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Consider a distributed system with $n$ processors out of which $f$ can be Byzantine faulty. In the approximate agreement task, each processor $i$ receives an input value $x_i$ and has to decide on an output value $y_i$ such that - the output values are in the convex hull of the non-faulty processors input values, - the output values are within distance $d$ of each other. Classically, the values are assumed to be from an $m$-dimensional Euclidean space, where $m ge 1$. In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph $G$ and the goal is to output vertices that are within distance $d$ of each other in $G$, but still remain in the graph-induced convex hull of the input values. For $d=0$, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any $d ge 1$, we show that the task is solvable in asynchronous systems when $G$ is chordal and $n > (omega+1)f$, where $omega$ is the clique number of~$G$. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures.
In this paper we address Approximate Agreement problem in the Mobile Byzantine faults model. Our contribution is threefold. First, we propose the the first mapping from the existing variants of Mobile Byzantine models to the Mixed-Mode faults model.T
Approximate agreement is one of the few variants of consensus that can be solved in a wait-free manner in asynchronous systems where processes communicate by reading and writing to shared memory. In this work, we consider a natural generalisation of
As Byzantine Agreement (BA) protocols find application in large-scale decentralized cryptocurrencies, an increasingly important problem is to design BA protocols with improved communication complexity. A few existing works have shown how to achieve s
In this paper we will present the Multidimensional Byzantine Agreement (MBA) Protocol, a leaderless Byzantine agreement protocol defined for complete and synchronous networks that allows a network of nodes to reach consensus on a vector of relevant i
In the Lattice Agreement (LA) problem, originally proposed by Attiya et al. cite{Attiya:1995}, a set of processes has to decide on a chain of a lattice. More precisely, each correct process proposes an element $e$ of a certain join-semi lattice $L$ a