The Power of Deferral: Maintaining a Constant-Competitive Steiner Tree Online
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 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 of dual balls and lengths of tree edges; since dual balls are associated with points and hence do not move around (in contrast to edges), we can closely monitor the edge lengths based on the dual radii. Showing that these dual radii cannot change too rapidly is the technical heart of the paper, and allows us to give a hard bound on the number of swaps per arrival, while maintaining a constant-competitive tree at all times. Previous results for this problem gave an algorithm that performed an amortized constant number of swaps: for each n, the number of swaps in the first $n$ steps was O(n). We also give a simpler tight analysis for this amortized case.