Streaming Complexity Separations for Dense and Sparse Graphs

2026-05-10

AI Summary

This work investigates the fundamental gap in space complexity between dense and sparse graphs when streaming algorithms must output an approximate maximum cut solution—not merely its value. Leveraging communication complexity and information-theoretic lower bounds within the streaming model, the paper establishes the first tight separation: for dense graphs, Θ(n/ε²) space is necessary and sufficient, whereas for sparse graphs with Θ(n/ε²) edges, any algorithm requires Ω(n log(ε²n)/ε²) space, a bound that is tight for all ε = ω(1/√n). This result demonstrates a pronounced complexity separation between dense and sparse instances in settings where the actual cut structure must be recovered, and it extends to constant-arity constraint satisfaction problems and related similarity tasks.

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 $Ω(n)$ space is necessary. In contrast, for graphs with $Θ(n/\varepsilon^2)$ edges, the situation is markedly different: we show that the problem requires $Ω(n \log(\varepsilon^2 n)/\varepsilon^2)$ space for any $\varepsilon=ω(1/\sqrt{n})$, which is tight for the full range of $\varepsilon$. We also give an $Ω(n \log n/\varepsilon^2)$-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.

Problem

• streaming complexity
• Maximum Cut
• dense graphs
• sparse graphs
• space complexity

Innovation

• streaming algorithms
• space complexity
• Maximum Cut
• dense vs sparse graphs
• lower bounds