Mohammad Bavarian

Tighter Relations between Sensitivity and Other Complexity Measures

The sensitivity conjecture of Nisan and Szegedy [12] asks whether the maximum sensitivity of a Boolean function is polynomially related to the other major complexity measures of Boolean functions. Despite major advances in analysis of Boolean functions in the past decade, the problem remains wide open with no positive result toward the conjecture since the work of Kenyon and Kutin from 2004 [11].

On the Role of Shared Randomness in Simultaneous Communication

Two parties wish to carry out certain distributed computational tasks, and they are given access to a source of correlated random bits. It allows the parties to act in a correlated manner, which can be quite useful. But what happens if the shared randomness is not perfect? In this work, we initiate the study of the power of different sources of shared randomness in communication complexity. This is done in the setting of simultaneous message passing (SMP) model of communication complexity, which is one of the most suitable models for studying the resource of shared randomness. Toward characterising the power of various sources of shared randomness, we introduce a measure for the quality of a source - we call it collision complexity. Our results show that the collision complexity tightly characterises the power of a (shared) randomness resource in the SMP model. Of independent interest is our demonstration that even the weakest sources of shared randomness can in some cases increase the power of SMP substantially: the equality function can be solved very efficiently with virtually any nontrivial shared randomness.

On the Sum of L1 Influences

For a function f over the discrete cube, the total L1 influence of f is defined as the sum of the L1 norm of the discrete derivatives of f in all n directions. In this work, we show that in the case of bounded functions this quantity can be upper bounded by a polynomial in the degree of f (independently of dimension n), resolving affirmatively an open problem of Aaronson and Ambainis (ITCS 2011). We also give an application of our theorem to graph theory, and discuss the connection between the study of bounded functions over the cube and the quantum query complexity of partial functions where Aaronson and Ambainis encountered this question.

Hardness amplification for entangled games via anchoring

We study the parallel repetition of one-round games involving players that can use quantum entanglement. A major open question in this area is whether parallel repetition reduces the entangled value of a game at an exponential rate - in other words, does an analogue of Razs parallel repetition theorem hold for games with players sharing quantum entanglement? Previous results only apply to special classes of games. We introduce a class of games we call anchored. We then introduce a simple transformation on games called anchoring, inspired in part by the Feige-Kilian transformation, that turns any (multiplayer) game into an anchored game. Unlike the Feige-Kilian transformation, our anchoring transformation is completeness preserving. We prove an exponential-decay parallel repetition theorem for anchored games that involve any number of entangled players. We also prove a threshold version of our parallel repetition theorem for anchored games. Together, our parallel repetition theorems and anchoring transformation provide the first hardness amplification techniques for general entangled games. We give an application to the games version of the Quantum PCP Conjecture.