-8969}, year = {2020}, volume = {153}, editor = {Felber, Pascal and Friedman, Roy and Gilbert, Seth and Miller, Avery}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2019.29}, URN = {urn:nbn:de:0030-drops-118158}, doi = {10.4230/LIPIcs.OPODIS.2019.29}, annote = {Keywords: Lattice Agreement, Generalized Lattice Agreement, Replicated State Machine, Consensus} }

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

This paper studies the lattice agreement problem and the generalized lattice agreement problem in distributed message passing systems. In the lattice agreement problem, given input values from a lattice, processes have to non-trivially decide output values that lie on a chain. We consider the lattice agreement problem in both synchronous and asynchronous systems. For synchronous lattice agreement, we present two algorithms which run in log(f) and min{O(log^2 h(L)), O(log^2 f)} rounds, respectively, where h(L) denotes the height of the input sublattice L, f < n is the number of crash failures the system can tolerate, and n is the number of processes in the system. These algorithms have significant better round complexity than previously known algorithms. The algorithm by Attiya et al. [Attiya et al. DISC, 1995] takes log(n) synchronous rounds, and the algorithm by Mavronicolasa [Mavronicolasa, 2018] takes min{O(h(L)), O(sqrt(f))} rounds. For asynchronous lattice agreement, we propose an algorithm which has time complexity of 2*min{h(L), f + 1} message delays which improves on the previously known time complexity of O(n) message delays.
The generalized lattice agreement problem defined by Faleiro et al in [Faleiro et al. PODC, 2012] is a generalization of the lattice agreement problem where it is applied for the replicated state machine. We propose an algorithm which guarantees liveness when a majority of the processes are correct in asynchronous systems. Our algorithm requires min{O(h(L)), O(f)} units of time in the worst case which is better than O(n) units of time required by the algorithm in [Faleiro et al. PODC, 2012].

Xiong Zheng, Changyong Hu, and Vijay K. Garg. Lattice Agreement in Message Passing Systems. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 41:1-41:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{zheng_et_al:LIPIcs.DISC.2018.41, author = {Zheng, Xiong and Hu, Changyong and Garg, Vijay K.}, title = {{Lattice Agreement in Message Passing Systems}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {41:1--41:17}, 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.2018.41}, URN = {urn:nbn:de:0030-drops-98301}, doi = {10.4230/LIPIcs.DISC.2018.41}, annote = {Keywords: Lattice Agreement, Replicated State Machine, Consensus} }

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