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Register a new userNondeterministic Communication Complexity with Help and Graph Functions
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Added by
Adi Shraibman
Publication date
2017
fields
Informatics Engineering
and research's language is
English
Authors
Adi Shraibman
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We define nondeterministic communication complexity in the model of communication complexity with help of Babai, Hayes and Kimmel. We use it to prove logarithmic lower bounds on the NOF communication complexity of explicit graph functions, which are complementary to the bounds proved by Beame, David, Pitassi and Woelfel.
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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 $epsilon > 0$.
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.
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.
Help bits are some limited trusted information about an instance or instances of a computational problem that may reduce the computational complexity of solving that instance or instances. In this paper, we study the value of help bits in the settings of randomized and average-case complexity. Amir, Beigel, and Gasarch (1990) show that for constant $k$, if $k$ instances of a decision problem can be efficiently solved using less than $k$ bits of help, then the problem is in P/poly. We extend this result to the setting of randomized computation: We show that the decision problem is in P/poly if using $ell$ help bits, $k$ instances of the problem can be efficiently solved with probability greater than $2^{ell-k}$. The same result holds if using less than $k(1 - h(alpha))$ help bits (where $h(cdot)$ is the binary entropy function), we can efficiently solve $(1-alpha)$ fraction of the instances correctly with non-vanishing probability. We also extend these two results to non-constant but logarithmic $k$. In this case however, instead of showing that the problem is in P/poly we show that it satisfies $k$-membership comparability, a notion known to be related to solving $k$ instances using less than $k$ bits of help. Next we consider the setting of average-case complexity: Assume that we can solve $k$ instances of a decision problem using some help bits whose entropy is less than $k$ when the $k$ instances are drawn independently from a particular distribution. Then we can efficiently solve an instance drawn from that distribution with probability better than $1/2$. Finally, we show that in the case where $k$ is super-logarithmic, assuming $k$-membership comparability of a decision problem, one cannot prove that the problem is in P/poly by a black-box proof.
In this work we introduce, both for classical communication complexity and query complexity, a modification of the partition bound introduced by Jain and Klauck [2010]. We call it the public-coin partition bound. We show that (the logarithm to the base two of) its communication complexity and query complexi
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