An Oblivious O(1)-Approximation for Single Source Buy-at-Bulk

TL;DR

This paper introduces a polynomial-time algorithm providing an O(1)-approximate solution for the single-source buy-at-bulk problem with unknown concave costs, improving upon the previous logarithmic approximation bounds.

Contribution

It presents a novel algorithmic framework that achieves a constant-factor approximation for all concave cost functions simultaneously, using the ellipsoid method and a new separation oracle.

Findings

Achieves an O(1)-approximation for all cost functions

Supports a distribution over at most 1+log|D| trees

Improves the approximation ratio from O(log |D|) to a constant

Abstract

We consider the single-source (or single-sink) buy-at-bulk problem with an unknown concave cost function. We want to route a set of demands along a graph to or from a designated root node, and the cost of routing x units of flow along an edge is proportional to some concave, non-decreasing function f such that f(0) = 0. We present a polynomial time algorithm that finds a distribution over trees such that the expected cost of a tree for any f is within an O(1)-factor of the optimum cost for that f. The previous best simultaneous approximation for this problem, even ignoring computation time, was O(log |D|), where D is the multi-set of demand nodes. We design a simple algorithmic framework using the ellipsoid method that finds an O(1)-approximation if one exists, and then construct a separation oracle using a novel adaptation of the Guha, Meyerson, and Munagala algorithm for the single-sink buy-at-bulk problem that proves an O(1) approximation is possible for all f. The number of trees in the support of the distribution constructed by our algorithm is at most 1+log |D|.