Optimal Bounds for Dominating Set in Graph Streams

We resolve the space complexity of one-pass streaming algorithms for Minimum Dominating Set (MDS) in both insertion-only and insertion-deletion streams (up to poly-logarithmic factors) where an input graph is revealed by a sequence of edge updates. Recently, streaming algorithms for the related Set Cover problem have received significant attention. Even though MDS can be viewed as a special case of Set Cover, it is however harder to solve in the streaming setting since the input stream consists of individual edges rather than entire vertex-neighborhoods, as is the case in Set Cover.
We prove the following results (n is the number of vertices of the input graph):
1) In insertion-only streams, we give a one-pass semi-streaming algorithm (meaning Õ(n) space) with approximation factor Õ(√n). We also prove that every one-pass streaming algorithm with space o(n) has an approximation factor of Ω(n/log n).
Combined with a result by [Assadi et al., STOC'16] for Set Cover which, translated to MDS, shows that space Θ̃(n² / α) is necessary and sufficient for computing an α-approximation for every α = o(√n), this completely settles the space requirements for MDS in the insertion-only setting.
2) In insertion-deletion streams, we prove that space Ω(n² / (α log n)) is necessary for every approximation factor α ≤ Θ(n / log³ n). Combined with the Set Cover algorithm of [Assadi et al., STOC'16], which can be adapted to MDS even in the insertion-deletion setting to give an α-approximation in Õ(n² / α) space, this completely settles the space requirements for MDS in the insertion-deletion setting.