On-line network synthesis

We consider on-line network synthesis problems. Let N = {1, , n} be a set of n sites. Traffic flow requirements between pairs of sites are revealed one by one. Whenever a new request rij = rji (i < j) between sites i and j is revealed, an on-line algorithm must install the additional necessary capacity without decreasing the existing network capacity such that all the traffic requirements are met. The objective is to minimize the total capacity installed by the algorithm. The performance of an on-line algorithm is measured by the competitive ratio, defined to be the worst-case ratio between the total capacity by the on-line algorithm and the total optimal (off-line) capacity assuming we have a priori information on all the requirements initially. We distinguish between two on-line versions of the problem depending on whether the entire set of sites is known a priori or not. For the first version where the entire set of sites is unknown, we present a best possible algorithm along with a matching lower bound. For the second version where the entire set of sites is known a priori, we present a best possible algorithm for n ⩽ 6.