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.
Highlights
We propose a 2(1 − 1/h)-approximation algorithm for cycle-star hub network design problems with h hubs and/or a cycle-metric labeling problem with h labels
Hub-and-spoke networks arise in airline transportation systems, delivery systems and telecommunication systems
A single allocation hub-and-spoke network design problem is essentially equivalent to a metric labeling problem introduced by Kleinberg and Tardos in [21], which has connections to Markov random field and classification problems that arise in computer vision and related areas
Summary
We propose a 2(1 − 1/h)-approximation algorithm for cycle-star hub network design problems with h hubs and/or a cycle-metric labeling problem with h labels. Ando and Matsui [5] deal with the case in which all the nodes are embedded in a 2-dimensional plane and the transportation cost of an edge per unit flow is proportional to the Euclidean distance between the end nodes of the edge They proposed a randomized (1 + 2/π)-approximation algorithm. A single allocation hub-and-spoke network design problem is essentially equivalent to a metric labeling problem introduced by Kleinberg and Tardos in [21], which has connections to Markov random field and classification problems that arise in computer vision and related areas They proposed an O(log h log log h)approximation algorithm where h is the number of labels (hubs). Our results give an important class of the metric labeling problem, which has a polynomial time approximation algorithm with a constant approximation ratio
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