arXiv subjects

Chandrima Kayal

Publications and source records attributed to Chandrima Kayal.

11 recordsLinked to original sources

arXiv2026

Bounds for Hardness Condensation in the Query Model

For any Boolean function $f:\{0,1\}^n \to \{0,1\}$ with a complexity measure having value $k \ll n$, is it possible to restrict the function $f$ to $\Theta(k)$ variables while keeping the complexity preserved at $\Theta(k)$? This question, in the context of query complexity, was recently studied by G{\"{o}}{\"{o}}s, Newman, Riazanov and Sokolov (STOC 2024). They showed, among other results, that query complexity can not be condensed losslessly. They asked if complexity measures like block sensitivity or unambiguous certificate complexity can be condensed losslessly? In this work, we show that decision tree measures like block sensitivity and certificate complexity, cannot be condensed losslessly. That is, there exists a Boolean function $f$ such that any restriction of $f$ to $O(\mathcal{M}(f))$ variables has $\mathcal{M}(\cdot)$-complexity at most $\tilde{O}(\mathcal{M}(f)^{2/3})$, where $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{D}\}$. This also improves upon a result of G{\"{o}}{\"{o}}s, Newman, Riazanov and Sokolov (STOC 2024). We also complement the negative results on lossless condensation with positive results about lossy condensation. In particular, we show that for every Boolean function $f$ there exists a restriction of $f$ to $O(\mathcal{M}(f))$ variables such that its $\mathcal{M}(\cdot)$-complexity is at least $\Omega(\mathcal{M}(f)^{1/2})$, where $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{UC}_{min},\mathsf{UC}_1,\mathsf{UC},\mathsf{D},\widetilde{\mathsf{deg}},\lambda\}$. We also show a slightly weaker positive result for randomized and quantum query complexity.

cs.CC↗

arXiv2026

Spectral Shadows: When Communication Complexity Meets Linear Invariance Testing

In this short note, we initiate the study of the Linear Isomorphism Testing Problem in the setting of communication complexity, a natural linear algebraic generalization of the classical Equality problem. Given Boolean functions $f, g : \mathbb{F}_2^n \to \{-1, +1\}$, Alice and Bob are tasked with determining whether $f$ and $g$ are equivalent up to a nonsingular linear transformation of the input variables, or far from being so. This problem has been extensively investigated in several models of computation, including standard algorithmic and property testing frameworks, owing to its fundamental connections with combinatorial circuit design, complexity theory, and cryptography. However, despite its broad relevance, it has remained unexplored in the context of communication complexity, a gap we address in this work. Our main results demonstrate that the approximate spectral norm of the input functions plays a central role in governing the communication complexity of this problem. We design a simple deterministic protocol whose communication cost is polynomial in the approximate spectral norm, and complement it with nearly matching lower bounds (up to a quadratic gap). In the randomised setting with private coins, we present an even more efficient protocol, though equally simple, that achieves a quadratically improved dependence on the approximate spectral norm compared to the deterministic case, and we prove that such a dependence is essentially unavoidable. These results identify the approximate spectral norm as a key complexity measure for testing linear invariance in the communication complexity framework. As a core technical ingredient, we establish new junta theorems for Boolean functions with small approximate spectral norm, which may be of independent interest in Fourier analysis and learning theory.

cs.DS↗

arXiv2025

Separations between Combinatorial Measures for Transitive Functions

The role of symmetry in Boolean functions $f:\{0,1\}^n \to \{0,1\}$ has been extensively studied in complexity theory. For example, symmetric functions, that is, functions that are invariant under the action of $S_n$, is an important class of functions in the study of Boolean functions. A function $f:\{0,1\}^n \to \{0,1\}$ is called transitive (or weakly-symmetric) if there exists a transitive group $G$ of $S_n$ such that $f$ is invariant under the action of $G$ - that is the function value remains unchanged even after the bits of the input of $f$ are moved around according to some permutation $σ\in G$. Understanding various complexity measures of transitive functions has been a rich area of research for the past few decades. In this work, we study transitive functions in light of several combinatorial measures. We look at the maximum separation between various pairs of measures for transitive functions. Such study for general Boolean functions has been going on for past many years. The best-known results for general Boolean functions have been nicely compiled by Aaronson et. al (STOC, 2021). The separation between a pair of combinatorial measures is shown by constructing interesting functions that demonstrate the separation. But many of the celebrated separation results are via the construction of functions (like "pointer functions" from Ambainis et al. (JACM, 2017) and "cheat-sheet functions" Aaronson et al. (STOC, 2016)) that are not transitive. Hence, we don't have such separation between the pairs of measures for transitive functions. In this paper we show how to modify some of these functions to construct transitive functions that demonstrate similar separations between pairs of combinatorial measures.

cs.CC↗

arXiv2025

Testing Isomorphism of Boolean Functions over Finite Abelian Groups

Let $f$ and $g$ be Boolean functions over a finite Abelian group $\mathcal{G}$, where $g$ is fully known, and we have {\em query access} to $f$, that is, given any $x \in \mathcal{G}$ we can get the value $f(x)$. We study the tolerant isomorphism testing problem: given $ε\geq 0$ and $τ> 0$, we seek to determine, with minimal queries, whether there exists an automorphism $σ$ of $\mathcal{G}$ such that the fractional Hamming distance between $f \circ σ$ and $g$ is at most $ε$, or whether for all automorphisms $σ$, the distance is at least $ε+ τ$. We design an efficient tolerant testing algorithm for this problem, with query complexity $\mathrm{poly}\left( s, 1/τ\right)$, where $s$ bounds the spectral norm of $g$. Additionally, we present an improved algorithm when $g$ is Fourier sparse. Our approach uses key concepts from Abelian group theory and Fourier analysis, including the annihilator of a subgroup, Pontryagin duality, and a pseudo inner-product for finite Abelian groups. We believe these techniques will find further applications in property testing.

cs.CC↗

arXiv2025

Approximate Degree Composition for Recursive Functions

Determining the approximate degree composition for Boolean functions remains a significant unsolved problem in Boolean function complexity. In recent decades, researchers have concentrated on proving that approximate degree composes for special types of inner and outer functions. An important and extensively studied class of functions are the recursive functions, i.e.~functions obtained by composing a base function with itself a number of times. Let $h^d$ denote the standard $d$-fold composition of the base function $h$. The main result of this work is to show that the approximate degree composes if either of the following conditions holds: (I) The outer function $f:\{0,1\}^n\to \{0,1\}$ is a recursive function of the form $h^d$, with $h$ being any base function and $d= Ω(\log\log n)$. (II) The inner function is a recursive function of the form $h^d$, with $h$ being any constant arity base function (other than AND and OR) and $d= Ω(\log\log n)$, where $n$ is the arity of the outer function. In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be \emph{efficiently eliminated} if the inner or outer function is a recursive function.

cs.CC↗

arXiv2024

About almost covering subsets of the hypercube

Let $\mathbb{F}$ be a field, and consider the hypercube $\{ 0, 1 \}^{n}$ in $\mathbb{F}^{n}$. Sziklai and Weiner (Journal of Combinatorial Theory, Series A 2022) showed that if a polynomial $P ( X_{1}, \dots, X_{n} ) \in \mathbb{F}[ X_{1}, \dots, X_{n}]$ vanishes on every point of the hypercube $\{0,1\}^{n}$ except those with at most $r$ many ones then the degree of the polynomial will be at least $n-r$. This is a generalization of Alon and F\"uredi's fundamental result (European Journal of Combinatorics 1993) about polynomials vanishing on every point of the hypercube except at the origin (point with all zero coordinates). Sziklai and Weiner proved their interesting result using M\"{o}bius inversion formula and the Zeilberger method for proving binomial equalities. In this short note, we show that a stronger version of Sziklai and Weiner's result can be derived directly from Alon and F\"{u}redi's result.

math.CO↗

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.

cs.CC↗

arXiv2023

Spectral Norm, Economical Sieve, and Linear Invariance Testing of Boolean Functions

Given Boolean functions $ f, g : \mathbb{F}_2^n \to \{-1,+1\} $, we say they are {\em linearly isomorphic} if there exists $ A \in \mathrm{GL}_n(\mathbb{F}_2) $ such that $ f(x)=g(Ax) $ for all $ x $. We study this problem in the tolerant property testing framework under the known--unknown model, where $ g $ is given explicitly and $ f $ is accessible only via oracle queries, meaning the algorithm may adaptively request the value of $ f(x) $ for inputs $ x \in \mathbb{F}_2^n $ of its choice. Given parameters $ \epsilon \ge 0 $ and $ \omega>0 $, the goal is to distinguish whether there exists $ A \in \mathrm{GL}_n(\mathbb{F}_{2})$ such that the normalized Hamming distance between $ f $ and $ g(Ax) $ is at most $ \epsilon $, or whether for every $ A \in \mathrm{GL}_n(\mathbb{F}_2) $ the distance is at least $ \epsilon+\omega $. Our main result is a tolerant tester making $ \widetilde{O} \left( \left( m/\omega \right)^4 \right) $ queries to $ f $, where $ m $ is an upper bound on the spectral norm of $ g $, improving the previous $ \widetilde{O} \left( \left( m/\omega \right)^{24} \right) $ bound of Wimmer and Yoshida. We complement this with a nearly matching lower bound of $ \Omega(m^2) $ for constant $ \omega $ (for example, $ \omega=1/4 $), improving the prior $ \Omega(\log m) $ lower bound of Grigorescu, Wimmer and Xie. A key technical ingredient on the algorithmic side is a query-efficient local list corrector. For the lower bound, we give a reduction from communication complexity using a novel subclass of Maiorana--McFarland functions from symmetric-key cryptography.

cs.CC↗

arXiv2023

On higher multiplicity hyperplane and polynomial covers for symmetry preserving subsets of the hypercube

Alon and Füredi (European J. Combin. 1993) gave a tight bound for the following hyperplane covering problem: find the minimum number of hyperplanes required to cover all points of the n-dimensional hypercube {0,1}^n except the origin. Their proof is among the early instances of the polynomial method, which considers a natural polynomial (a product of linear factors) associated to the hyperplane arrangement, and gives a lower bound on its degree, whilst being oblivious to the (product) structure of the polynomial. Thus, their proof gives a lower bound for a weaker polynomial covering problem, and it turns out that this bound is tight for the stronger hyperplane covering problem. In a similar vein, solutions to some other hyperplane covering problems were obtained, via solutions of corresponding weaker polynomial covering problems, in some special cases in the works of the fourth author (Electron. J. Combin. 2022), and the first three authors (Discrete Math. 2023). In this work, we build on these and solve a hyperplane covering problem for general symmetric sets of the hypercube, where we consider hyperplane covers with higher multiplicities. We see that even in this generality, it is enough to solve the corresponding polynomial covering problem. Further, this seems to be the limit of this approach as far as covering symmetry preserving subsets of the hypercube is concerned. We gather evidence for this by considering the class of blockwise symmetric sets of the hypercube (which is a strictly larger class than symmetric sets), and note that the same proof technique seems to only solve the polynomial covering problem.

math.CO↗

arXiv2023

On the Composition of Randomized Query Complexity and Approximate Degree

For any Boolean functions $f$ and $g$, the question whether $R(f\circ g) = \tildeΘ(R(f)R(g))$, is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether $\widetilde{deg}(f\circ g) = \tildeΘ(\widetilde{deg}(f)\cdot\widetilde{deg}(g))$. These questions are two of the most important and well-studied problems, and yet we are far from answering them satisfactorily. It is known that the measures compose if one assumes various properties of the outer function $f$ (or inner function $g$). This paper extends the class of outer functions for which $\text{R}$ and $\widetilde{\text{deg}}$ compose. A recent landmark result (Ben-David and Blais, 2020) showed that $R(f \circ g) = Ω(noisyR(f)\cdot R(g))$. This implies that composition holds whenever $noisyR(f) = \TildeΘ(R(f))$. We show two results: (1)When $R(f) = Θ(n)$, then $noisyR(f) = Θ(R(f))$. (2) If $\text{R}$ composes with respect to an outer function, then $\text{noisyR}$ also composes with respect to the same outer function. On the other hand, no result of the type $\widetilde{deg}(f \circ g) = Ω(M(f) \cdot \widetilde{deg}(g))$ (for some non-trivial complexity measure $M(\cdot)$) was known to the best of our knowledge. We prove that $\widetilde{deg}(f\circ g) = \widetildeΩ(\sqrt{bs(f)} \cdot \widetilde{deg}(g)),$ where $bs(f)$ is the block sensitivity of $f$. This implies that $\widetilde{\text{deg}}$ composes when $\widetilde{\text{deg}}(f)$ is asymptotically equal to $\sqrt{\text{bs}(f)}$. It is already known that both $\text{R}$ and $\widetilde{\text{deg}}$ compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function.

cs.CC↗

arXiv2022

Almost covering all the layers of hypercube with multiplicities

Given a hypercube $\mathcal{Q}^{n} := \{0,1\}^{n}$ in $\mathbb{R}^{n}$ and $k \in \{0, \dots, n\}$, the $k$-th layer $\mathcal{Q}^{n}_{k}$ of $\mathcal{Q}^{n}$ denotes the set of all points in $\mathcal{Q}^{n}$ whose coordinates contain exactly $k$ many ones. For a fixed $t \in \mathbb{N}$ and $k \in \{0, \dots, n\}$, let $P \in \mathbb{R}\left[x_{1}, \dots, x_{n}\right]$ be a polynomial that has zeroes of multiplicity at least $t$ at all points of $\mathcal{Q}^{n} \setminus \mathcal{Q}^{n}_{k}$, and $P$ has zeros of multiplicity exactly $t-1$ at all points of $\mathcal{Q}^{n}_{k}$. In this short note, we show that $$deg(P) \geq \max\left\{ k, n-k\right\}+2t-2.$$Matching the above lower bound we give an explicit construction of a family of hyperplanes $H_{1}, \dots, H_{m}$ in $\mathbb{R}^{n}$, where $m = \max\left\{ k, n-k\right\}+2t-2$, such that every point of $\mathcal{Q}^{n}_{k}$ will be covered exactly $t-1$ times, and every other point of $\mathcal{Q}^{n}$ will be covered at least $t$ times. Note that putting $k = 0$ and $t=1$, we recover the much celebrated covering result of Alon and F\"uredi (European Journal of Combinatorics, 1993). Using the above family of hyperplanes we disprove a conjecture of Venkitesh (The Electronic Journal of Combinatorics, 2022) on exactly covering symmetric subsets of hypercube $\mathcal{Q}^{n}$ with hyperplanes. To prove the above results we have introduced a new measure of complexity of a subset of the hypercube called index complexity which we believe will be of independent interest. We also study a new interesting variant of the restricted sumset problem motivated by the ideas behind the proof of the above result.

math.CO↗