On Fixed-Parameter Tractability of Weighted 0-1 Timed Matching Problem on Temporal Graphs

TL;DR

This paper investigates the computational complexity of the weighted 0-1 timed matching problem on temporal graphs, establishing its NP-completeness and W[1]-hardness under various parameters, and proposing an FPT algorithm for specific cases.

Contribution

It proves the problem's NP-completeness and W[1]-hardness, and introduces a fixed-parameter tractable algorithm based on maximum degree and treewidth.

Findings

NP-complete on graphs with bounded treewidth

W[1]-hard when parameterized by solution size

FPT algorithm for bounded degree and treewidth

Abstract

Temporal graphs are introduced to model systems where the relationships among the entities of the system evolve over time. In this paper, we consider the temporal graphs where the edge set changes with time and all the changes are known a priori. The underlying graph of a temporal graph is a static graph consisting of all the vertices and edges that exist for at least one timestep in the temporal graph. The concept of 0-1 timed matching in temporal graphs was introduced by Mandal and Gupta [DAM2022] as an extension of the matching problem in static graphs. A 0-1 timed matching of a temporal graph is a non-overlapping subset of the edge set of that temporal graph. The problem of finding the maximum 0-1 timed matching is proved to be NP-complete on multiple classes of temporal graphs. We study the fixed-parameter tractability of the maximum 0-1 timed matching problem. We prove that the problem remains to be NP-complete even when the underlying static graph of the temporal graph has a bounded treewidth. Furthermore, we establish that the problem is W[1]-hard when parameterized by the solution size. Finally, we present a fixed-parameter tractable (FPT) algorithm to address the problem when the problem is parameterized by the maximum vertex degree and the treewidth of the underlying graph of the temporal graph.