Journal ArticleDOI
On the "log rank"-conjecture in communication complexity
Ran Raz,Boris Spieker +1 more
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The existence of a non-constant gap between the communication complexity of a function and the logarithm of the rank of its input matrix is shown and an Ω(nloglogn) lower bound for the graph connectivity problem in the non-deterministic case is proved.Abstract:
We show the existence of a non-constant gap between the communication complexity of a function and the logarithm of the rank of its input matrix. We consider the following problem: each of two players gets a perfect matching between twon-element sets of vertices. Their goal is to decide whether or not the union of the two matcliings forms a Hamiltonian cycle. We prove:
Our result also supplies a superpolynomial gap between the chromatic number of a graph and the rank of its adjacency matrix. Another conclusion from the second result is an Ω(nloglogn) lower bound for the graph connectivity problem in the non-deterministic case. We make use of the theory of group representations for the first result. The second result is proved by an information theoretic argument.read more
Citations
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MonographDOI
Computational Complexity: A Modern Approach
Sanjeev Arora,Boaz Barak +1 more
TL;DR: This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory and can be used as a reference for self-study for anyone interested in complexity.
Book ChapterDOI
Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth
TL;DR: Two new approaches rooted in linear algebra, based on matrix rank and determinants, which provide deterministic c tw | V | O ( 1 ) time algorithms, also for weighted and counting versions of connectivity problems are presented.
Journal ArticleDOI
On rank vs. communication complexity
Noam Nisan,Avi Wigderson +1 more
TL;DR: This paper gives an example exhibiting the largest gap known and proves two related theorems about the relationship between the communication complexity of a boolean function and the rank of the associated matrix.
Book
Lower Bounds in Communication Complexity
Troy Lee,Adi Shraibman +1 more
TL;DR: Lower Bounds in Communication Complexity focuses on showing lower bounds on the communication complexity of explicit functions, and treats different variants of communication complexity, including randomized, quantum, and multiparty models.
Proceedings ArticleDOI
Distributed verification and hardness of distributed approximation
Atish Das Sarma,Stephan Holzer,Liah Kor,Amos Korman,Danupon Nanongkai,Gopal Pandurangan,David Peleg,Roger Wattenhofer +7 more
TL;DR: In this paper, the authors study the verification problem in distributed networks, and give almost tight lower bounds on the running time of distributed verification algorithms for many fundamental problems such as connectivity, spanning connected subgraph, and s-t cut verification.
References
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Proceedings ArticleDOI
Las Vegas is better than determinism in VLSI and distributed computing (Extended Abstract)
Kurt Mehlhorn,Erik M. Schmidt +1 more
TL;DR: A new method for proving lower bounds on the complexity of VLSI - computations and more generally distributed computations which only applies to deterministic computations.
Las Vegas Is better than Determinism in VLSI and Distributed Computing
Kurt Mehlhorn,Erik M. Schmidt +1 more
Journal ArticleDOI
On rank vs. communication complexity
Noam Nisan,Avi Wigderson +1 more
TL;DR: This paper gives an example exhibiting the largest gap known and proves two related theorems about the relationship between the communication complexity of a boolean function and the rank of the associated matrix.
Proceedings ArticleDOI
On the communication complexity of graph properties
TL;DR: The bounds imply improved lower bounds for the VLSI complexity of these decision problems and sharp bounds for a generalized decision tree model which is related to the notion of evasiveness.
Journal ArticleDOI
Whitney Number Inequalities for Geometric Lattices
TL;DR: In this paper, it was shown that a middle term can be interpolated in the above inequality if and only if the lattice L is modular, and two further results concerning matchings of lattice elements of rank k into those of rank or = r - k were given.