Red-Blue-Partitioned MST, TSP, and Matching

TL;DR

This paper studies partitioned network optimization problems involving red-blue node colorings and the construction of spanning trees, TSP tours, and matchings, providing improved approximation guarantees and hardness results.

Contribution

It offers new approximation bounds and hardness results for nine NP-hard problems involving partitioned network structures and objectives.

Findings

Improved approximation guarantees for partitioned spanning trees, TSP, and matchings.

Strengthened hardness results for the nine problem settings.

Enhanced understanding of the computational complexity of partitioned network problems.

Abstract

Arkin et al.~\cite{ArkinBCCJKMM17} recently introduced \textit{partitioned pairs} network optimization problems: given a metric-weighted graph on $n$ pairs of nodes, the task is to color one node from each pair red and the other blue, and then to compute two separate \textit{network structures} or disjoint (node-covering) subgraphs of a specified sort, one on the graph induced by the red nodes and the other on the blue nodes. Three structures have been investigated by \cite{ArkinBCCJKMM17}---\textit{spanning trees}, \textit{traveling salesperson tours}, and \textit{perfect matchings}---and the three objectives to optimize for when computing such pairs of structures: \textit{min-sum}, \textit{min-max}, and \textit{bottleneck}. We provide improved approximation guarantees and/or strengthened hardness results for these nine NP-hard problem settings.