Temporal Network Optimization Subject to Connectivity Constraints

In this work we consider

temporal networks

, i.e. networks defined by a

labeling

λ

assigning to each edge of an

underlying graph

G

a set of

discrete

time-labels. The labels of an edge, which are natural numbers, indicate the discrete time moments at which the edge is available. We focus on

path problems

of temporal networks. In particular, we consider

time-respecting

paths, i.e. paths whose edges are assigned by

λ

a strictly increasing sequence of labels. We begin by giving two efficient algorithms for computing shortest time-respecting paths on a temporal network. We then prove that there is a

natural analogue of Menger’s theorem

holding for arbitrary temporal networks. Finally, we propose two

cost minimization parameters

for temporal network design. One is the

temporality

of

G

, in which the goal is to minimize the maximum number of labels of an edge, and the other is the

temporal cost

of

G

, in which the goal is to minimize the total number of labels used. Optimization of these parameters is performed subject to some

connectivity constraint

. We prove several lower and upper bounds for the temporality and the temporal cost of some very basic graph families such as rings, directed acyclic graphs, and trees.