One Tree Suffices: A Simultaneous O(1)-Approximation for Single-Sink Buy-at-Bulk

TL;DR

This paper introduces a simple, fast algorithm that constructs a single tree achieving a constant-factor approximation for the buy-at-bulk problem across all concave cost functions, a feat previously thought impossible.

Contribution

The paper presents the first known single tree that provides a simultaneous constant-factor approximation for all concave cost functions in the buy-at-bulk problem.

Findings

Achieves a 47.45-approximation for all concave functions.

Improves the approximation ratio compared to function-specific optimizations.

Demonstrates the existence of trees with universal approximation guarantees.

Abstract

We study the single-sink buy-at-bulk problem with an unknown cost function. We wish to route flow from a set of demand nodes to a root node, where the cost of routing x total flow along an edge is proportional to f(x) for some concave, non-decreasing function f satisfying f(0)=0. We present a simple, fast, combinatorial algorithm that takes a set of demands and constructs a single tree T such that for all f the cost f(T) is a 47.45-approximation of the optimal cost for that f. This is within a factor of 2.33 of the best approximation ratio currently achievable when the tree can be optimized for a specific function. Trees achieving simultaneous O(1)-approximations for all concave functions were previously not known to exist regardless of computation time.