3 research outputs found
On Temporal Graph Exploration
A temporal graph is a graph in which the edge set can change from step to
step. The temporal graph exploration problem TEXP is the problem of computing a
foremost exploration schedule for a temporal graph, i.e., a temporal walk that
starts at a given start node, visits all nodes of the graph, and has the
smallest arrival time. In the first part of the paper, we consider only
temporal graphs that are connected at each step. For such temporal graphs with
nodes, we show that it is NP-hard to approximate TEXP with ratio
for any . We also provide an explicit
construction of temporal graphs that require steps to be
explored. We then consider TEXP under the assumption that the underlying graph
(i.e. the graph that contains all edges that are present in the temporal graph
in at least one step) belongs to a specific class of graphs. Among other
results, we show that temporal graphs can be explored in steps if the underlying graph has treewidth and in
steps if the underlying graph is a grid. In the second part of the
paper, we replace the connectedness assumption by a weaker assumption and show
that -edge temporal graphs with regularly present edges and with random
edges can always be explored in steps and steps with high
probability, respectively. We finally show that the latter result can be used
to obtain a distributed algorithm for the gossiping problem.Comment: This is an extended version of an ICALP 2015 pape
On the Size and the Approximability of Minimum Temporally Connected Subgraphs
We consider temporal graphs with discrete time labels and investigate the
size and the approximability of minimum temporally connected spanning
subgraphs. We present a family of minimally connected temporal graphs with
vertices and edges, thus resolving an open question of (Kempe,
Kleinberg, Kumar, JCSS 64, 2002) about the existence of sparse temporal
connectivity certificates. Next, we consider the problem of computing a minimum
weight subset of temporal edges that preserve connectivity of a given temporal
graph either from a given vertex r (r-MTC problem) or among all vertex pairs
(MTC problem). We show that the approximability of r-MTC is closely related to
the approximability of Directed Steiner Tree and that r-MTC can be solved in
polynomial time if the underlying graph has bounded treewidth. We also show
that the best approximation ratio for MTC is at least and at most , for
any constant , where is the number of temporal edges and
is the maximum degree of the underlying graph. Furthermore, we prove
that the unweighted version of MTC is APX-hard and that MTC is efficiently
solvable in trees and -approximable in cycles
Some hesitant fuzzy geometric operators and their application to multiple attribute group decision making
Hesitant fuzzy set (HFS), a generalization of fuzzy set (FS), permits the membership degree of an element of a set to be represented as several possible values between 0 and 1. In this paper, motivated by the extension principle of HFs, we export Einstein operations on FSs to HFs, and develop some new aggregation operators, such as the hesitant fuzzy Einstein weighted geometric operator, hesitant fuzzy Einstein ordered weighted geometric operator, and hesitant fuzzy Einstein hybrid weighted geometric operator, for aggregating hesitant fuzzy elements. In addition, we discuss the correlations between the proposed aggregation operators and the existing ones respectively. Finally, we apply the hesitant fuzzy Einstein weighted geometric operator to multiple attribute group decision making with hesitant fuzzy information. Some numerical examples are given to illustrate the proposed aggregation operators.
First published online: 09 Jun 201