DeepAI
Log In Sign Up

The complexity of computing optimum labelings for temporal connectivity

02/11/2022
by   Nina Klobas, et al.
0

A graph is temporally connected if there exists a strict temporal path, i.e. a path whose edges have strictly increasing labels, from every vertex u to every other vertex v. In this paper we study temporal design problems for undirected temporally connected graphs. The basic setting of these optimization problems is as follows: given a connected undirected graph G, what is the smallest number |λ| of time-labels that we need to add to the edges of G such that the resulting temporal graph (G,λ) is temporally connected? As it turns out, this basic problem, called MINIMUM LABELING (ML), can be optimally solved in polynomial time. However, exploiting the temporal dimension, the problem becomes more interesting and meaningful in its following variations, which we investigate in this paper. First we consider the problem MIN. AGED LABELING (MAL) of temporally connecting the graph when we are given an upper-bound on the allowed age (i.e. maximum label) of the obtained temporal graph (G,λ). Second we consider the problem MIN. STEINER LABELING (MSL), where the aim is now to have a temporal path between any pair of "terminals" vertices which lie in a subset R⊆ V. This relaxed problem resembles STEINER TREE in static graphs. However, due to the requirement of strictly increasing labels in a temporal path, STEINER TREE is not a special case of MSL. Finally we consider the age-restricted version of MSL, namely MIN. AGED STEINER LABELING (MASL). Our main results are threefold: we prove that (i) MAL becomes NP-complete on undirected graphs, while (ii) MASL becomes W[1]-hard with respect to the number |R| of terminals. On the other hand we prove that (iii) although the age-unrestricted problem MSL is NP-hard, it is in FPT with respect to the number |R| of terminals. That is, adding the age restriction, makes the above problems strictly harder.

READ FULL TEXT

page 1

page 2

page 3

page 4

02/12/2021

The Complexity of Transitively Orienting Temporal Graphs

In a temporal network with discrete time-labels on its edges, entities a...
01/10/2021

Shortest non-separating st-path on chordal graphs

Many NP-Hard problems on general graphs, such as maximum independence se...
03/04/2021

Computing Subset Feedback Vertex Set via Leafage

Chordal graphs are characterized as the intersection graphs of subtrees ...
05/29/2020

Parameterized Complexity of Min-Power Asymmetric Connectivity

We investigate parameterized algorithms for the NP-hard problem Min-Powe...
11/07/2020

Sharp Thresholds in Random Simple Temporal Graphs

A graph whose edges only appear at certain points in time is called a te...
02/25/2022

Making Life More Confusing for Firefighters

It is well known that fighting a fire is a hard task. The Firefighter pr...
09/28/2018

Temporal Cliques admit Sparse Spanners

Let G=(G,λ) be a labeled graph on n vertices with λ:E_G→N a locally inj...