Online Buy-at-Bulk Network Design.

We present the first online algorithms for the nonuniform, multicommodity buy-atbulk (MC-BB) network design problem. Our competitive ratios qualitatively match the best known approximation factors for the corresponding offline problems. In particular, we show (a) a polynomial time online algorithm with a polylogarithmic competitive ratio for the MC-BB problem in undirected edge-weighted graphs, (b) a quasi-polynomial time online algorithm with a polylogarithmic competitive ratio for the MC-BB problem in undirected node-weighted graphs, (c) for any fixed epsilon > 0, a polynomial time online algorithm with a competitive ratio of (O) over tilde k(1/2+epsilon)) (where k is the number of demands, and the tilde hides polylog factors) for MC-BB in directed graphs, and (d) algorithms with matching competitive ratios for the prize-collecting variant of all the preceding problems. Prior to our work, a logarithmic competitive ratio was known for undirected, edge-weighted graphs only for the special case of uniform costs [B. Awerbuch and Y. Azar, FOCS, 1997, pp. 542-547], and a polylogarithmic-competitive algorithm was known for the edge-weighted single-sink problem [A. Meyerson, Procedings of SPAA, 2004, pp. 275-280]. We believe no online algorithm was known in the node-weighted and directed settings, even for uniform costs. Our main technical contribution is an online reduction theorem of MC-BB problems to their single-sink counterparts. We use the concept of junction-tree solutions from [C. Chekuri, M. T. Hajiaghayi, G. Kortsarz, and M. R. Salavatipour, Proceedings of FOCS, 2006, pp. 677-686], which play an important role in solving the offline versions of the problem via a greedy subroutine-an inherently offline procedure. We use just the existence of good junction-trees for our reduction.