Network Optimization on Partitioned Pairs of Points

TL;DR

This paper addresses the problem of optimally coloring pairs of points in a metric space to minimize network costs, providing complexity results and approximation algorithms for various network structures and objectives.

Contribution

It introduces a framework for partitioning pairs of points into two networks with optimized costs, and offers the first constant factor approximation algorithms for these problems.

Findings

Some problems are NP-hard.

Constant factor approximation algorithms are developed.

Applicable to spanning trees, tours, and matchings.

Abstract

Given $n$ pairs of points, $\mathcal{S} = \{\{p_1, q_1\}, \{p_2, q_2\}, \dots, \{p_n, q_n\}\}$, in some metric space, we study the problem of two-coloring the points within each pair, red and blue, to optimize the cost of a pair of node-disjoint networks, one over the red points and one over the blue points. In this paper we consider our network structures to be spanning trees, traveling salesman tours or matchings. We consider several different weight functions computed over the network structures induced, as well as several different objective functions. We show that some of these problems are NP-hard, and provide constant factor approximation algorithms in all cases.