Dominating Set is a fundamental problem in graph theory: given a graph, find a minimum-weight subset of vertices such that every vertex is either selected or shares an edge with a selected vertex.
In online settings where vertices arrive sequentially, comparing algorithms against an offline optimum with full knowledge of the input leads to extremely strong lower bounds, where even a simple star graph shows that any online algorithm must have competitive ratio Ω(n), with n being the number of vertices, matching the trivial strategy of selecting all vertices.
We study the incremental dominating set problem, where the optimal algorithm is constrained to the same choices available to online algorithms. This introduces a benchmark that enables a meaningful comparison between algorithms.
We present the first results for vertex-weighted graphs and randomized algorithms in this model. For incremental dominating set, we give an O(Δ)-competitive deterministic algorithm and an O(log²Δ)-competitive randomized algorithm, where Δ is the maximum degree in the graph.
We extend these results to the Connected Dominating Set problem, which requires the dominating set to be connected. When the neighborhood of each arriving vertex is known in advance, we can improve the competitive ratio of deterministic algorithms to be polylogarithmic.
Finally, we establish matching lower bounds, showing that all our results are optimal up to constant factors.
