Deterministic Independent Sets in the Semi-Streaming Model

We consider the independent set problem in the semi-streaming model. For any input graph $G=(V, E)$ with $n$ vertices, an independent set is a set of vertices with no edges between any two elements. In the semi-streaming model, $G$ is presented as a stream of edges and any algorithm must use $\tilde O(n)$ bits of memory to output a large independent set at the end of the stream. Prior work has designed various semi-streaming algorithms for finding independent sets. Due to the hardness of finding maximum and maximal independent sets in the semi-streaming model, the focus has primarily been on finding independent sets in terms of certain parameters, such as the maximum degree $\Delta$. In particular, there is a simple randomized algorithm that obtains independent sets of size $\frac n{\Delta+1}$ in expectation, which can also be achieved with high probability using more complicated algorithms. For deterministic algorithms, the best bounds are significantly weaker. In fact, the best we currently know is a straightforward algorithm that finds an $\tilde\Omega\left(\frac n{\Delta^2}\right)$ size independent set. We show that this straightforward algorithm is nearly optimal by proving that any deterministic semi-streaming algorithm can only output an $\tilde O\left(\frac n{\Delta^2}\right)$ size independent set. Our result proves a strong separation between the power of deterministic and randomized semi-streaming algorithms for the independent set problem.