The monotone circuit complexity of Boolean functions
Combinatorica
The probabilistic communication complexity of set intersection
SIAM Journal on Discrete Mathematics
Fixed-parameter tractability and completeness II: on completeness for W[1]
Theoretical Computer Science
Communication complexity
Reductions in streaming algorithms, with an application to counting triangles in graphs
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Some complexity questions related to distributive computing(Preliminary Report)
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
A characterization of easily testable induced subgraphs
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Approximating Maximum Clique by Removing Subgraphs
SIAM Journal on Discrete Mathematics
On graph problems in a semi-streaming model
Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
Complexity classes in communication complexity theory
SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science
Efficient semi-streaming algorithms for local triangle counting in massive graphs
Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining
Mining Large Networks with Subgraph Counting
ICDM '08 Proceedings of the 2008 Eighth IEEE International Conference on Data Mining
Graph Sparsification in the Semi-streaming Model
ICALP '09 Proceedings of the 36th Internatilonal Collogquium on Automata, Languages and Programming: Part II
Graph Distances in the Data-Stream Model
SIAM Journal on Computing
Estimating clustering indexes in data streams
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Streaming algorithms for independent sets
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Intractability of min- and max-cut in streaming graphs
Information Processing Letters
K-median clustering, model-based compressive sensing, and sparse recovery for earth mover distance
Proceedings of the forty-third annual ACM symposium on Theory of computing
Approximate counting of cycles in streams
ESA'11 Proceedings of the 19th European conference on Algorithms
Better inapproximability results for maxclique, chromatic number and min-3lin-deletion
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Space-bounded communication complexity
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
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We consider the classic clique (or, equivalently, the independent set) problem in two settings. In the streaming model, edges are given one by one in an adversarial order, and the algorithm aims to output a good approximation under space restrictions. In the communication complexity setting, two players, each holds a graph on n vertices, and they wish to use a limited amount of communication to distinguish between the cases when the union of the two graphs has a low or a high clique number. The settings are related in that the communication complexity gives a lower bound on the space complexity of streaming algorithms. We give several results that illustrate different tradeoffs between clique separability and the required communication/space complexity under randomization. The main result is a lower bound of $\Omega(\frac{n^2}{r^2\log^2{n}})$-space for any r-approximate randomized streaming algorithm for maximum clique. A simple random sampling argument shows that this is tight up to a logarithmic factor. For the case when r=o(logn), we present another lower bound of $\Omega(\frac{n^2}{r^4})$. In particular, it implies that any constant approximation randomized streaming algorithm requires Ω(n2) space, even if the algorithm runs in exponential time. Finally, we give a third lower bound that holds for the extremal case of s−1 vs. $\mathcal{R}(s)-1$, where $\mathcal{R}(s)$ is the s-th Ramsey number. This is the extremal setting of clique numbers that can be separated. The proofs involve some novel combinatorial structures and sophisticated combinatorial constructions.