Steiner Network Problems on Temporal Graphs

TL;DR

This paper introduces the NP-hard temporal Steiner network problem, demonstrating its complexity, exploring special cases with monotonic changes, and providing approximation algorithms and an ILP model for applications like computational biology.

Contribution

The paper formalizes the $k$-Temporal Steiner Network problem, proves its tight approximation bounds, and connects it to existing problems, offering new algorithms and models for practical applications.

Findings

$k$-TSN is NP-hard to approximate within a factor of $k - \\epsilon$.

Monotonic $k$-TSN reduces to Priority and Directed Steiner Tree problems.

An ILP model based on network flows is developed for $k$-TSN.

Abstract

We introduce a temporal Steiner network problem in which a graph, as well as changes to its edges and/or vertices over a set of discrete times, are given as input; the goal is to find a minimal subgraph satisfying a set of $k$ time-sensitive connectivity demands. We show that this problem, $k$-Temporal Steiner Network ($k$-TSN), is NP-hard to approximate to a factor of $k - \epsilon$, for every fixed $k \geq 2$ and $\epsilon > 0$. This bound is tight, as certified by a trivial approximation algorithm. Conceptually this demonstrates, in contrast to known results for traditional Steiner problems, that a time dimension adds considerable complexity even when the problem is offline. We also discuss special cases of $k$-TSN in which the graph changes satisfy a monotonicity property. We show approximation-preserving reductions from monotonic $k$-TSN to well-studied problems such as Priority Steiner Tree and Directed Steiner Tree, implying improved approximation algorithms. Lastly, $k$-TSN and its variants arise naturally in computational biology; to facilitate such applications, we devise an integer linear program for $k$-TSN based on network flows.