An Oblivious Spanning Tree for Buy-at-Bulk Network Design Problems

TL;DR

This paper introduces a novel oblivious spanning tree construction for buy-at-bulk network design in doubling-dimension graphs, independent of demand sets and fusion costs, with provable approximation guarantees.

Contribution

It is the first to propose a single, oblivious spanning tree solution for the buy-at-bulk network design problem, independent of demand and cost functions.

Findings

Provides a polynomial-time algorithm with $ ext{log}^3 D ext{log} n$ approximation.

Achieves $O( ext{log}^3 D)$-approximation for Steiner trees with constant fusion costs.

First to address oblivious spanning trees in this context.

Abstract

We consider the problem of constructing a single spanning tree for the single-source buy-at-bulk network design problem for doubling-dimension graphs. We compute a spanning tree to route a set of demands (or data) along a graph to or from a designated root node. The demands could be aggregated at (or symmetrically distributed to) intermediate nodes where the fusion-cost is specified by a non-negative concave function $f$. We describe a novel approach for developing an oblivious spanning tree in the sense that it is independent of the number of data sources (or demands) and cost function at intermediate nodes. To our knowledge, this is the first paper to propose a single spanning tree solution to this problem (as opposed to multiple overlay trees). There has been no prior work where the tree is oblivious to both the fusion cost function and the set of sources (demands). We present a deterministic, polynomial-time algorithm for constructing a spanning tree in low doubling graphs that guarantees $\log^{3}D\cdot\log n$-approximation over the optimal cost, where $D$ is the diameter of the graph and $n$ the total number of nodes. With constant fusion-cost function our spanning tree gives a $O(\log^3 D)$-approximation for every Steiner tree to the root.