Disentangling the Computational Complexity of Network Untangling

TL;DR

This paper analyzes the computational complexity of network untangling, a problem involving covering temporal graph edges with vertex intervals, revealing the parameterized complexity landscape for various problem parameters.

Contribution

It provides a comprehensive parameterized complexity analysis of the network untangling problem across multiple parameters, clarifying tractability boundaries.

Findings

Almost complete classification of fixed-parameter tractability for all parameter combinations.

Identification of NP-hardness for both problem variants.

Delimitation of the complexity landscape for temporal graph untangling.

Abstract

We study the network untangling problem introduced by Rozenshtein, Tatti, and Gionis [DMKD 2021], which is a variant of Vertex Cover on temporal graphs -- graphs whose edge set changes over discrete time steps. They introduce two problem variants. 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.