Eric Blais, Clement Canonne, Tom Gur

We present a new methodology for proving distribution testing lower bounds, establishing a connection between distribution testing and the simultaneous message passing (SMP) communication model. Extending the framework of Blais, Brody, and Matulef [BBM12], we show a simple way to reduce (private-coin) SMP problems to distribution testing problems. This method ... more >>>

Elchanan Mossel, Sampath Sampath Kannan, Grigory Yaroslavtsev

We initiate a systematic study of linear sketching over $\mathbb F_2$. For a given Boolean function $f \colon \{0,1\}^n \to \{0,1\}$ a randomized $\mathbb F_2$-sketch is a distribution $\mathcal M$ over $d \times n$ matrices with elements over $\mathbb F_2$ such that $\mathcal Mx$ suffices for computing $f(x)$ with high ... more >>>

Anat Ganor, Karthik C. S.

We show a communication complexity lower bound for finding a correlated equilibrium of a two-player game. More precisely, we define a two-player $N \times N$ game called the 2-cycle game and show that the randomized communication complexity of finding a 1/poly($N$)-approximate correlated equilibrium of the 2-cycle game is $\Omega(N)$. For ... more >>>

Young Kun Ko, Arial Schvartzman

In the recent paper of~\cite{BR16}, the authors show that, for any constant $10^{-15} > \varepsilon > 0$ the communication complexity of $\varepsilon$-approximate Nash equilibria in $2$-player $n \times n$ games is $n^{\Omega(\varepsilon)}$, resolving the long open problem of whether or not there exists a polylogarithmic communication protocol. In this paper ... more >>>

Paul Beame, Noah Fleming, Russell Impagliazzo, Antonina Kolokolova, Denis Pankratov, Toniann Pitassi, Robert Robere

We introduce and develop a new semi-algebraic proof system, called Stabbing Planes that is in the style of DPLL-based modern SAT solvers. As with DPLL, there is only one rule: the current polytope can be subdivided by

branching on an inequality and its "integer negation.'' That is, we can (nondeterministically ...
more >>>

Arkadev Chattopadhyay, Michal Koucky, Bruno Loff, Sagnik Mukhopadhyay

We develop a technique for proving lower bounds in the setting of asymmetric communication, a model that was introduced in the famous works of Miltersen (STOC'94) and Miltersen, Nisan, Safra and Wigderson (STOC'95). At the core of our technique is a novel simulation theorem: Alice gets a $p \times n$ ... more >>>

Daniel Kane, Roi Livni, Shay Moran, Amir Yehudayoff

This work introduces a model of distributed learning in the spirit of Yao's communication complexity model. We consider a two-party setting, where each of the players gets a list of labelled examples and they communicate in order to jointly perform some learning task. To naturally fit into the framework of ... more >>>

Iftach Haitner, Noam Mazor, Rotem Oshman, Omer Reingold, Amir Yehudayoff

Key-agreement protocols whose security is proven in the random oracle model are an important alternative to the more common public-key based key-agreement protocols. In the random oracle model, the parties and the eavesdropper have access to a shared random function (an "oracle"), but they are limited in the number of ... more >>>

Andrei Romashchenko, Marius Zimand

We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings

$x$ and $y$ is equal, up to logarithmic precision, to the length of the longest shared secret key that

two parties, one having $x$ and the complexity profile of the pair and the ...
more >>>

Jayadev Acharya, Clement Canonne, Himanshu Tyagi

Independent samples from an unknown probability distribution $\mathbf{p}$ on a domain of size $k$ are distributed across $n$ players, with each player holding one sample. Each player can communicate $\ell$ bits to a central referee in a simultaneous message passing (SMP) model of communication to help the referee infer a ... more >>>

Dmitry Itsykson, Alexander Knop, Andrei Romashchenko, Dmitry Sokolov

In 2004 Atserias, Kolaitis and Vardi proposed OBDD-based propositional proof systems that prove unsatisfiability of a CNF formula by deduction of identically false OBDD from OBDDs representing clauses of the initial formula. All OBDDs in such proofs have the same order of variables. We initiate the study of OBDD based ... more >>>

Mark Bun, Nikhil Mande, Justin Thaler

The communication class $UPP^{cc}$ is a communication analog of the Turing Machine complexity class $PP$. It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds.

For a communication problem ... more >>>

Amit Chakrabarti, Prantar Ghosh

We give new algorithms in the annotated data streaming setting---also known as verifiable data stream computation---for certain graph problems. This setting is meant to model outsourced computation, where a space-bounded verifier limited to sequential data access seeks to overcome its computational limitations by engaging a powerful prover, without needing to ... more >>>

Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir, Toniann Pitassi

Lifting theorems are theorems that relate the query complexity of a function $f:\left\{ 0,1 \right\}^n\to \left\{ 0,1 \right\}$ to the communication complexity of the composed function $f\circ g^n$, for some “gadget” $g:\left\{ 0,1 \right\}^b\times \left\{ 0,1 \right\}^b\to \left\{ 0,1 \right\}$. Such theorems allow transferring lower bounds from query complexity to ... more >>>

Sivaramakrishnan Natarajan Ramamoorthy, Cyrus Rashtchian

Recently, Dvir, Golovnev, and Weinstein have shown that sufficiently strong lower bounds for linear data structures would imply new bounds for rigid matrices. However, their result utilizes an algorithm that requires an $NP$ oracle, and hence, the rigid matrices are not explicit. In this work, we derive an equivalence between ... more >>>

Rahul Ilango, Bruno Loff, Igor Carboni Oliveira

Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an ... more >>>

Shachar Lovett, Raghu Meka, Jiapeng Zhang

Lifting theorems are a generic way to lift lower bounds in query complexity to lower bounds in communication complexity, with applications in diverse areas, such as combinatorial optimization, proof complexity, game theory. Lifting theorems rely on a gadget, where smaller gadgets give stronger lower bounds. However, existing proof techniques are ... more >>>

Amit Chakrabarti, Prantar Ghosh, Justin Thaler

We study graph computations in an enhanced data streaming setting, where a space-bounded client reading the edge stream of a massive graph may delegate some of its work to a cloud service. We seek algorithms that allow the client to verify a purported proof sent by the cloud service that ... more >>>

Ian Mertz, Toniann Pitassi

Query-to-communication lifting theorems translate lower bounds on query complexity to lower bounds for the corresponding communication model. In this paper, we give a simplified proof of deterministic lifting (in both the tree-like and dag-like settings). Whereas previous proofs used sophisticated Fourier analytic techniques, our proof uses elementary counting together with ... more >>>

Anup Bhattacharya, Sourav Chakraborty, Arijit Ghosh, Gopinath Mishra, Manaswi Paraashar

The disjointness problem - where Alice and Bob are given two subsets of $\{1, \dots, n\}$ and they have to check if their sets intersect - is a central problem in the world of communication complexity. While both deterministic and randomized communication complexities for this problem are known to be ... more >>>

Nikhil Mande, Swagato Sanyal

We study parity decision trees for Boolean functions. The motivation of our study is the log-rank conjecture for XOR functions and its connection to Fourier analysis and parity decision tree complexity. Our contributions are as follows. Let $f : \mathbb{F}_2^n \to \{-1, 1\}$ be a Boolean function with Fourier support ... more >>>

Susanna de Rezende, Jakob Nordström, Marc Vinyals

We obtain the first true size-space trade-offs for the cutting planes proof system, where the upper bounds hold for size and total space for derivations with constant-size coefficients, and the lower bounds apply to length and formula space (i.e., number of inequalities in memory) even for derivations with exponentially large ... more >>>

Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh Saxena, Zhao Song, Huacheng Yu

We give an almost quadratic $n^{2-o(1)}$ lower bound on the space consumption of any $o(\sqrt{\log n})$-pass streaming algorithm solving the (directed) $s$-$t$ reachability problem. This means that any such algorithm must essentially store the entire graph. As corollaries, we obtain almost quadratic space lower bounds for additional fundamental problems, including ... more >>>

Klim Efremenko, Gillat Kol, Raghuvansh Saxena

Interactive error correcting codes are codes that encode a two party communication protocol to an error-resilient protocol that succeeds even if a constant fraction of the communicated symbols are adversarially corrupted, at the cost of increasing the communication by a constant factor. What is the largest fraction of corruptions that ... more >>>

Nikhil Mande, Swagato Sanyal

We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the Inner Product on a constant number of inputs. Let $IP$ denote Inner Product on ... more >>>

Marcel Dall'Agnol, Tom Gur, Subhayan Roy Moulik, Justin Thaler

We initiate the systematic study of QMA algorithms in the setting of property testing, to which we refer as QMA proofs of proximity (QMAPs). These are quantum query algorithms that receive explicit access to a sublinear-size untrusted proof and are required to accept inputs having a property $\Pi$ and reject ... more >>>

Christian Ikenmeyer, Balagopal Komarath, Nitin Saurabh

We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games.

Using this game, we ... more >>>

Ron D. Rothblum, Michael Ezra

The inner product function $\langle x,y \rangle = \sum_i x_i y_i \bmod 2$ can be easily computed by a (linear-size) ${AC}^0(\oplus)$ circuit: that is, a constant depth circuit with AND, OR and parity (XOR) gates. But what if we impose the restriction that the parity gates can only be on ... more >>>

Oded Goldreich

This is a purely pedagogical text.

We advocate using KW-games as a teaser (or ``riddle'') for a complexity theoretic course.

In particular, stating the KW-game for a familiar NP-complete problem such as 3-Colorability and asking to prove that it requires more than polylogarithmic communication poses a seemingly tractable question ...
more >>>

Artur Ignatiev, Ivan Mihajlin, Alexander Smal

In this paper, we prove a super-cubic lower bound on the size of a communication protocol for generalized Karchmer-Wigderson game for some explicit function $f: \{0,1\}^n\to \{0,1\}^{\log n}$. Lower bounds for original Karchmer-Wigderson games correspond to De Morgan formula lower bounds, thus the best known size lower bound is cubic. ... more >>>

Guangxu Yang, Jiapeng Zhang

The notion of lifting theorems is a generic method to lift hardness of one-party functions to two-party lower bounds in communication model. It has many applications in different areas such as proof complexity, game theory, combinatorial optimization. Among many lifting results, a central idea is called Raz-McKenize simulation (FOCS 1997). ... more >>>

Klim Efremenko, Bernhard Haeupler, Yael Kalai, Pritish Kamath, Gillat Kol, Nicolas Resch, Raghuvansh Saxena

Given a Boolean circuit $C$, we wish to convert it to a circuit $C'$ that computes the same function as $C$ even if some of its gates suffer from adversarial short circuit errors, i.e., their output is replaced by the value of one of their inputs [KLM97]. Can we ... more >>>

Huacheng Yu

In this paper, we prove a strong XOR lemma for bounded-round two-player randomized communication. For a function $f:\mathcal{X}\times \mathcal{Y}\rightarrow\{0,1\}$, the $n$-fold XOR function $f^{\oplus n}:\mathcal{X}^n\times \mathcal{Y}^n\rightarrow\{0,1\}$ maps $n$ input pairs $(X_1,\ldots,X_n,Y_1,\ldots,Y_n)$ to the XOR of the $n$ output bits $f(X_1,Y_1)\oplus \cdots \oplus f(X_n, Y_n)$. We prove that if every ... more >>>

Scott Aaronson, Harry Buhrman, William Kretschmer

Relational problems (those with many possible valid outputs) are different from decision problems, but it is easy to forget just how different. This paper initiates the study of FBQP/qpoly, the class of relational problems solvable in quantum polynomial-time with the help of polynomial-sized quantum advice, along with its analogues for ... more >>>

Itai Dinur

The random-query model was introduced by Raz and Zhan at ITCS 2020 as a new model of space-bounded computation. In this model, a branching program of length $T$ and width $2^{S}$ attempts to compute a function $f:\{0,1\}^n \rightarrow \{0,1 \}$. However, instead of receiving direct access to the input bits ... more >>>

Ari Karchmer

Carmosino et al. (2016) demonstrated that natural proofs of circuit lower bounds imply algorithms for learning circuits with membership queries over the uniform distribution. Indeed, they exercised this implication to obtain a quasi-polynomial time learning algorithm for ${AC}^0[p]$ circuits, for any prime $p$, by leveraging the existing natural proofs from ... more >>>

Mi-Ying Huang, Xinyu Mao, Guangxu Yang, Jiapeng Zhang

Constructing key-agreement protocols in the random oracle model (ROM) is a viable method to assess the feasibility of developing public-key cryptography within Minicrypt. Unfortunately, as shown by Impagliazzo and Rudich (STOC 1989) and Barak and Mahmoody (Crypto 2009), such protocols can only guarantee limited security: any $\ell$-query protocol can be ... more >>>

Vladimir Podolskii, Dmitrii Sluch

Boolean function $F(x,y)$ for $x,y \in \{0,1\}^n$ is an XOR function if $F(x,y) = f(x\oplus y)$ for some function $f$ on $n$ input bits, where $\oplus$ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing Fourier analytic technique. For total XOR functions it is known ... more >>>

Guangxu Yang, Jiapeng Zhang

Collision problems are important problems in complexity theory and cryptography with diverse applications. Previous fruitful works have mainly focused on query models. Driven by various applications, several works by Bauer, Farshim and Mazaheri (CRYPTO 2018), Itsykson and Riazanov (CCC 2021), Göös and Jain (RANDOM 2022) independently proposed the communication version ... more >>>

Shuo Wang, Guangxu Yang, Jiapeng Zhang

Set-disjointness is one of the most fundamental and widely studied problems in the area of communication complexity. In this problem, each party $i$ receives a set $S_i\subseteq [N]$. The goal is to determine whether $\bigcap S_i$ is empty via communication between players. The decision version simply asks if the common ... more >>>

Huacheng Yu, Wei Zhan

Given a distribution over $[n]^n$ such that any $k$ coordinates need $k/\log^{O(1)}n$ bits of communication to sample, we prove that any map that samples this distribution from uniform cells requires locality $\Omega(\log(n/k)/\log\log(n/k))$. In particular, we show that for any constant $\delta>0$, there exists $\varepsilon=2^{-\Omega(n^{1-\delta})}$ such that $\Omega(\log n/\log\log n)$ non-adaptive ... more >>>

Pavel Hrubes

Given a non-negative real matrix $M$ of non-negative rank at least $r$, can we witness this fact by a small submatrix of $M$? While Moitra (SIAM J. Comput. 2013) proved that this cannot be achieved exactly, we show that such a witnessing is possible approximately: an $m\times n$ matrix always ... more >>>

Xinyu Mao, Guangxu Yang, Jiapeng Zhang

The notion of query-to-communication lifting theorems is a generic framework to convert query lower bounds into two-party communication lower bounds. Though this framework is very generic and beautiful, it has some inherent limitations such as it only applies to lifted functions. In order to address this issue, we propose gadgetless ... more >>>

Yahel Manor, Or Meir

Lifting theorems are theorems that bound the communication complexity

of a composed function $f\circ g^{n}$ in terms of the query complexity

of $f$ and the communication complexity of $g$. Such theorems

constitute a powerful generalization of direct-sum theorems for $g$,

and have seen numerous applications in recent years.

We prove ... more >>>

Klim Efremenko, Gillat Kol, Dmitry Paramonov, Raghuvansh Saxena

Kol and Raz [STOC 2013] showed how to simulate any alternating two-party communication protocol designed to work over the noiseless channel, by a protocol that works over a stochastic channel that corrupts each sent symbol with probability $\epsilon>0$ independently, with only a $1+\mathcal{O}(\sqrt{\H(\epsilon)})$ blowup to the communication. In particular, this ... more >>>