# Tagged Questions

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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### Communication complexity protocols depending on inputs

Classical communication complexity requires one protocol (binary tree with edges labeled by bits Alice and Bob send) to solve the problem for every pair of inputs. What if we allow Alice and Bob to ...
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### Distribution attaining minimum discrepancy of disjointness function

Is it true that for the optimal distribution $\nu$ (not necessarily uniform) that attains minimum discrepancy $\mathsf{disc}(\mathsf{DISJ}_n)$ for the disjointness function $\mathsf{DISJ}_n$ we have ...
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### Methods for proving deterministic communication complexity lower bounds

I am familiar with the classic techniques for proving deterministic communication complexity lower bounds for boolean functions in the 2-party model: To the best of my knowledge, these are fooling ...
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### Nondeterministic communication complexity of Hamming distance

It is something that I think should be known: what is nondeterministic communication complexity of following task: is $H(x,y) \geq k$? There is an obvious upper bound $k \log(n)$. I would expect ...
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### Problems still “hard” in the SMP/Referee model with shared randomness?

In the referee (SMP: Simultaneous Message Passing) model introduced by Yao (see e.g. [1]), Alice and Bob have respectively inputs $x\in X$ and $y\in Y$, and wish to communicate with a third-party, the ...
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### Communication complexity of edit resilient synchronization

Supposing we have two strings $A$ and $B$ that are both edit distance $\tau$ from each other at two different sites is there a communication complexity model where synchronizing such strings have been ...
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### Lower Bound for Nonzero Terms of a Polynomial Fully Sensitive at 0

Every boolean function $f:\{0,1\}^n\to \{0,1\}$ can be uniquely represented as a multilinear polynomial $p=\sum_{S\subseteq [n]}c_S \chi_s$ where $\chi_s=\prod_{i\in S}x_i$. A boolean function is ...
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### Communication Complexity with real numbers

I'm looking into communication complexity with real numbers. One problem if we want to define this is that one can encode many real numbers $0.a_1a_2a_3... , 0.b_1b_2b_3..., 0.c_1c_2c_3...$ using only ...
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### A direct-sum theorem for the non-deterministic communication complexity of inequality?

A non-deterministic protocol for the inequality function is a protocol that behaves as follows: Alice and Bob get strings $x,y\in\{0,1\}^n$ respectively, and an untrusted prover is trying to convince ...
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### One-way randomized communication complexity of approximate Hamming distance

If Alice and Bob both have $n$ bit strings, consider a one-way randomized communication problem where Bob has to output with some good probability a number which is within a $(1+\epsilon)$ factor of ...
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### Communication problems for which a deterministic direct-sum theorem is not known to hold

It is an old open problem whether a direct-sum theorem holds for deterministic communication complexity, that is, whether solving $t$ independent instances of a problem is $t$ times harder than ...
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### Information complexity of query algorithms?

Information complexity has been a very useful tool in communication complexity, mainly used to lower bound the communication complexity of distributed problems. Is there an analogue of information ...
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### Communication complexity of Independent Set game?

Consider the following communication game. Independent Set game Let $[n] = \{0,1,\dots,n-1\}$ and let $r$ be a positive integer smaller than $n/(1+\log n)$. Alice receives a set $X$ of edges, each ...
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### Low rank Log rank conjecture

What is known about log rank conjecture in special situations of $O(\log N)$ rank $0/1$ matrix of size $N\times N$? Is there at least a conditional result showing better than $O(\sqrt{\log N})$ bound?
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### Research problems in communication complexity

There have been many open challenges questions in this forum. For instance, Research and open challenges in Programming Language Theory What are current open problems in compiler theory? ...
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### On a possible probabilistic log-rank conjecture

The communication matrix of a function $f$ is a $2^{n}\times 2^{n}$ matrix $M_f$ where the indices of the rows (columns) correspond to the inputs of Alice (Bob), and each entry $M_f (a,b)$ is the ...
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### 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 ...
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### 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 ...
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### $CIS_G$ 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 ...
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### 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 $A$, ...
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### 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 ...
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### 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 $A$, ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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