Multistage Graph Problems on a Global Budget

TL;DR

This paper explores multistage graph problems on temporal graphs with a global budget constraint, revealing new complexity results and tractability insights for classical problems.

Contribution

It introduces a global budget framework for multistage problems on temporal graphs and analyzes their parameterized complexity, uncovering surprising tractability results.

Findings

Global multistage Vertex Cover is more tractable than Matching.

Some NP-hard problems are fixed-parameter tractable under the new model.

Polynomial-time solvable problems can be harder than NP-hard ones in this setting.

Abstract

Time-evolving or temporal graphs gain more and more popularity when studying the behavior of complex networks. In this context, the multistage view on computational problems is among the most natural frameworks. Roughly speaking, herein one studies the different (time) layers of a temporal graph (effectively meaning that the edge set may change over time, but the vertex set remains unchanged), and one searches for a solution of a given graph problem for each layer. The twist in the multistage setting is that the solutions found must not differ too much between subsequent layers. We relax on this already established notion by introducing a global instead of the local budget view studied so far. More specifically, we allow for few disruptive changes between subsequent layers but request that overall, that is, summing over all layers, the degree of change is moderate. Studying several classical graph problems (both NP-hard and polynomial-time solvable ones) from a parameterized complexity angle, we encounter both fixed-parameter tractability and parameterized hardness results. Somewhat surprisingly, we find that sometimes the global multistage versions of NP-hard problems such as Vertex Cover turn out to be computationally more tractable than the ones of polynomial-time solvable problems such as Matching.