On the Size and the Approximability of Minimum Temporally Connected Subgraphs

TL;DR

This paper investigates the size and approximability of minimum temporally connected subgraphs in temporal graphs, providing new bounds, complexity results, and algorithms for related connectivity problems.

Contribution

It resolves an open question on sparse temporal connectivity certificates and establishes complexity and approximation bounds for minimum temporal connectivity problems.

Findings

Existence of sparse temporal connectivity certificates with (n^2) edges.

r-MTC approximability linked to Directed Steiner Tree.

MTC is APX-hard and solvable in trees, 2-approximable in cycles.

Abstract

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 $n$ vertices and $\Omega(n^2)$ 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 $O(2^{\log^{1-\epsilon} n})$ and at most $O(\min\{n^{1+\epsilon}, (\Delta M)^{2/3+\epsilon}\})$, for any constant $\epsilon > 0$, where $M$ is the number of temporal edges and $\Delta$ 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 $2$-approximable in cycles.