Dual polynomials and communication complexity of XOR functions.

Yang Liu, Shengyu Zhang

Ming Lam Leung, Yang Li, Shengyu Zhang

Vladimir V. Podolskii, Sluch, Dmitrii

Boolean function $F(x,y)$ for $x,y \in \{0,1\}^n$ is an XOR function if $F(x,y)=f(x\oplus y)$ for some function $f$ on $n$ input bits, where $\oplus$ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing Fourier analytic technique. For total XOR functions it is known that deterministic communication complexity of $F$ is closely related to parity decision tree complexity of $f$. Montanaro and Osbourne (2009) observed that one-sided communication complexity $D_{cc}^{\rightarrow}(F)$ of $F$ is exactly equal to nonadaptive parity decision tree complexity $NADT^{\oplus}(f)$ of $f$. Hatami et al. (2018) showed that unrestricted communication complexity of $F$ is polynomially related to parity decision tree complexity of $f$. We initiate the studies of a similar connection for partial functions. We show that in case of one-sided communication complexity whether these measures are equal, depends on the number of undefined inputs of $f$. On the one hand, if $D_{cc}^{\rightarrow}(F)=t$ and $f$ is undefined on at most $O(\frac{2^{n-t}}{\sqrt{n-t}})$, then $NADT^{\oplus}(f)=t$. On the other hand, for a wide range of values of $D_{cc}^{\rightarrow}(F)$ and $NADT^{\oplus}(f)$ (from constant to $n-2$) we provide partial functions for which $D_{cc}^{\rightarrow}(F) < NADT^{\oplus}(f)$. In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-sided communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on Boolean Fourier analysis and the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.

Vladimir V. Podolskii, Dmitrii Sluch

Boolean function F(x,y) for x,y ∈ {0,1}ⁿ is an XOR function if F(x,y) = f(x⊕ y) for some function f on n input bits, where ⊕ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing the Fourier analytic technique. For total XOR functions, it is known that deterministic communication complexity of F is closely related to parity decision tree complexity of f. Montanaro and Osbourne (2009) observed that one-way communication complexity D_{cc}^{→}(F) of F is exactly equal to non-adaptive parity decision tree complexity NADT^{⊕}(f) of f. Hatami et al. (2018) showed that unrestricted communication complexity of F is polynomially related to parity decision tree complexity of f. We initiate the study of a similar connection for partial functions. We show that in the case of one-way communication complexity whether these measures are equal, depends on the number of undefined inputs of f. More precisely, if D_{cc}^{→}(F) = t and f is undefined on at most O((2^{n-t})/(√{n-t})) inputs, then NADT^{⊕}(f) = t. We also provide stronger bounds in extreme cases of small and large complexity. We show that the restriction on the number of undefined inputs in these results is unavoidable. That is, for a wide range of values of D_{cc}^{→}(F) and NADT^{⊕}(f) (from constant to n-2) we provide partial functions (with more than Ω((2^{n-t})/(√{n-t})) undefined inputs, where t = D_{cc}^{→}) for which D_{cc}^{→}(F) < NADT^{⊕}(f). In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-way communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on the Boolean Fourier analysis and thus, the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.

Marius Junge, Carlos Palazuelos, Ignacio Villanueva

In this work we introduce an intermediate setting between quantum nonlocality and communication complexity problems. More precisely, we study the value of XOR games $G$ when Alice and Bob are allowed to use a limited amount of one-way classical communication $ω_{o.w.-c}(G)$ (resp. one-way quantum communication $ω_{o.w.-c}^*(G)$), where $c$ denotes the number of bits (resp. qubits). The key quantity here is the quotient $ω_{o.w.-c}^*(G)/ω_{o.w.-c}(G)$. We provide a universal way to obtain Bell inequality violations of general Bell functionals from XOR games for which the quotient $ω_{o.w.-c}^*(G)/ω_{o.w.-2c}(G)$ is larger than 1. This allows, in particular, to find (unbounded) Bell inequality violations from communication complexity problems in the same spirit as the recent work by Buhrman et al. (2016). We also provide an example of a XOR game for which the previous quotient is optimal (up to a logarithmic factor) in terms of the amount of information $c$. Interestingly, this game has only polynomially many inputs per player. For the related problem of separating the classical vs quantum communication complexity of a function, the known examples attaining exponential separation require exponentially many inputs per party.

Rafail Ostrovsky, William E. Skeith