# Questions tagged [communication-complexity]

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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### Expected vs worst-case communication complexity

In the set disjointness problem of 2-party communication complexity, Alice and Bob are both given an $n$-bit string as input; denoted by $X$ for Alice's input, and $Y$ for Bob's input. They need to ...
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### Matrix rank approximation

I am aware of the problem of low rank approximation of matrices which has been studied in various models of computation. My question is the following: What is the status of approximating rank of a ...
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### Why not include private randomness in internal communication information definition?

I am using https://www.cs.toronto.edu/~toni/Courses/CommComplexity2014/Lectures/lecture12.pdf as a reference. This isn't exactly a research question but I can't find a good place to ask it. Suppose ...
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### Newman's lemma for distributional communication complexity

This may be obvious — sorry if it is. Newman's lemma (Newman91] shows that any public-coin communication protocol to compute a Boolean function $f\colon \{0,1\}^n\times\{0,1\}^n\to\{0,1\}$ can be ...
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### What is the exact communication complexity of subtree disjointness?

A classic textbook example for communication complexity is when A and B both receive a subtree of a an $n$-node tree (that they both know), and they need to output whether their subtrees are disjoint ...
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### Communication complexity of approximating the size of set intersection

Consider the set-intersection problem: Alice and Bob each get a subset of $\left\{ 1,\ldots, n\right\}$, and they would like to know whether their sets intersect. This is a canonical problem of ...
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### Is this graph communication game known?

Let $X_m=[m]=\{0,1,\dots,m-1\}$ and let $Y_m=[2m]\setminus [m]$. Given is a complete bipartite graph $G_m$, with parts $X_m$ and $Y_m$ and edges $\{x,y\}$ for every $x\in X_m$ and $y\in Y_m$. Alice ...
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### Is there a name for this concept in Communication Complexity?

Let Alice and Bob compute boolean function $f(x_1,\dots,x_{2n})$. Select a random subset $\mathcal I\subseteq\{1,\dots,2n\}$ of cardinality $n$ and let $\mathcal J=\{1,\dots,2n\}\backslash\mathcal I$....
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### Regular languages and constant communication complexity

Let $L \subseteq A^*$ be a language, and define $f_L\colon A^* \times A^* \to \{0, 1\}$ by $f_L(x, y) = 1$ iff $x\cdot y \in L$. I'm searching for a reference for: Proposition. $L$ is regular iff ...
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### Evaluating boolean formula without knowing all values

I am looking for research approaches for the following problem: assume we have a set of $m$ computers, each carries a bit, and a Boolean formula $\varphi$ over those $m$ variables. The computers are ...
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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. ), 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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### Gap-Hamming with different “threshold” (i.e., not $n/2$)

Following a previous question, I'm trying to get a better understanding of the parameters at play in $\textsf{Gap-Hamming}$. In the "standard" setting, we have $x,y\in\{0,1\}^n$ and the partial ...
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### One way communication complexity of multi exact matching

Let Alice have $n$ binary strings, each of length $n$ and let Bob have one binary string of length $n$. Bob has to output $1$ if his string matches any of Alice's exactly and $0$ otherwise. Clearly ...
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### One-way randomized complexity of (variants of) Gap-Hamming-Distance?

The $\textsf{GapHammingDistance}$ problem over $\{0,1\}^n$ is defined as follows: Alice (resp. Bob) is given an input $x\in\{0,1\}^n$ (resp, $y\in\{0,1\}^n$), under the promise that their Hamming ...
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### Log Rank Conjecture Collaborative Approach [closed]

Recently a post was made in Mathoverflow seeking possible avenues for collaborative projects. I made a proposal for Log Rank conjecture in https://mathoverflow.net/questions/219638/proposals-for-...
### Number of $0/1$-monochromatic rectangles and communication complexity
What is the relation between number $0$-monochromatic rectangles in characteristic matrix and communication complexity? What is the relation between number $1$-monochromatic rectangles in ...