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Hartmut Klauck

Publications and source records attributed to Hartmut Klauck.

At least 19 recordsLinked to original sources
arXiv2026

A Hierarchy for Constant Communication Complexity

Similarly to the Chomsky hierarchy, we offer a classification of communication complexity measures such that these measures are organized into equivalence classes. Different from previous attempts of this endeavor, we consider two communication complexity measures as equivalent, if, when one is constant, then the other is constant as well, and vice versa. Most previous considerations of similar topics have been using polylogarithmic input length as a defining characteristic of equivalence. In this paper, two measures ${\cal C}_1, {\cal C}_2$ are constant-equivalent, if and only if for all total Boolean (families of) functions $f:\{0, 1\}^n\times\{0, 1\}^n\rightarrow \{0, 1\}$ we have ${\cal C}_1(f)=O(1)$ if and only if ${\cal C}_2(f)=O(1)$. We identify five equivalence classes according to the above equivalence relation. Interestingly, the classification is counter-intuitive in that powerful models of communication are grouped with weak ones, and seemingly weaker models end up on the top of the hierarchy.

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arXiv2023

Distributed Algorithms for Large-Scale Graphs

Motivated by the increasing need for fast processing of large-scale graphs, we study a number of fundamental graph problems in a message-passing model for distributed computing, called $k$-machine model, where we have $k$ machines that jointly perform computations on $n$-node graphs. The graph is assumed to be partitioned in a balanced fashion among the $k$ machines, a common implementation in many real-world systems. Communication is point-to-point via bandwidth-constrained links, and the goal is to minimize the round complexity, i.e., the number of communication rounds required to finish a computation. We present a generic methodology that allows to obtain efficient algorithms in the $k$-machine model using distributed algorithms for the classical CONGEST model of distributed computing. Using this methodology, we obtain algorithms for various fundamental graph problems such as connectivity, minimum spanning trees, shortest paths, maximal independent sets, and finding subgraphs, showing that many of these problems can be solved in $\tilde{O}(n/k)$ rounds; this shows that one can achieve speedup nearly linear in $k$. To complement our upper bounds, we present lower bounds on the round complexity that quantify the fundamental limitations of solving graph problems distributively. We first show a lower bound of $Ω(n/k)$ rounds for computing a spanning tree of the input graph. This result implies the same bound for other fundamental problems such as computing a minimum spanning tree, breadth-first tree, or shortest paths tree. We also show a $\tilde Ω(n/k^2)$ lower bound for connectivity, spanning tree verification and other related problems. The latter lower bounds follow from the development and application of novel results in a random-partition variant of the classical communication complexity model.

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arXiv2019

The Power of One Clean Qubit in Communication Complexity

We study quantum communication protocols, in which the players' storage starts out in a state where one qubit is in a pure state, and all other qubits are totally mixed (i.e. in a random state), and no other storage is available (for messages or internal computations). This restriction on the available quantum memory has been studied extensively in the model of quantum circuits, and it is known that classically simulating quantum circuits operating on such memory is hard when the additive error of the simulation is exponentially small (in the input length), under the assumption that the polynomial hierarchy does not collapse. We study this setting in communication complexity. The goal is to consider larger additive error for simulation-hardness results, and to not use unproven assumptions. We define a complexity measure for this model that takes into account that standard error reduction techniques do not work here. We define a clocked and a semi-unclocked model, and describe efficient simulations between those. We characterize a one-way communication version of the model in terms of weakly unbounded error communication complexity. Our main result is that there is a quantum protocol using one clean qubit only and using $O(\log n)$ qubits of communication, such that any classical protocol simulating the acceptance behaviour of the quantum protocol within additive error $1/poly(n)$ needs communication $Ω(n)$. We also describe a candidate problem, for which an exponential gap between the one-clean-qubit communication complexity and the randomized complexity is likely to hold, and hence a classical simulation of the one-clean-qubit model within {\em constant} additive error might be hard in communication complexity. We describe a geometrical conjecture that implies the lower bound.

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arXiv2019

The aBc Problem and Equator Sampling Renyi Divergences

We investigate the problem of approximating the product $a^TBc$, where $a,c\in S^{n-1}$ and $B\in O_n$, in models of communication complexity and streaming algorithms. The worst meaningful approximation is to simply decide whether the product is 1 or -1, given the promise that it is either. We call that problem the aBc problem. This is a modification of computing approximate inner products, by allowing a basis change. While very efficient streaming algorithms and one-way communication protocols are known for simple inner products (approximating $a^Tc$) we show that no efficient one-way protocols/streaming algorithms exist for the aBc problem. In communication complexity we consider the 3-player number-in-hand model. 1) In communication complexity $a^TBc$ can be approximated within additive error $ε$ with communication $O(\sqrt n/ε^2)$ by a one-way protocol Charlie to Bob to Alice. 2) The $aBc$ problem has a streaming algorithm that uses space $O(\sqrt n \log n)$. 3) Any one-way communication protocol for $aBc$ needs communication at least $Ω(n^{1/3})$, and we prove a tight results regarding a communication tradeoff: if Charlie and Bob communicate over many rounds such that Charlie communicates $o(n^{2/3})$ and Bob $o(n^{1/3})$, and then the transcript is sent to Alice, the error will be large. 4) To establish our lower bound we show concentration results for Renyi divergences under the event of restricting a density function on the sphere to a random equator and subsequently normalizing the restricted density function. This extends previous results by Klartag and Regev for set sizes to Renyi divergences of arbitrary density functions. 5) We show a strong concentration result for conditional Renyi divergences on bipartite systems for all $α>1$.

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arXiv2017

Quadratically Tight Relations for Randomized Query Complexity

Let $f:\{0,1\}^n \rightarrow \{0,1\}$ be a Boolean function. The certificate complexity $C(f)$ is a complexity measure that is quadratically tight for the zero-error randomized query complexity $R_0(f)$: $C(f) \leq R_0(f) \leq C(f)^2$. In this paper we study a new complexity measure that we call expectational certificate complexity $EC(f)$, which is also a quadratically tight bound on $R_0(f)$: $EC(f) \leq R_0(f) = O(EC(f)^2)$. We prove that $EC(f) \leq C(f) \leq EC(f)^2$ and show that there is a quadratic separation between the two, thus $EC(f)$ gives a tighter upper bound for $R_0(f)$. The measure is also related to the fractional certificate complexity $FC(f)$ as follows: $FC(f) \leq EC(f) = O(FC(f)^{3/2})$. This also connects to an open question by Aaronson whether $FC(f)$ is a quadratically tight bound for $R_0(f)$, as $EC(f)$ is in fact a relaxation of $FC(f)$. In the second part of the work, we upper bound the distributed query complexity $D^μ_ε(f)$ for product distributions $μ$ by the square of the query corruption bound ($\mathrm{corr}_ε(f)$) which improves upon a result of Harsha, Jain and Radhakrishnan [2015]. A similar statement for communication complexity is open.

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arXiv2015

Equality, Revisited

We develop a new lower bound method for analysing the complexity of the Equality function (EQ) in the Simultaneous Message Passing (SMP) model of communication complexity. The new technique gives tight lower bounds of $Ω(\sqrt n)$ for both EQ and its negation NE in the non-deterministic version of quantum-classical SMP, where Merlin is also quantum $-$ this is the strongest known version of SMP where the complexity of both EQ and NE remain high (previously known techniques seem to be insufficient for this). Besides, our analysis provides to a unified view of the communication complexity of EQ and NE, allowing to obtain tight characterisation in all previously studied and a few newly introduced versions of SMP, including all possible combination of either quantum or randomised Alice, Bob and Merlin in the non-deterministic case. Some of our results highlight that NE is easier than EQ in the presence of classical proofs, whereas the problems have (roughly) the same complexity when a quantum proof is present.

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arXiv2015

Correlation in Hard Distributions in Communication Complexity

We study the effect that the amount of correlation in a bipartite distribution has on the communication complexity of a problem under that distribution. We introduce a new family of complexity measures that interpolates between the two previously studied extreme cases: the (standard) randomised communication complexity and the case of distributional complexity under product distributions. We give a tight characterisation of the randomised complexity of Disjointness under distributions with mutual information $k$, showing that it is $Θ(\sqrt{n(k+1)})$ for all $0\leq k\leq n$. This smoothly interpolates between the lower bounds of Babai, Frankl and Simon for the product distribution case ($k=0$), and the bound of Razborov for the randomised case. The upper bounds improve and generalise what was known for product distributions, and imply that any tight bound for Disjointness needs $Ω(n)$ bits of mutual information in the corresponding distribution. We study the same question in the distributional quantum setting, and show a lower bound of $Ω((n(k+1))^{1/4})$, and an upper bound, matching up to a logarithmic factor. We show that there are total Boolean functions $f_d$ on $2n$ inputs that have distributional communication complexity $O(\log n)$ under all distributions of information up to $o(n)$, while the (interactive) distributional complexity maximised over all distributions is $Θ(\log d)$ for $6n\leq d\leq 2^{n/100}$. We show that in the setting of one-way communication under product distributions, the dependence of communication cost on the allowed error $ε$ is multiplicative in $\log(1/ε)$ -- the previous upper bounds had the dependence of more than $1/ε$.

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arXiv2014

New Bounds for the Garden-Hose Model

We show new results about the garden-hose model. Our main results include improved lower bounds based on non-deterministic communication complexity (leading to the previously unknown $Θ(n)$ bounds for Inner Product mod 2 and Disjointness), as well as an $O(n\cdot \log^3 n)$ upper bound for the Distributed Majority function (previously conjectured to have quadratic complexity). We show an efficient simulation of formulae made of AND, OR, XOR gates in the garden-hose model, which implies that lower bounds on the garden-hose complexity $GH(f)$ of the order $Ω(n^{2+ε})$ will be hard to obtain for explicit functions. Furthermore we study a time-bounded variant of the model, in which even modest savings in time can lead to exponential lower bounds on the size of garden-hose protocols.

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arXiv2014

Can Quantum Communication Speed Up Distributed Computation?

The focus of this paper is on {\em quantum distributed} computation, where we investigate whether quantum communication can help in {\em speeding up} distributed network algorithms. Our main result is that for certain fundamental network problems such as minimum spanning tree, minimum cut, and shortest paths, quantum communication {\em does not} help in substantially speeding up distributed algorithms for these problems compared to the classical setting. In order to obtain this result, we extend the technique of Das Sarma et al. [SICOMP 2012] to obtain a uniform approach to prove non-trivial lower bounds for quantum distributed algorithms for several graph optimization (both exact and approximate versions) as well as verification problems, some of which are new even in the classical setting, e.g. tight randomized lower bounds for Hamiltonian cycle and spanning tree verification, answering an open problem of Das Sarma et al., and a lower bound in terms of the weight aspect ratio, matching the upper bounds of Elkin [STOC 2004]. Our approach introduces the {\em Server model} and {\em Quantum Simulation Theorem} which together provide a connection between distributed algorithms and communication complexity. The Server model is the standard two-party communication complexity model augmented with additional power; yet, most of the hardness in the two-party model is carried over to this new model. The Quantum Simulation Theorem carries this hardness further to quantum distributed computing. Our techniques, except the proof of the hardness in the Server model, require very little knowledge in quantum computing, and this can help overcoming a usual impediment in proving bounds on quantum distributed algorithms.

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arXiv2014

Two Results about Quantum Messages

We show two results about the relationship between quantum and classical messages. Our first contribution is to show how to replace a quantum message in a one-way communication protocol by a deterministic message, establishing that for all partial Boolean functions $f:\{0,1\}^n\times\{0,1\}^m\to\{0,1\}$ we have $D^{A\to B}(f)\leq O(Q^{A\to B,*}(f)\cdot m)$. This bound was previously known for total functions, while for partial functions this improves on results by Aaronson, in which either a log-factor on the right hand is present, or the left hand side is $R^{A\to B}(f)$, and in which also no entanglement is allowed. In our second contribution we investigate the power of quantum proofs over classical proofs. We give the first example of a scenario, where quantum proofs lead to exponential savings in computing a Boolean function. The previously only known separation between the power of quantum and classical proofs is in a setting where the input is also quantum. We exhibit a partial Boolean function $f$, such that there is a one-way quantum communication protocol receiving a quantum proof (i.e., a protocol of type QMA) that has cost $O(\log n)$ for $f$, whereas every one-way quantum protocol for $f$ receiving a classical proof (protocol of type QCMA) requires communication $Ω(\sqrt n/\log n)$.

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arXiv2014

An Improved Interactive Streaming Algorithm for the Distinct Elements Problem

The exact computation of the number of distinct elements (frequency moment $F_0$) is a fundamental problem in the study of data streaming algorithms. We denote the length of the stream by $n$ where each symbol is drawn from a universe of size $m$. While it is well known that the moments $F_0,F_1,F_2$ can be approximated by efficient streaming algorithms, it is easy to see that exact computation of $F_0,F_2$ requires space $Ω(m)$. In previous work, Cormode et al. therefore considered a model where the data stream is also processed by a powerful helper, who provides an interactive proof of the result. They gave such protocols with a polylogarithmic number of rounds of communication between helper and verifier for all functions in NC. This number of rounds $\left(O(\log^2 m) \;\text{in the case of} \;F_0 \right)$ can quickly make such protocols impractical. Cormode et al. also gave a protocol with $\log m +1$ rounds for the exact computation of $F_0$ where the space complexity is $O\left(\log m \log n+\log^2 m\right)$ but the total communication $O\left(\sqrt{n}\log m\left(\log n+ \log m \right)\right)$. They managed to give $\log m$ round protocols with $\operatorname{polylog}(m,n)$ complexity for many other interesting problems including $F_2$, Inner product, and Range-sum, but computing $F_0$ exactly with polylogarithmic space and communication and $O(\log m)$ rounds remained open. In this work, we give a streaming interactive protocol with $\log m$ rounds for exact computation of $F_0$ using $O\left(\log m \left(\,\log n + \log m \log\log m\,\right)\right)$ bits of space and the communication is $O\left( \log m \left(\,\log n +\log^3 m (\log\log m)^2 \,\right)\right)$. The update time of the verifier per symbol received is $O(\log^2 m)$.

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arXiv2012

Fooling One-Sided Quantum Protocols

We use the venerable "fooling set" method to prove new lower bounds on the quantum communication complexity of various functions. Let f:X x Y-->{0,1} be a Boolean function, fool^1(f) its maximal fooling set size among 1-inputs, Q_1^*(f) its one-sided error quantum communication complexity with prior entanglement, and NQ(f) its nondeterministic quantum communication complexity (without prior entanglement; this model is trivial with shared randomness or entanglement). Our main results are the following, where logs are to base 2: * If the maximal fooling set is "upper triangular" (which is for instance the case for the equality, disjointness, and greater-than functions), then we have Q_1^*(f)>=(1/2)log fool^1(f) - 1/2, which is essentially optimal by superdense coding. No super-constant lower bound for equality seems to follow from earlier techniques. * For all f we have Q_1^*(f)>=(1/4)log fool^1(f) - 1/2, which is optimal up to a factor of 2. * NQ(f)>=log \fool^1(f)/2 + 1. We do not know if the factor 1/2 is needed in this result, but it cannot be replaced by 1: we give an example where NQ(f)~0.613 log fool^1(f).

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arXiv2012

New bounds on the classical and quantum communication complexity of some graph properties

We study the communication complexity of a number of graph properties where the edges of the graph $G$ are distributed between Alice and Bob (i.e., each receives some of the edges as input). Our main results are: * An Omega(n) lower bound on the quantum communication complexity of deciding whether an n-vertex graph G is connected, nearly matching the trivial classical upper bound of O(n log n) bits of communication. * A deterministic upper bound of O(n^{3/2}log n) bits for deciding if a bipartite graph contains a perfect matching, and a quantum lower bound of Omega(n) for this problem. * A Theta(n^2) bound for the randomized communication complexity of deciding if a graph has an Eulerian tour, and a Theta(n^{3/2}) bound for the quantum communication complexity of this problem. The first two quantum lower bounds are obtained by exhibiting a reduction from the n-bit Inner Product problem to these graph problems, which solves an open question of Babai, Frankl and Simon. The third quantum lower bound comes from recent results about the quantum communication complexity of composed functions. We also obtain essentially tight bounds for the quantum communication complexity of a few other problems, such as deciding if G is triangle-free, or if G is bipartite, as well as computing the determinant of a distributed matrix.

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arXiv2011

On Arthur Merlin Games in Communication Complexity

We show several results related to interactive proof modes of communication complexity. First we show lower bounds for the QMA-communication complexity of the functions Inner Product and Disjointness. We describe a general method to prove lower bounds for QMA-communication complexity, and show how one can 'transfer' hardness under an analogous measure in the query complexity model to the communication model using Sherstov's pattern matrix method. Combining a result by Vereshchagin and the pattern matrix method we find a communication problem with AM-communication complexity $O(\log n)$, PP-communication complexity $Ω(n^{1/3})$, and QMA-communication complexity $Ω(n^{1/6})$. Hence in the world of communication complexity noninteractive quantum proof systems are not able to efficiently simulate co-nondeterminism or interaction. These results imply that the related questions in Turing machine complexity theory cannot be resolved by 'algebrizing' techniques. Finally we show that in MA-protocols there is an exponential gap between one-way protocols and two-way protocols (this refers to the interaction between Alice and Bob). This is in contrast to nondeterministic, AM-, and QMA-protocols, where one-way communication is essentially optimal.

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arXiv2010

A Strong Direct Product Theorem for Disjointness

A strong direct product theorem states that if we want to compute $k$ independent instances of a function, using less than $k$ times the resources needed for one instance, then the overall success probability will be exponentially small in $k$. We establish such a theorem for the randomized communication complexity of the Disjointness problem, i.e., with communication $const\cdot kn$ the success probability of solving $k$ instances of size $n$ can only be exponentially small in $k$. We show that this bound even holds for $AM$ communication protocols with limited ambiguity. This also implies a new lower bound for Disjointness in a restricted 3-player NOF protocol, and optimal communication-space tradeoffs for Boolean matrix product. Our main result follows from a solution to the dual of a linear programming problem, whose feasibility comes from a so-called Intersection Sampling Lemma that generalizes a result by Razborov.

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arXiv2010

Optimal Direct Sum Results for Deterministic and Randomized Decision Tree Complexity

A Direct Sum Theorem holds in a model of computation, when solving some k input instances together is k times as expensive as solving one. We show that Direct Sum Theorems hold in the models of deterministic and randomized decision trees for all relations. We also note that a near optimal Direct Sum Theorem holds for quantum decision trees for boolean functions.

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arXiv2009

The Partition Bound for Classical Communication Complexity and Query Complexity

We describe new lower bounds for randomized communication complexity and query complexity which we call the partition bounds. They are expressed as the optimum value of linear programs. For communication complexity we show that the partition bound is stronger than both the rectangle/corruption bound and the γ_2/generalized discrepancy bounds. In the model of query complexity we show that the partition bound is stronger than the approximate polynomial degree and classical adversary bounds. We also exhibit an example where the partition bound is quadratically larger than polynomial degree and classical adversary bounds.

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