Polylogarithmic Approximation Algorithm for Non-Uniform Multicommodity Buy-at-Bulk
We consider the non-uniform multicommodity buy-at-bulknetworkdesign problem. In this problem we are given a graph $G(V,E)$withtwo cost functions on the edges, a buy cost $b:E\longrightarrow \RR^+$and a rent cost$r:E\longrightarrow\RR^+$, and a set of source-sink pairs$s_i,t_i\in V$ ($1\leq i\leq \alpha$)with each pair $i$ having a positivedemand $\delta_i$. Our goal is to designa minimum cost network $G(V,E')$such that for every $1\leq i\leq\alpha$, $s_i$ and $t_i$ are in thesameconnected component in $G(V,E')$. Thetotal cost of $G(V,E')$ is the sum ofbuy costs of the edges in $E'$plus sum of total demand going through everyedge in $E'$ times therent cost of that edge. Since the costs of differentedges can bedifferent, we say that the problem is non-uniform. Thefirstnon-trivial approximation algorithm for this problem is due toCharikarand Karagiozova (STOC' 05) whose algorithm has anapproximation guarantee of$\exp(O(\sqrt{\log n\log\log n}))$,when all $\delta_i=1$ and$\exp(O(\sqrt{\log N\log\log N}))$ for the generaldemand case where $N$ isthe sum of all demands. We improve upon this result, bypresenting the firstpolylogarithmic (specifically, $O(\log^4 n)$ for unit demandsand $O(\log^4N)$ for the general demands)approximation for this problem. The algorithmrelies on a recent result\cite{HKS1} for the buy-at-bulk $k$-Steiner treeproblem.