Approximation Algorithm for Cycle-Star Hub Network Design Problems and Cycle-Metric Labeling Problems

TL;DR

This paper introduces a new approximation algorithm for the cycle-star hub network design problem, which minimizes transportation costs in cycle-based hub networks, and relates it to cycle-metric labeling problems.

Contribution

It presents a $2(1-1/h)$-approximation algorithm for the cycle-star hub network problem using linear relaxation and dependent rounding, connecting it to cycle-metric labeling.

Findings

Achieves a $2(1-1/h)$ approximation ratio.

Uses convex combination of Monge matrices for analysis.

Applicable to telecommunications and airline network design.

Abstract

We consider a single allocation hub-and-spoke network design problem which allocates each non-hub node to exactly one of given hub nodes so as to minimize the total transportation cost. This paper deals with a case in which the hubs are located in a cycle, which is called a cycle-star hub network design problem. The problem is essentially equivalent to a cycle-metric labeling problem. The problem is useful in the design of networks in telecommunications and airline transportation systems.We propose a $2(1-1/h)$-approximation algorithm where $h$ denotes the number of hub nodes. Our algorithm solves a linear relaxation problem and employs a dependent rounding procedure. We analyze our algorithm by approximating a given cycle-metric matrix by a convex combination of Monge matrices.