Deterministic Streaming Lower Bounds for Approximate Maximum Clique and Maximum Independent Set

We study the canonical $\mathsf{\text{Maximum}}\mathsf{\text{Clique}}$ and $\mathsf{\text{Maximum}}\mathsf{\text{Independent}}\mathsf{\text{Set}}$ problems in the one-pass edge-arrival graph streaming setting. Here, the edges of some input graph $G=(V,E)$ are presented one at a time (possibly including deletions), before an algorithm needs to produce either a large clique or independent set at the end of the stream, with the focus being on space complexity. We are interested in finding $\beta$-approximate solutions, for any $\beta \ge 1$. Previous work gave an algorithm using $\tilde{O}\left(n^2\mathrm{/}{\beta }^2\right)$ bits of space, together with a corresponding $\tilde{\mathrm{\Omega }}\left(n^2\mathrm{/}{\beta }^2\right)$ two-party communication lower bound [Halldórsson et al., ICALP’12], seeming to resolve the problem. However, their algorithm crucially relies on randomness, and the best known deterministic algorithm remains a folklore derandomisation using $O\left(n^2\mathrm{/}\beta \right)$ bits of space, leaving a (deterministic) gap of size $\tilde{O}(\beta )$. We resolve this deterministic gap with an (almost) tight lower bound: any deterministic algorithm for either problem must use $\mathrm{\Omega }\left(\frac{n^2}{\beta \cdot \log n}\right)$ bits of space. Our proof is via a two-party one-way communication lower bound, and highlights the power of randomness when approaching either of these problems.