Separation of the Factorization Norm and Randomized Communication Complexity

While exponential separations are known between quantum and randomized communication complexity for partial functions (Raz, STOC 1999), the best known separation between these measures for a total function is quadratic, witnessed by the disjointness function. We give the first super-quadratic separation between quantum and randomized communication complexity for a total function, giving an example exhibiting a power 2.5 gap. We further present a 1.5 power separation between exact quantum and randomized communication complexity, improving on the previous ~1.15 separation by Ambainis (STOC 2013). Finally, we present a nearly optimal quadratic separation between randomized communication complexity and the logarithm of the partition number, improving upon the previous best power 1.5 separation due to G\"o\"os, Jayram, Pitassi, and Watson. Our results are the communication analogues of separations in query complexity proved using the recent cheat sheet framework of Aaronson, Ben-David, and Kothari (STOC 2016). Our main technical results are randomized communication and information complexity lower bounds for a family of functions, called lookup functions, that generalize and port the cheat sheet framework to communication complexity.

The disjointness problem - where Alice and Bob are given two subsets of {1, … , 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 Θ(n), it is also known that if the sets are assumed to be drawn from some restricted set systems then the communication complexity can be much lower. In this work, we explore how communication complexity measures change with respect to the complexity of the underlying set system. The complexity measure for the set system that we use in this work is the Vapnik–Chervonenkis (VC) dimension. More precisely, on any set system with VC dimension bounded by d, we analyze how large can the deterministic and randomized communication complexities be, as a function of d and n. The d-sparse set disjointness problem, where the sets have size at most d, is one such set system with VC dimension d. The deterministic and the randomized communication complexities of the d-sparse set disjointness problem have been well studied and is known to be Θ (d log ({n}/{d})) and Θ(d), respectively, in the multi-round communication setting. In this paper, we address the question of whether the randomized communication complexity is always upper bounded by a function of the VC dimension of the set system, and does there always exist a gap between the deterministic and randomized communication complexity for set systems with small VC dimension. In this paper, we construct two natural set systems of VC dimension d, motivated from geometry. Using these set systems we show that the deterministic and randomized communication complexity can be Θ(dlog (n/d)) for set systems of VC dimension d and this matches the deterministic upper bound for all set systems of VC dimension d. We also study the deterministic and randomized communication complexities of the set intersection problem when sets belong to a set system of bounded VC dimension. We show that there exists set systems of VC dimension d such that both deterministic and randomized (one-way and multi-round) complexities for the set intersection problem can be as high as Θ(dlog (n/d)), and this is tight among all set systems of VC dimension d.

We investigate the power of the most important lower bound technique in randomized communication complexity, which is based on an evaluation of the maximal size of approximately monochromatic rectangles, with respect to arbitrary distributions on the inputs. While it is known that the 0-error version of this bound is polynomially tight for deterministic communication, nothing in this direction is known for constant error and randomized communication complexity. We first study a one-sided version of this bound and obtain that its value lies between the MA- and AM- complexities of the considered function. Hence the lower bound actually works for a (communication) complexity class between MA/spl cap/co - MA and AM/spl cap/co - AM, and allows to show that the MA-complexity of the disjointness problem is /spl Omega/(/spl radic/n). Following this we consider the conjecture that the lower bound method is polynomially tight for randomized communication complexity. First we disprove a distributional version of this conjecture. Then we give a combinatorial characterization of the value of the lower bound method, in which the optimization over all distributions is absent. This characterization is done by what we call a bounded error uniform threshold cover, and reduces showing tightness of the bound to the construction of an efficient protocol for a specific communication problem. We then study relaxations of bounded error uniform threshold covers, namely approximate majority covers and majority covers, and exhibit exponential separations between them. Each of these covers captures a lower bound method previously used for randomized communication complexity.

The EQUALITY problem is usually one’s first encounter with communication complexity and is one of the most fundamental problems in the field. Although its deterministic and randomized communication complexity were settled decades ago, we find several new things to say about the problem by focusing on three subtle aspects. The first is to consider the expected communication cost (at a worst-case input) for a protocol that uses limited interaction—i.e., a bounded number of rounds of communication—and whose error probability is zero or close to it. The second is to treat the false negative error rate separately from the false positive error rate. The third is to consider the information cost of such protocols. We obtain asymptotically optimal rounds-versus-cost tradeoffs for EQUALITY: both expected communication cost and information cost scale as Theta(log log ... log n), with r-1 logs, where r is the number of rounds. These bounds hold even when the false negative rate approaches 1. For the case of zero-error communication cost, we obtain essentially matching bounds, up to a tiny additive constant. We also provide some applications.

A counter-example to the probabilistic universal graph conjecture via randomized communication complexity

In this work we study the direct-sum problem with respect to communication complexity: Consider a relation f defined over $\{0,1\}^{n} \times \{0,1\}^{n}$. Can the communication complexity of simultaneously computing f on $\ell $ instances $(x_{1}, y_{1}), \dotsc , (x_{\ell}, y_{\ell})$ be smaller than the communication complexity of separately computing f on the $\ell $ instances? Let the amortized communication complexity of f be the communication complexity of simultaneously computing f on $\ell $ instances divided by $\ell $. We study the properties of the amortized communication complexity. We show that the amortized communication complexity of a relation can be smaller than its communication complexity. More precisely, we present a partial function whose (deterministic) communication complexity is $\Theta (\log n)$ and amortized (deterministic) communication complexity is $O(1)$. Similarly, for randomized protocols we present a function whose randomized communication complexity is $\Theta (\log n)$ and amortized randomized communication complexity is $O(1)$. We also give a general lower bound on the amortized communication complexity of any functionf in terms of its communication complexity $C(f)$: for every function f the amortized communication complexity of f is $\Omega (\sqrt{C(f)} - \log n)$.

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 communication complexity is likely to hold, and hence a classical simulation of the one-clean-qubit model within constant additive error might be hard in communication complexity. We describe a geometrical conjecture that implies the lower bound.

There are very few computing models for which the power of randomized computing is as well understood as for communication protocols and their communication complexity. Since the communication complexity is strongly related to several complexity measures of distinct basic models of computation, there exist possibilities to transform some results about randomized communication protocols to other computing models, and so communication complexity has established itself as a powerful instrument for the study of randomization in complexity theory. The aim of this work is to survey the fundamental results about randomized communication complexity with the focus on the comparison of the efficiency of deterministic, nondeterministic and randomized communication.KeywordsRandomized computingcommunication complexitytwoparty protocols

Sign-rank and discrepancy are two central notions in communication complexity. The seminal work of Babai, Frankl, and Simon from 1986 initiated an active line of research that investigates the gap between these two notions. In this article, we establish the strongest possible separation by constructing a boolean matrix whose sign-rank is only 3, and yet its discrepancy is 2-Ω(n). We note that every matrix of sign-rank 2 has discrepancy n-O(1).Our result in particular implies that there are boolean functions with O(1) unbounded error randomized communication complexity while having Ω(n) weakly unbounded error randomized communication complexity.

We study the 2-party randomized communication complexity of read-once AC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> formulae. For balanced AND-OR trees T with n inputs and depth d, we show that the communication complexity of the function f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> (x, y) = T(x omicron y) is Omega(n/4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> ) where (x omicron y) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> is defined so that the resulting tree also has alternating levels of AND and OR gates. For each bit of x, y, the operation omicron is either AND or OR depending on the gate in T to which it is an input. Using this, we show that for general AND-OR trees T with n inputs and depth d, the communication complexity of f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> (x, y) is n/2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Omega(d</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">log</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d)</sup> . These results generalize classical results on the communication complexity of set-disjointness (where T is an OR -gate) and recent results on the communication complexity of the TRIBES functions (where T is a depth-2 read-once formula). Our techniques build on and extend the information complexity methodology for proving lower bounds on randomized communication complexity. Our analysis for trees of depth d proceeds in two steps: (1) reduction to measuring the information complexity of binary depth-d trees, and (2) proving lower bounds on the information complexity of binary trees. In order to execute this program, we carefully construct input distributions under which both these steps can be carried out simultaneously. We believe the tools we develop will prove useful in further studies of information complexity in particular, and communication complexity in general.

The disjointness problem—where Alice and Bob are given twosubsets of {1, dots, n} and they have to check if their setsintersect—is a central problem in the world of communicationcomplexity. While both deterministic and randomized communicationcomplexities for this problem are known to be Theta(n), it isalso known that if the sets are assumed to be drawn from somerestricted set systems then the communication complexity can be muchlower. In this work, we explore how communication complexitymeasures change with respect to the complexity of the underlying setsystem. The complexity measure for the set system that we use inthis work is the Vapnik—Chervonenkis (VC) dimension. Moreprecisely, on any set system with VC dimension bounded by d, weanalyze how large can the deterministic and randomized communicationcomplexities be, as a function of d and n. The d-sparse setdisjointness problem, where the sets have size at most d, is onesuch set system with VC dimension d. The deterministic and therandomized communication complexities of the d-sparse setdisjointness problem have been well studied and are known to beTheta left( d log left({n}/{d}right)right) and Theta(d),respectively, in the multi-round communication setting. In thispaper, we address the question of whether the randomizedcommunication complexity of the disjointness problem is always upperbounded by a function of the VC dimension of the set system, anddoes there always exist a gap between the deterministic andrandomized communication complexities of the disjointness problemfor set systems with small VC dimension.We construct two natural set systems of VC dimension d, motivated from geometry. Using these set systems, we show that the deterministic and randomized communication complexity can be widetilde{Theta}left(dlog left( n/d right)right) for set systems of VC dimension d and this matches the deterministic upper bound for all set systems of VC dimension d. We also study the deterministic and randomized communication complexities of the set intersection problem when sets belong to a set system of bounded VC dimension. We show that there exist set systems of VC dimension d such that both deterministic and randomized (one-way and multi-round) complexities for the set intersection problem can be as high as Thetaleft( dlog left( n/d right) right).

The fooling pairs method is one of the standard methods for proving lower bounds for deterministic two-player communication complexity. We study fooling pairs in the context of randomized communication complexity. We show that every fooling pair induces far away distributions on transcripts of private-coin protocols. We use the above to conclude that the private-coin randomized \(\varepsilon \)-error communication complexity of a function f with a fooling set \(\mathcal S\) is at least order \(\log \frac{\log |\mathcal S|}{\varepsilon }\). This relationship was earlier known to hold only for constant values of \(\varepsilon \). The bound we prove is tight, for example, for the equality and greater-than functions.

In a recent breakthrough result, Chattopadhyay, Mande and Sherif [ECCC TR18-17] showed an exponential separation between the log approximate rank and randomized communication complexity of a total function $f$, hence refuting the log approximate rank conjecture of Lee and Shraibman [2009]. We provide an alternate proof of their randomized communication complexity lower bound using the information complexity approach. Using the intuition developed there, we derive a polynomially-related quantum communication complexity lower bound using the quantum information complexity approach, thus providing an exponential separation between the log approximate rank and quantum communication complexity of $f$. Previously, the best known separation between these two measures was (almost) quadratic, due to Anshu, Ben-David, Garg, Jain, Kothari and Lee [CCC, 2017]. This settles one of the main question left open by Chattopadhyay, Mande and Sherif, and refutes the quantum log approximate rank conjecture of Lee and Shraibman [2009]. Along the way, we develop a Shearer-type protocol embedding for product input distributions that might be of independent interest.

We prove that several measures in communication complexity are equivalent, up to polynomial factors in the logarithm of the rank of the associated matrix: deterministic communication complexity, randomized communication complexity, information cost and zero-communication cost. This shows that in order to prove the log-rank conjecture, it suffices to show that low-rank matrices have efficient protocols in any of the aforementioned measures.

The equality problem is usually one's first encounter with communication complexity and is one of the most fundamental problems in the field. Although its deterministic and randomized communication complexity were settled decades ago, we find several new things to say about the problem by focusing on three subtle aspects. The first is to consider the expected communication cost (at a worst-case input) for a protocol that uses limited interaction--i.e., a bounded number of rounds of communication--and whose error probability is zero or close to it. The second is to treat the false negative error rate separately from the false positive error rate. The third is to consider the information cost of such protocols. We obtain asymptotically optimal rounds-versus-cost tradeoffs for equality: both expected communication complexity and information complexity scale as $$\Theta ({{\mathrm{ilog}}}^{r-1} n)$$ź(ilogr-1n), where r is the number of rounds and $${{\mathrm{ilog}}}^k n = \log \log \cdots \log n$$ilogkn=loglogźlogn, with k logs. These bounds hold even when the false negative rate approaches 1. For the case of zero-error communication cost, we obtain essentially matching bounds, up to a tiny additive constant. We also provide some applications. As an application of our information cost bounds, we obtain new bounded-round randomized lower bounds for the Intersection problem, in which there are two players who hold subsets $$S,T \subseteq [n]$$S,T⊆[n]. In many realistic scenarios, the sizes of S and T are significantly smaller than n, so we impose the constraint that $$|S|, |T| \le k$$|S|,|T|≤k. We study the minimum number of bits the parties need to communicate in order to compute the entire intersection set $$S \cap T$$SźT, using r rounds. We show that any r-round protocol has information cost (and thus communication cost) $$\Omega (k {{\mathrm{ilog}}}^r k)$$Ω(kilogrk) bits. We also give an O(r)-round protocol achieving $$O(k{{\mathrm{ilog}}}^r k)$$O(kilogrk) bits, which for $$r = \log ^* k$$r=logźk gives a protocol with O(k) bits of communication. This is in contrast to other basic problems such as computing the union or symmetric difference, for which $$\Omega (k\log (n/k))$$Ω(klog(n/k)) bits of communication is required for any number of rounds.