Abstract
In the multilevel aggregation problem (MLAP), requests arrive at the nodes of an edge-weighted tree J and have to be served eventually. A service is defined as a subtree X of J that contains the root of J.
This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times.
The objective is to minimize the total waiting cost of all requests plus the total cost of all service subbees. MLAP is a generalization of some well-studied optimization problems; for example, for trees of depth 1, MLAP is equivalent to the Transmission Control Protocol acknowledgment problem, whereas for trees of depth 2, it is equivalent to the joint replenishment problem.
Aggregation problems for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and supply chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests.
Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant-competitive online algorithms for trees of depth more than 2.
Addressing this open problem, we give the first constant-competitive online algorithm for trees of arbitrary (fixed) depth. The competitive ratio is O(D(4)2(D)), where D is the depth of J.
The algorithm works for arbitrary waiting cost functions, including the variant with deadlines.