Abstract
In the online Steiner tree problem, a sequence of points is revealed one-by-one: when a point arrives, we only have time to add a single edge connecting this point to the previous ones, and we want to minimize the total length of edges added. Here, a tight bound has been known for two decades: the greedy algorithm maintains a tree whose cost is $O(\log n)$ times the Steiner tree cost, and this is the best possible. But suppose, in addition to the new edge we add, we have time to change a single edge from the previous set of edges: can we do much better? Can we, e.g., maintain a tree that is constant-competitive? We answer this question in the affirmative. We give a primal-dual algorithm that makes only a single swap per step (in addition to adding the edge connecting the new point to the previous ones), and such that the tree's cost is only a constant times the optimal cost. Our dual-based analysis is quite different from previous primal-only analyses. In particular, we give a correspondence between radii...
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