Competitive online multicommodity routing

We study online multicommodity routing problems in networks, in which commodities have to be routed sequentially. The flow of each commodity can be split on several paths. Arcs are equipped with load dependent price functions defining routing costs, which have to be minimized. We discuss a greedy online algorithm that routes each commodity by minimizing a convex cost function that depends on the previously routed flow. We present a competitive analysis of this algo-rithm showing that for affine price functions this algorithm is 4K 2 (1+K)2 competitive, where K is the number of commodities. For networks with two nodes and parallel arcs, this algorithm is optimal. Without restrictions on the price functions and network, no algorithm is competitive. We then investigate a variant in which the demands have to be routed unsplittably. In this case, it is NP-hard to compute the offline optimum. The variant of the greedy algorithm that produces unsplittable flows is (3+2 2)-competitive, and we prove a lower bound of 2 for the competitive ratio of any deterministic online algorithm