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Register a new userThe communication complexity of XOR games via summing operators
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The discrepancy method is widely used to find lower bounds for communication complexity of XOR games. It is well known that these bounds can be far from optimal. In this context Disjointness is usually mentioned as a case where the method fails to give good bounds, because the increment of the value of the game is linear (rather than exponential) in the number of communicated bits. We show in this paper the existence of XOR games where the discrepancy method yields bounds as poor as one desires. Indeed, we show the existence of such games with any previously prescribed value. To prove this result we apply the theory of p-summing operators, a central topic in Banach space theory. We show in the paper other applications of this theory to the study of the communication complexity of XOR games.
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We show a new duality between the polynomial margin complexity of $f$ and the discrepancy of the function $f circ textsf{XOR}$, called an $textsf{XOR}$ function. Using this duality, we develop polynomial based techniques for understanding the bounded error ($textsf{BPP}$) and the weakly-unbounded error ($textsf{PP}$) communication complexities of $textsf{XOR}$ functions. We show the following. A weak form of an interesting conjecture of Zhang and Shi (Quantum Information and Computation, 2009) (The full conjecture has just been reported to be independently settled by Hatami and Qian (Arxiv, 2017). However, their techniques are quite different and are not known to yield many of the results we obtain here). Zhang and Shi assert that for symmetric functions $f : {0, 1}^n rightarrow {-1, 1}$, the weakly unbounded-error complexity of $f circ textsf{XOR}$ is essentially characterized by the number of points $i$ in the set ${0,1, dots,n-2}$ for which $D_f(i) eq D_f(i+2)$, where $D_f$ is the predicate corresponding to $f$. The number of such points is called the odd-even degree of $f$. We show that the $textsf{PP}$ complexity of $f circ textsf{XOR}$ is $Omega(k/ log(n/k))$. We resolve a conjecture of a different Zhang characterizing the Threshold of Parity circuit size of symmetric functions in terms of their odd-even degree. We obtain a new proof of the exponential separation between $textsf{PP}^{cc}$ and $textsf{UPP}^{cc}$ via an $textsf{XOR}$ function. We provide a characterization of the approximate spectral norm of symmetric functions, affirming a conjecture of Ada et al. (APPROX-RANDOM, 2012) which has several consequences. Additionally, we prove strong $textsf{UPP}$ lower bounds for $f circ textsf{XOR}$, when $f$ is symmetric and periodic with period $O(n^{1/2-epsilon})$, for any constant $epsilon > 0$.
We study the communication complexity of computing functions $F:{0,1}^ntimes {0,1}^n rightarrow {0,1}$ in the memoryless communication model. Here, Alice is given $xin {0,1}^n$, Bob is given $yin {0,1}^n$ 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. (ITCS13), space bounded communication complexity by Brody et al. (ITCS13) and the overlay communication complexity by Papakonstantinou et al. (CCC14). 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 $c log n$? Note that $cgeq 2+varepsilon$ would imply a $Omega(n^{2+varepsilon})$ lower bound for general formula size, improving upon the best lower bound by Nev{c}iporuk in 1966.
In this note, we study the relation between the parity decision tree complexity of a boolean function $f$, denoted by $mathrm{D}_{oplus}(f)$, and the $k$-party number-in-hand multiparty communication complexity of the XOR functions $F(x_1,ldots, x_k)= f(x_1opluscdotsoplus x_k)$, denoted by $mathrm{CC}^{(k)}(F)$. It is known that $mathrm{CC}^{(k)}(F)leq kcdotmathrm{D}_{oplus}(f)$ because the players can simulate the parity decision tree that computes $f$. In this note, we show that [mathrm{D}_{oplus}(f)leq Obig(mathrm{CC}^{(4)}(F)^5big).] Our main tool is a recent result from additive combinatorics due to Sanders. As $mathrm{CC}^{(k)}(F)$ is non-decreasing as $k$ grows, the parity decision tree complexity of $f$ and the communication complexity of the corresponding $k$-argument XOR functions are polynomially equivalent whenever $kgeq 4$. Remark: After the first version of this paper was finished, we discovered that Hatami and Lovett had already discovered the same result a few years ago, without writing it up.
The classical communication complexity of testing closeness of discrete distributions has recently been studied by Andoni, Malkin and Nosatzki (ICALP19). In this problem, two players each receive $t$ samples from one distribution over $[n]$, and the goal is to decide whether their two distributions are equal, or are $epsilon$-far apart in the $l_1$-distance. In the present paper we show that the quantum communication complexity of this problem is $tilde{O}(n/(tepsilon^2))$ qubits when the distributions have low $l_2$-norm, which gives a quadratic improvement over the classical communication complexity obtained by Andoni, Malkin and Nosatzki. We also obtain a matching lower bound by using the pattern matrix method. Let us stress that the samples received by each of the parties are classical, and it is only communication between them that is quantum. Our results thus give one setting where quantum protocols overcome classical protocols for a testing problem with purely classical samples.
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 $Theta(sqrt{n(k+1)})$ for all $0leq kleq n$. 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 $Omega(n)$ bits of mutual information in the corresponding distribution. We study the same question in the distributional quantum setting, and show a lower bound of $Omega((n(k+1))^{1/4})$, 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 $Theta(log d)$ for $6nleq dleq 2^{n/100}$. We show that in the setting of one-way communication under product distributions, the dependence of communication cost on the allowed error $epsilon$ is multiplicative in $log(1/epsilon)$ -- the previous upper bounds had the dependence of more than $1/epsilon$.
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