Streaming Complexity Separations for Dense and Sparse Graphs

TL;DR

This paper establishes fundamental space complexity separations for the Maximum Cut problem in streaming models, contrasting dense and sparse graphs, and extends techniques to related problems like Densest Subgraph and CSPs.

Contribution

It provides tight bounds and demonstrates a sharp separation in streaming space complexity for approximate Maximum Cut in dense versus sparse graphs.

Findings

Dense graphs require O(n/ε^2) space, sparse graphs need Ω(n log(ε^2 n)/ε^2) space

Deterministic algorithms need Ω(n log n/ε^2) space for (1-ε) approximation

Similar techniques apply to Densest Subgraph and certain CSPs, improving space bounds

Abstract

We identify a sharp separation in the streaming space complexity of Maximum Cut when the algorithm must output an approximate cut (rather than only the approximate value). For dense graphs, we show that $O(n/\varepsilon^2)$ space is sufficient and that $\Omega(n)$ space is necessary. In contrast, for graphs with $\Theta(n/\varepsilon^2)$ edges, the situation is markedly different: we show that the problem requires $\Omega(n \log(\varepsilon^2 n)/\varepsilon^2)$ space for any $\varepsilon=\omega(1/\sqrt{n})$, which is tight for the full range of $\varepsilon$. We also give an $\Omega(n \log n/\varepsilon^2)$-space lower bound against deterministic algorithms for outputting a $(1-\varepsilon)$ approximation to the value of the maximum cut. Using similar techniques we prove an analogous sharp separation in the streaming space complexity of Densest Subgraph and show that for every constant-arity CSP over a constant-size alphabet and the Similarity problem the space complexity in dense streams can be improved by shaving a logarithmic factor.