In communication complexity the input of a function $f:X\times Y\rightarrow Z$ is distributed between two players Alice and Bob. If Alice knows only $x\in X$ and Bob only $y\in Y$, how much information must Alice and Bob share to be able to elicit the value of $f(x,y)$? Do we need $\ell$ more resources to solve $\ell$ instances of a problem? This question is the direct sum question and has been studied in many computational models. In this paper we focus on the case of 2-party deterministic communication complexity and give a counterexample to the direct sum conjecture in its strongest form. To do so we exhibit a family of functions for which the complexity of solving $\ell$ instances is less than $(1 -ε)\ell$ times the complexity of solving one instance for some small enough $ε>0$. We use a customised method in the analysis of our family of total functions, showing that one can force the alternation of rounds between players. This idea allows us to exploit the integrality of the complexity measure to create an increasing gap between the complexity of solving the instances independently with that of solving them together.
In communication complexity the input of a function f:X×Y→Z is distributed between two players Alice and Bob. If Alice knows only x∈X and Bob only y∈Y, how much information must Alice and Bob share to be able to elicit the value of f(x,y)? Do we need ℓ more resources to solve ℓ instances of a problem? This question is the direct sum question and has been studied in many computational models. In this paper we focus on the case of 2-party deterministic communication complexity and give a counterexample to the direct sum conjecture in its strongest form. To do so we exhibit a family of functions for which the complexity of solving ℓ instances is less than (1−ε)ℓ times the complexity of solving one instance for some small enough ε>0. We use a customised method in the analysis of our family of total functions, showing that one can force the alternation of rounds between players. This idea allows us to exploit the integrality of the complexity measure to create an increasing gap between the complexity of solving the instances independently with that of solving them together.
We study the communication complexity of computing functions F: {0,1}ⁿ × {0,1}ⁿ → {0,1} in the memoryless communication model. Here, Alice is given x ∈ {0,1}ⁿ, Bob is given y ∈ {0,1}ⁿ and their goal is to compute F(x,y) subject to the following constraint: at every round, Alice receives a message from Bob and her reply to Bob solely depends on the message received and her input x (in particular, her reply is independent of the information from the previous rounds); the same applies to Bob. The cost of computing F in this model is the maximum number of bits exchanged in any round between Alice and Bob (on the worst case input x,y). In this paper, we also consider variants of our memoryless model wherein one party is allowed to have memory, the parties are allowed to communicate quantum bits, only one player is allowed to send messages. We show that some of these different variants of our memoryless communication model capture the garden-hose model of computation by Buhrman et al. (ITCS'13), space-bounded communication complexity by Brody et al. (ITCS'13) and the overlay communication complexity by Papakonstantinou et al. (CCC'14). Thus the memoryless communication complexity model provides a unified framework to study all these space-bounded communication complexity models. We establish the following main results: (1) We show that the memoryless communication complexity of F equals the logarithm of the size of the smallest bipartite branching program computing F (up to a factor 2); (2) We show that memoryless communication complexity equals garden-hose model of computation; (3) We exhibit various exponential separations between these memoryless communication models. We end with an intriguing open question: can we find an explicit function F and universal constant c > 1 for which the memoryless communication complexity is at least c log n? Note that c ≥ 2+ε would imply a Ω(n^{2+ε}) lower bound for general formula size, improving upon the best lower bound by [Nečiporuk, 1966].
We study the communication complexity of computing functions F:{0,1}n×{0,1}n→{0,1} in the memoryless communication model. Here, Alice is given x∈{0,1}n, Bob is given y∈{0,1}n and their goal is to compute F(x,y) subject to the following constraint: at every round, Alice receives a message from Bob and her reply to Bob solely depends on the message received and her input x; the same applies to Bob. The cost of computing F in this model is the maximum number of bits exchanged in any round between Alice and Bob (on the worst case input x,y). In this paper, we also consider variants of our memoryless model wherein one party is allowed to have memory, the parties are allowed to communicate quantum bits, only one player is allowed to send messages. We show that our memoryless communication model capture the garden-hose model of computation by Buhrman et al. (ITCS'13), space bounded communication complexity by Brody et al. (ITCS'13) and the overlay communication complexity by Papakonstantinou et al. (CCC'14). Thus the memoryless communication complexity model provides a unified framework to study space-bounded communication models. We establish the following: (1) We show that the memoryless communication complexity of F equals the logarithm of the size of the smallest bipartite branching program computing F (up to a factor 2); (2) We show that memoryless communication complexity equals garden-hose complexity; (3) We exhibit various exponential separations between these memoryless communication models. We end with an intriguing open question: can we find an explicit function F and universal constant c>1 for which the memoryless communication complexity is at least clogn? Note that c≥2+ε would imply a Ω(n2+ε) lower bound for general formula size, improving upon the best lower bound by Nečiporuk in 1966.
A fundamental question in computer science is: Is it harder to solve n instances independently than to solve them simultaneously? This question, known as the direct sum question or direct sum theorem, has been paid much attention in several research fields. Despite its importance, however, little has been discovered in many other research fields. In this paper, we introduce a novel framework that extends to classical/quantum query complexity, PAC-learning for machine learning, statistical estimation theory, and more. Within this framework, we establish several fundamental direct sum theorems. The main contributions of this paper include: (i) establishing a complete characterization of the amortized query/oracle complexities, and (ii) proving tight direct sum theorems when the error is small. Note that in our framework, every oracle access needs to be performed \emph{classically} even in the quantum setting. This can be thought of one limitation of this work. As a direct consequence of our results, we obtain the following: (A) The first known asymptotic separation of the randomized query complexity. Specifically, we show that there is a function f:{0,1}k→{0,1} and small error ε>0 such that solving n instances simultaneously requires the query complexity O~(nk) but solving one instance with the same error has the complexity Ω~(k). In communication complexity this type of separation was previously given in~Feder, Kushilevitz, Naor and Nisan (1995). (B) The query complexity counterpart of the ``information = amortized communication" relation, one of the most influential results in communication complexity shown by Braverman and Rao (2011) and further investigated by Braverman (2015). We hope that our results will provide further interesting applications in the future.
Communication complexity is the amount of communication needed to compute a function when the function inputs are distributed over multiple parties. In its simplest form, one-way communication complexity, Alice and Bob compute a function f(x,y) , where x is given to Alice and y is given to Bob, and only one message from Alice to Bob is allowed. A fundamental question in quantum information is the relationship between one-way quantum and classical communication complexities, i.e., how much shorter the message can be if Alice is sending a quantum state instead of bit strings? We make some progress towards this question with the following results.Let f:X×Y→Z∪{⊥} be a partial function and μ be a distribution with support contained in f−1(Z) . Denote d=|Z| . Let Rϵ1,μ(f) be the classical one-way communication complexity of f ; Qϵ1,μ(f) be the quantum one-way communication complexity of f and Qϵ1,μ,∗(f) be the entanglement-assisted quantum one-way communication complexity of f , each with distributional error (average error over μ ) at most ϵ . We show:1) If μ is a product distribution, η>0 and 0≤ϵ≤1−1/d , then, R2ϵ−dϵ2/(d−1)+η1,μ(f)≤2Qϵ1,μ,∗(f)+O(loglog(1/η)). 2)If μ is a non-product distribution and Z={0,1} , then ∀ϵ,η>0 such that ϵ/η+η<0.5 , R3η1,μ(f)=O(Qϵ1,μ(f)⋅CS(f)/η3), where CS(f)=maxyminz&am
We study the role of interaction in the Common Randomness Generation (CRG) and Secret Key Generation (SKG) problems. In the CRG problem, two players, Alice and Bob, respectively get samples X1,X2,… and Y1,Y2,… with the pairs (X1,Y1), (X2,Y2), … being drawn independently from some known probability distribution μ. They wish to communicate so as to agree on L bits of randomness. The SKG problem is the restriction of the CRG problem to the case where the key is required to be close to random even to an eavesdropper who can listen to their communication (but does not have access to the inputs of Alice and Bob). In this work, we study the relationship between the amount of communication and the number of rounds of interaction in both the CRG and the SKG problems. Specifically, we construct a family of distributions μ=μr,n,L, parametrized by integers r, n and L, such that for every r there exists a constant b=b(r) for which CRG (respectively SKG) is feasible when (Xi,Yi)∼μr,n,L with r+1 rounds of communication, each consisting of O(logn) bits, but when restricted to r/2−3 rounds of interaction, the total communication must exceed Ω(n/logb(n)) bits. Prior to our work no separations were known for r≥2.
We study a new type of separations between quantum and classical communication complexity, separations that are obtained using quantum protocols where all parties are efficient , in the sense that they can be implemented by small quantum circuits, with oracle access to their inputs. Our main result qualitatively matches the strongest known separation between quantum and classical communication complexity Gavinsky (2016) and is obtained using a quantum protocol where all parties are efficient. More precisely, we give an explicit partial Boolean function f over inputs of length N , such that: f can be computed by a simultaneous-message quantum protocol with communication complexity polylog( N ) (where at the beginning of the protocol Alice and Bob also have polylog( N ) entangled EPR pairs). Any classical randomized protocol for f , with any number of rounds, has communication complexity at least Ω~(N1/4) . All parties in the quantum protocol of Item (1) (Alice, Bob and the referee) can be implemented by quantum circuits of size polylog( N ) (where Alice and Bob have oracle access to their inputs). Items (1), (2) qualitatively match the strongest known separation between quantum and classical communication complexity, proved by Gavinsky (2016). Item (3) is new. (Our result is incomparable to the one of Gavinsky. While he obtained a quantitatively better lower bound of Ω(N1/2) in the classical case, the referee in his quantum protocol is inefficient). Exponential separations of quantum and classical communication complexity have been studied in numerous previous works, but to the best of our knowledge the efficiency of the parties in the quantum protocol has not been addressed, and in most previous separations the quantum parties seem to be inefficient. The only separations that we know of that have efficient quantum parties are the recent separations that are based on lifting Göös et al. (2017), Chattopadhyay et al. (2019a). However, these separations seem to require quantum protocols with at least two rounds of communication, so they imply a separation of two-way quantum and classical communication complexity, but they do not give the stronger separations of simultaneous-message quantum communication complexity vs. two-way classical communication complexity (or even one-way quantum communication complexity vs. two-way classical communication complexity). Our proof technique is completely new, in the context of communication complexity, and is based on techniques from Raz & Tal (2019). Our function f is based on a lift of the forrelation problem, using xor as a gadget.