arXiv subjects

Farzan Byramji

Publications and source records attributed to Farzan Byramji.

6 recordsLinked to original sources

arXiv2026

Hard-to-Sample Distributions from Robust Extractors

We provide a unified method for constructing explicit distributions which are difficult for restricted models of computation to generate. Our constructions are based on a new notion of robust extractors, which are extractors that remain sound even when a small number of points violate the min-entropy constraint. Using such objects, we show that for a broad range of sampling models (e.g., low-depth circuits, small-space sources, etc.), every output of the model has distance $1 - o(1)$ from our target distribution, qualitatively recovering essentially all previously known hardness results. Our work extends that of Viola (SICOMP '14), who developed an earlier unified framework based on traditional extractors to rule out sampling with very small error. As a further application of our technique, we leverage a recent extractor construction of Chattopadhyay, Goodman, and Gurumukhani (ITCS '24) to present the first explicit distribution with distance $1 - o(1)$ from the output of any low-degree $\mathbb{F}_2$-polynomial source. We note that a similar bound was obtained concurrently and independently by Khodabandeh and Shinkar (ECCC '26). We also describe a potential avenue toward proving a similar hardness result for $\mathsf{AC^0}[\oplus]$ circuits.

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arXiv2026

On the Advantage of Adaptivity for Sampling with Cell Probes

We construct an explicit distribution $\mathbf{D}$ over $\{0,1\}^N$ that exhibits an essentially optimal separation between adaptive and non-adaptive cell-probe sampling. The distribution can be sampled exactly when each output bit is allowed two adaptive probes to an arbitrarily long sequence of independent uniform symbols from $[N]$. In contrast, any non-adaptive sampler requires $\widetildeΩ(N)$ non-adaptive cell probes to generate a distribution with total variation distance less than $1-o(1)$ from $\mathbf{D}$. This provides a $2$-vs-$\widetildeΩ(N)$ separation for sampling with adaptive versus non-adaptive cell probes, improving upon the $2$-vs-$\widetildeΩ(\log N)$ separation of Yu and Zhan (ITCS '24) and the $(\log N)^{O(1)}$-vs-$N^{Ω(1)}$ separation of Alekseev, Göös, Myasnikov, Riazanov, and Sokolov (STOC '26).

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arXiv2026

Quantum-Classical Equivalence for AND-Functions

A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for AND-functions of the form $$ F(x,y) = f(x_1 \land y_1, \ldots, x_n \land y_n), $$ when the outer function $f$ is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all AND-functions has remained open and has been posed by several authors. In this work, we settle this problem in a strong way. We show that for every Boolean function $f$, the bounded-error quantum and classical deterministic communication complexities of the function $f \circ \mathrm{AND}_2$ are polynomially related, up to polylogarithmic factors in $n$. We prove this by showing that both are characterized--up to polynomial loss--by the logarithm of the De Morgan sparsity of $f$. Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett (2025) on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general AND-functions.

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arXiv2025

Lower Bounds for Bit Pigeonhole Principles in Bounded-Depth Resolution over Parities

We prove lower bounds for proofs of the bit pigeonhole principle (BPHP) and its generalizations in bounded-depth resolution over parities (Res$(\oplus)$). For weak BPHP$_n^m$ with $m = cn$ pigeons (for any constant $c>1$) and $n$ holes, for all $ε>0$, we prove that any depth $N^{1.5 - ε}$ proof in Res$(\oplus)$ must have exponential size, where $N = cn\log n$ is the number of variables. Inspired by recent work in TFNP on multicollision-finding, we consider a generalization of the bit pigeonhole principle, denoted $t$-BPHP$_n^m$, asserting that there is a map from $[m]$ to $[n]$ ($m > (t-1)n$) such that each $i \in [n]$ has fewer than $t$ preimages. We prove that any depth $N^{2-1/t-ε}$ proof in Res$(\oplus)$ of $t$-BPHP$_n^{ctn}$ (for any constant $c \geq 1$) must have exponential size. For the usual bit pigeonhole principle, we show that any depth $N^{2-ε}$ Res$(\oplus)$ proof of BPHP$_n^{n+1}$ must have exponential size. As a byproduct of our proof, we obtain that any randomized parity decision tree for the collision-finding problem with $n+1$ pigeons and $n$ holes must have depth $Ω(n)$, which matches the upper bound coming from a deterministic decision tree. We also prove a lifting theorem for bounded-depth Res$(\oplus)$ with a constant size gadget which lifts from $(p, q)$-DT-hardness, recently defined by Bhattacharya and Chattopadhyay. By combining our lifting theorem with the $(Ω(n), Ω(n))$-DT-hardness of the $n$-variate Tseitin contradiction over a suitable expander, proved by Bhattacharya and Chattopadhyay, we obtain an $N$-variate constant-width unsatisfiable CNF formula with $O(N)$ clauses for which any depth $N^{2-ε}$ Res$(\oplus)$ proof requires size $\exp(Ω(N^ε))$. Previously no superpolynomial lower bounds were known for Res$(\oplus)$ proofs when the depth is superlinear in the size of the formula.

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arXiv2024

Relations between monotone complexity measures based on decision tree complexity

In a recent result, Knop, Lovett, McGuire and Yuan (STOC 2021) proved the log-rank conjecture for communication complexity, up to log n factor, for any Boolean function composed with AND function as the inner gadget. One of the main tools in this result was the relationship between monotone analogues of well-studied Boolean complexity measures like block sensitivity and certificate complexity. The relationship between the standard measures has been a long line of research, with a landmark result by Huang (Annals of Mathematics 2019), finally showing that sensitivity is polynomially related to all other standard measures. In this article, we study the monotone analogues of standard measures like block sensitivity (mbs(f)), certificate complexity (MCC(f)) and fractional block sensitivity (fmbs(f)); and study the relationship between these measures given their connection with AND-decision tree and sparsity of a Boolean function. We show the following results: 1) Given a Boolean function $f : \{0, 1\}^{n} \rightarrow \{0, 1\}$, the ratio $fmbs(f^l )/mbs(f^l )$ is bounded by a function of n (and not l). A similar result was known for the corresponding standard measures (Tal, ITCS 2013). This result allows us to extend any upper bound by a well behaved measure on monotone block sensitivity to monotone fractional block sensitivity. 2) The question of the best possible upper bound on monotone block sensitivity by the logarithm of sparsity is equivalent to the natural question of best upper bound by degree on sensitivity. One side of this relationship was used in the proof by Knop, Lovett, McGuire and Yuan (STOC 2021). 3) For two natural classes of functions, symmetric and monotone, hitting set complexity (MCC) is equal to monotone sensitivity.

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arXiv2022

Query complexity of Boolean functions on slices

We study the deterministic query complexity of Boolean functions on slices of the hypercube. The $k^{th}$ slice $\binom{[n]}{k}$ of the hypercube $\{0,1\}^n$ is the set of all $n$-bit strings with Hamming weight $k$. We show that there exists a function on the balanced slice $\binom{[n]}{n/2}$ requiring $n - O(\log \log n)$ queries. We give an explicit function on the balanced slice requiring $n - O(\log n)$ queries based on independent sets in Johnson graphs. On the weight-2 slice, we show that hard functions are closely related to Ramsey graphs. Further we describe a simple way of transforming functions on the hypercube to functions on the balanced slice while preserving several complexity measures.

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