An FTP Algorithm for Temporal Graph Untangling

TL;DR

This paper introduces an FPT algorithm for the MinTimelineCover problem on temporal graphs, resolving an open question about its parameterized complexity by leveraging iterative compression and reduction to Digraph Pair Cut.

Contribution

The paper presents the first fixed-parameter tractable algorithm for MinTimelineCover on general temporal graphs, expanding understanding of its computational complexity.

Findings

The problem is NP-hard even in restricted cases.

The new algorithm is based on iterative compression and reduction techniques.

It fully characterizes the parameterized complexity of MinTimelineCover.

Abstract

Several classical combinatorial problems have been considered and analysed on temporal graphs. Recently, a variant of Vertex Cover on temporal graphs, called MinTimelineCover, has been introduced to summarize timeline activities in social networks. The problem asks to cover every temporal edge while minimizing the total span of the vertices (where the span of a vertex is the length of the timestamp interval it must remain active in, minus one). While the problem has been shown to be NP-hard even in very restricted cases, its parameterized complexity has not been fully understood. The problem is known to be in FPT under the span parameter only for graphs with two timestamps, but the parameterized complexity for the general case is open. We settle this open problem by giving an FPT algorithm that is based on a combination of iterative compression and a reduction to the Digraph Pair Cut problem, a powerful problem that has received significant attention recently.