Questions regarding the amount of communication needed to accomplish a computational task, when the information about the task is spread across several agents

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1answer
171 views

Is it possible to prove stronger bounds for the deterministic communication complexity compared to nondeterministic communication complexity?

Inspired by the questions Nondeterministic communication complexity of set disjointness?, I was wondering about the following: Is there an example of a function $f$ where the nondeterministic ...
6
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1answer
110 views

Nondeterministic communication complexity of set disjointness?

In the two-party setting, bounds of $\Theta(n)$ bits are known for deterministic and bounded-error randomized protocols for $\text{DISJ}_n$. (Here $\text{DISJ}_n$ is the $n$-element set disjointness ...
2
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0answers
106 views

$CIS_G$ problem deterministic lower bound

In notes, http://www.cs.toronto.edu/~toni/Courses/CommComplexity2/Lectures/clique-notes.pdf, it is mentioned towards end of section $4$ that http://dl.acm.org/citation.cfm?id=237817 shows that ...
2
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0answers
75 views

Rounds in Log rank conjecture

Given a Boolean function $f$ with characteristic matrix $M_f$ with rank $r$, a trivial algorithm that utilizes $r\log r$ bits ($\log r$ bits to index atmost $r$ distinct rows) by Alice to indicate its ...
6
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1answer
156 views

On the notion of positive rank

The positive rank of a square matrix is defined in Theorem $3$ of "Expressing Combinatorial Optimization Problems by Linear Programs" by Mihalis Yannakakis as follows: given a $n\times n$ matrix $A$, ...
2
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1answer
116 views

Real representation versus communication complexity

Suppose that Alice and Bob communicate to compute a function $f:\{0,1\}^n\times\{0,1\}^n\rightarrow\{0,1\}$. Does the minimal degree of a real polynomial/rational representation of $f$ play a role for ...
4
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1answer
155 views

On the notion of positive rank of a matrix

The positive rank of a square matrix is defined in Theorem $3$ of "Expressing Combinatorial Optimization Problems by Linear Programs" by Mihalis Yannakakis as follows: given a $n\times n$ matrix $A$, ...
11
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4answers
111 views

Minimum communication cost for zero knowledge proofs of three colorability

Goldreich et al.'s proof that three colorability has zero knowledge proofs uses bit commitment for an entire coloring of the graph in each round [1]. If a graph has $n$ vertices and $e$ edges, a ...
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0answers
37 views

Modern tools deterministic communication applications

Partition number, Fooling-set method along with rank method provide important tools to identify deterministic communication complexity of a function. These techniques were identified some decades ...
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300 views

Does Rabin/Yao exist (at least in a form that can be cited)?

In Andrew Chi-Chih Yao's classic 1979 paper he references "M. O. Rabin and A. C. Yao, in preparation". This is for the result that the bounded-error communication complexity of the equality function ...
0
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1answer
68 views

On connecting combinatorial rectangles

In communication complexity one important object is the combinatorial retangle. Given a $0-1$ square matrix $M$, do their exist permutations $\sigma,\pi$ such that $\sigma M\pi$ consists of only ...
3
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1answer
71 views

Classification of a specific problem

Is it known that $\mathsf{IP}\notin\mathsf{NP}^{cc}\cup\mathsf{coNP}^{cc}$ where $\mathsf{IP}$ is inner product communication complexity problem? Where is the classification of $\mathsf{IP}$ currently ...
2
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1answer
79 views

Partition Number of a Matrix

Given matrix $M\in\{0,1\}^{n\times n}$, let the minimum number of monochromatic rectangles it can be partitioned be $p$. Let the positive rank of $M$ be $\sigma$ and the rank be $r$. Is it known ...
3
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1answer
131 views

The relationship between QCMA and QMA in the Turing and Communication model

First my background about computational complexity is still beginner. Recent paper published by Klauck and Podder [KP14] show that for the first time an exponential gap between computing partial ...
8
votes
1answer
288 views

Lower bound for NFA accepting 3 letter language

Related to a recent question (Bounds on the size of the smallest NFA for L_k-distinct) Noam Nisan asked for a method to give a better lower bound for the size of an NFA than what we get from ...
19
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1answer
245 views

Deterministic communication complexity vs partition number

Background: Consider the usual two-party model of communication complexity where Alice and Bob are given $n$-bit strings $x$ and $y$ and have to compute some Boolean function $f(x,y)$, where ...
9
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1answer
372 views

Best sources for communication complexity

What are some of the best sources (books and papers) to motivate and learn communication complexity on its own and in connection with its relation to computational complexity theory?
11
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1answer
154 views

Any evidence that Linial, Shraibman lower bound on quantum communication complexity is not tight?

As far as I know, the factorization norm lower bound given by Linial and Shraibman is essentially the only lower bound known for quantum communication complexity (or at least it subsumes all others). ...
6
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1answer
130 views

Zero error randomized communication complexity vs deterministic communication complexity

It is known that for $\Theta(1)$ error the worst case definition of randomized communication complexity and average case definition are equivalent. But when the error is $0$, the worst case randomized ...
18
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2answers
595 views

Bounds on the size of the smallest NFA for L_k-distinct

Consider the language $L_{k-distinct}$ consisting of all $k$-letter strings over $\Sigma$ such that no two letters are equal: $$ L_{k-distinct} :=\{w = \sigma_1\sigma_2...\sigma_k \mid \forall ...
7
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1answer
100 views

Lower bounds on alternative models of multiparty communication complexity

I'm a newcomer to communication complexity, and so far I've read the chapter in Arora-Barak and some papers giving lower bounds in various applications. A priori the definition of multiparty ...
8
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3answers
253 views

Communication complexity of finding common element of two subsets

Suppose that Alice receives a subset $S \subseteq \{1,\dots,n\}$ and Bob receives $T \subseteq \{1,\dots,n\}$. It is promised that $\lvert S \cap T \rvert = 1$. What is the randomized communication ...
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1answer
155 views

Succinct Representation and Communication complexity

Succinct representation is often used to define NEXP or EXP complete problems. For example, when a graph is given as a circuit to compute the existence of edge between vertex $i,j$ for indices of ...
0
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1answer
145 views

Various conjectures which is similar to Log Rank conjecture

Log rank conjecture is one of the most famous open problems in the area of communication compleixty. Lets consider the two party cdommunication complexity. Alice and Bob have $n$ bit strings $a,b$ , ...
2
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0answers
74 views

Are there efficient black-box constructions of sigma-protocols for SAT?

Is there a known black-box construction for the following implication? non-interactive string commitment that stretches additively by an amount which does not depend on the string being ...
6
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2answers
282 views

Testing for equality with zero error

This question comes from what I asked in a comment here, although I realized that I don't actually care about which input is less than the other, if they're different. Alice and Bob have n-bit ...
14
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2answers
581 views

Testing for positivity instead of equality

Alice and Bob have n-bit strings, and want to figure out if they're equal while doing little communication. The standard randomized solution is to treat the n-bit strings as polynomials of degree $n$ ...
10
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1answer
278 views

Why does the log-rank conjecture use rank over the reals?

In communication complexity, the log-rank conjecture states that $$cc(M) = (\log rk(M))^{O(1)}$$ Where $cc(M)$ is the communication complexity of $M(x,y)$ and $rk(M)$ is the rank of $M$ (as a ...
3
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0answers
215 views

Streaming Algorithm Lower Bounds by Communication Complexity

I am learning the methods for proving lower bounds on streaming algorithms using communication complexity. My question is about a basic technique to prove lower bounds on streaming models using the ...
1
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1answer
228 views

“Send Once”-One way Multiparty Communication Complexity

There are plenty results on multiparty communication complexity, and one way protocol which anyone playing communicatin games is able to send one person, is a basic setting. I want to consider more ...
6
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1answer
370 views

Streaming Algorithms: Motivations for estimating frequency moments

The celebrated AMS paper "The space complexity of approximating the frequency moments" defines the problem as following: Let $a_1, a_2,\dotsc, a_m$ be a sequence of integers where each $a_j \in ...
14
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1answer
269 views

Best communication complexity lower bound of disjointness

It is well known that no deterministic two-party protocol can solve disjointness problem (DISJ) on $n$-bit inputs without sending $n+1$ bits in the worst case (see, e.g., the book by Kushilevitz and ...
7
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2answers
167 views

Delegating all of the work to the prover in $\mathcal{MA}$ protocols

An $\mathcal{MA}$ communication complexity protocol is communication complexity protocol that starts with an omniscient prover that sends a proof (that depends on the the specific input of the ...
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0answers
137 views

Communication complexity of random functions with limited independence

Let $X_0, \ldots, X_{2^n-1}$ be $k$-wise independent random $0/1$ variables over a sample space $\Omega$ and $Prob \left[ X_i = 1 \right] = p$ for every $i$ and some $0 < p < 1$. Let assume $n$ ...
8
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1answer
234 views

A balanced generalization of Hall’s theorem

Let $X$ and $Y$ be sets, and $\mathcal{B}$ be a partition of $X \times Y$. I would like to prove that there exists a distribution $\mathcal{D}$ over $X \times Y$ whose marginal is uniform over $X$, ...
3
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0answers
140 views

Lower Bound Methods in NonDet Communication Complexity

rank+($M$) is the minimum $r$ such that the following statement holds. The statement : there exists matrices $U,V$ such that $M = UV$ and $U$ has $r$ columns and $V$ has $r$ rows. Is rank+($M$) ...
5
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1answer
199 views

Average message complexity for the election problem on graphs

I am currently studying the election problem in distributed algorithms. There, I stumpled over one approach to implement a Chang-Roberts-like message extinction algorithm on graphs without requiring a ...
4
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0answers
174 views

Do the quantum communication complexity lower bounds hold when parties can send a “duplicated” qubits?

This question continues from the previous question where I mistakenly asked a question that is too general. In quantum communication complexity, we always assume that Alice and Bob have unlimited ...
0
votes
2answers
311 views

Are Alice and Bob allowed to copy qubits in quantum communication complexity model?

In quantum communication complexity, we always assume that Alice and Bob have unlimited computational power and are still prove lower bounds such as the $\Omega(n)$ lower bounds of parity. What ...
12
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0answers
405 views

Known upper bounds on the communication complexity of Karchmer-Wigderson games

In 1988 Karchmer and Wigderson established a nice characterization of the circuit depth $d$ (DeMorgan circuits) of a Boolean function $f \colon \{0,1\}^n\rightarrow\{0,1\}$: $d$ is exactly the number ...
10
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3answers
235 views

Bounds on approximating frequency moments

Let $a_1, a_2,\dotsc, a_m$ be a sequence of integers where each $a_j \in \{1,2,\dotsc,n\}$. For $i \in \{1,2,\dotsc,n\}$, let $m_i = |\{j : a_j = i\}|$. The $k$th frequency moment is defined to be ...
9
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2answers
478 views

Communication complexity with a referee

Assume a framework in communication complexity where we have two players A(lice) and B(ob) and a R(eferee). A and B don't communicate directly with each other. In each round of communication, each of ...
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0answers
290 views

Descriptive complexity of communication complexity classes

It is well known that some major complexity classes, like P or NP, admit a full logical characterization (e.g NP = existential 2nd order logic by Fagin's theorem). On the other hand, one can also ...
10
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2answers
353 views

Nonlocal Games and Quantum Communication

I'm currently on the look out for some good reference material relating non-local games with beneficial aspects in quantum communication. For instance, I am aware that non-local games are good at ...
7
votes
1answer
87 views

Effect of protocol ordering on multiparty comm. complexity

Brief Background In Multi-Party Protocols by Chandra, Lipton, and Furst [CFL83], a Ramsey-theoretic proof is used to show a lower bound (and later, a matching upper bound) for the predicate ...
4
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1answer
411 views

Monochromatic Rectangle Tiling

This problem originates from the tiling lowerbound method for communication complexity. In that method, there is a 0-1 matrix $M_{n \times n}$. A rectangle is defined as a submatrix $A \times B$ where ...
6
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1answer
532 views

Existence of zero-knowledge proof for location

N items have been placed at specific points on a map. A prize is awarded to the first person who turns in a list with the location of all N items. The location of each item must fall with a distance ...
10
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1answer
246 views

What is the largest gap between rank and approximate rank?

We know that the log of the rank of a 0-1 matrix is the lower bound of deterministic communication complexity, and the log of the approximate rank is the lower bound of randomized communication ...
4
votes
1answer
170 views

Why does deterministic recognition of DYCK(2) languages in the streaming model take linear space?

I was reading the paper "Recognizing Well-Paranthesized Expressions in the Streaming Model" by Magniez, Mathieu and Nayak where they give upper and lower bounds on the space required to recognize ...
12
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2answers
632 views

Best alien communication protocol?

Let's say we discover alien civilizations that are able to send and receive messages using an interstellar digital communications channel. (Say using modulated radio waves, laser pulses, ...