# Disentangling the Computational Complexity of Network Untangling

# Disentangling the Computational Complexity of Network Untangling

## Vincent Froese, Pascal Kunz, Philipp Zschoche

Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence

Main Track. Pages 2037-2043.
https://doi.org/10.24963/ijcai.2022/283

We study the recently introduced network untangling problem, a variant of Vertex Cover on temporal graphs---graphs whose edge set changes over discrete time steps. There are two versions of this problem. The goal is to select at most k time intervals for each vertex such that all time-edges are covered and (depending on the problem variant) either the maximum interval length or the total sum of interval lengths is minimized. This problem has data mining applications in finding activity timelines that explain the interactions of entities in complex networks.
Both variants of the problem are NP-hard. In this paper, we initiate a multivariate complexity analysis involving the following parameters: number of vertices, lifetime of the temporal graph, number of intervals per vertex, and the interval length bound. For both problem versions, we (almost) completely settle the parameterized complexity for all combinations of those four parameters, thereby delineating the border of fixed-parameter tractability.

Keywords:

Data Mining: Mining Spatial and/or Temporal Data

Data Mining: Mining Graphs

Data Mining: Networks