93 research outputs found
Byzantine Approximate Agreement on Graphs
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
1) the output values are in the convex hull of the non-faulty processors\u27 input values,
2) 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 >= 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 >= 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
The general position number and the iteration time in the P3 convexity
In this paper, we investigate two graph convexity parameters: the iteration
time and the general position number. Harary and Nieminem introduced in 1981
the iteration time in the geodesic convexity, but its computational complexity
was still open. Manuel and Klav\v{z}ar introduced in 2018 the general position
number of the geodesic convexity and proved that it is NP-hard to compute. In
this paper, we extend these parameters to the P3 convexity and prove that it is
NP-hard to compute them. With this, we also prove that the iteration number is
NP-hard on the geodesic convexity even in graphs with diameter two. These
results are the last three missing NP-hardness results regarding the ten most
studied graph convexity parameters in the geodesic and P3 convexities
The Detour Monophonic Convexity Number of a Graph
A set is detour monophonic convexif The detour monophonic convexity number is denoted by is the cardinality of a maximum proper detour monophonic convex subset of Some general properties satisfied by this concept are studied. The detour monophonic convexity number of certain classes of graphs are determined. It is shown that for every pair of integers and with there exists a connected graph such that and , where is the monophonic convexity number of
Equivalence between Hypergraph Convexities
Let G be a connected graph on V. A subset X of V is all-paths convex (or
ap
-convex) if X contains each vertex on every path joining two vertices in X and is monophonically convex (or m-convex) if X contains each vertex on every chordless path joining two vertices in X. First of all, we prove that
ap
-convexity and m-convexity coincide in G if and only if G is a tree. Next, in order to generalize this result to a connected hypergraph H, in addition to the hypergraph versions of
ap
-convexity and m-convexity, we consider canonical convexity (or c-convexity) and simple-path convexity (or
sp
-convexity) for which it is well known that m-convexity is finer than both c-convexity and
sp
-convexity and
sp
-convexity is finer than
ap
-convexity. After proving
sp
-convexity is coarser than c-convexity, we characterize the hypergraphs in which each pair of the four convexities above is equivalent. As a result, we obtain a convexity-theoretic characterization of Berge-acyclic hypergraphs and of γ-acyclic hypergraphs
Computing the hull number in toll convexity
A walk W between vertices u and v of a graph G is called a tolled walk between u and v if u, as well as v, has exactly one neighbour in W. A set S ⊆ V (G) is toll convex if the vertices contained in any tolled walk between two vertices of S are contained in S. The toll convex hull of S is the minimum toll convex set containing S. The toll hull number of G is the minimum cardinality of a set S such that the toll convex hull of S is V (G). The main contribution of this work is an algorithm for computing the toll hull number of a general graph in polynomial time
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