arXiv subjects

Arkadev Chattopadhyay

Publications and source records attributed to Arkadev Chattopadhyay.

At least 19 recordsLinked to original sources

arXiv2026

The Role of Symmetry in Quantum Query-to-Communication Simulation

Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}^n to {-1,1} and G in {AND_2, XOR_2}, the bounded-error quantum communication complexity of the composed function f o G equals O(Q(f) log n), where Q(f) denotes the bounded-error quantum query complexity of f. This is achieved by Alice running the optimal quantum query algorithm for f, using a round of O(log n) qubits of communication to implement each query. This is in contrast with the classical setting, where it is easy to show that R^{cc}(f o G) is at most 2R(f), where R^{cc} and R denote bounded-error communication and query complexity, respectively. We show that the O(log n) overhead is required for some functions in the quantum setting, and thus the BCW simulation is tight. We note here that prior to our work, the possibility of Q^{cc}(f o G) = O(Q(f)), for all f and all G in {AND_2, XOR_2}, had not been ruled out. More specifically, we show the following. - We show that the log n overhead is *not* required when f is symmetric, generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05). - In order to prove the above, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function. - In view of our first result above, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive, which is a weaker notion of symmetry. We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2. - We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model.

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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

Exponential Lower Bounds on the Size of ResLin Proofs of Nearly Quadratic Depth

Itsykson and Sokolov [IS14] identified resolution over parities, denoted by $\text{Res}(\oplus)$, as a natural and simple fragment of $\text{AC}^0[2]$-Frege for which no super-polynomial lower bounds on size of proofs are known. Building on a recent line of work ([EGI24], [BCD24], [AI25]), Efremenko and Itsykson [EI25] proved lower bounds of the form $\text{exp}(N^{Ω(1)})$, on the size of $\text{Res}(\oplus)$ proofs whose depth is upper bounded by $O(N\log N)$, where $N$ is the number of variables of the unsatisfiable CNF formula. The hard formula they used was Tseitin on an appropriately expanding graph, lifted by a $2$-stifling gadget. They posed the natural problem of proving super-polynomial lower bounds on the size of proofs that are $Ω(N^{1+ε})$ deep, for any constant $ε> 0$. We prove the first such lower bounds. In fact, we show that $\text{Res}(\oplus)$ refutations of Tseitin formulas on constant-degree expanders on $m$ vertices, lifted with Inner-Product gadget of size $O(\log m)$, must have size $\text{exp}(\tildeΩ(N^ε))$, as long as the depth of the $\text{Res}(\oplus)$ proofs are $O(N^{2-ε})$, for every $ε> 0$. Here $N=Θ(m\log m)$ is the number of variables of the lifted formula. An important ingredient in our work is to show that arbitrary distributions lifted with such gadgets fool safe affine spaces, an idea which originates in the earlier work of Bhattacharya, Chattopadhyay and Dvorak [BCD24].

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arXiv2025

Exact versus Approximate Representations of Boolean Functions in the De Morgan Basis

A seminal result of Nisan and Szegedy (STOC, 1992) shows that for any total Boolean function, the degree of the real polynomial that computes the function, and the minimal degree of a real polynomial that point-wise approximates the function, are at most polynomially separated. Extending this result from degree to other complexity measures like sparsity of the polynomial representation, or total weight of the coefficients, remains poorly understood. In this work, we consider this problem in the De Morgan basis, and prove an analogous result for the sparsity of the polynomials at a logarithmic scale. Our result further implies that the exact $\ell_1$ norm and its approximate variant are also similarly related to each other at a log scale. This is in contrast to the Fourier basis, where the analog of our results are known to be false. Our proof is based on a novel random restriction method. Unlike most existing random restriction methods used in complexity theory, our random restriction process is adaptive and is based on how various complexity measures simplify during the restriction process.

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arXiv2024

Exponential Separation Between Powers of Regular and General Resolution Over Parities

Proving super-polynomial lower bounds on the size of proofs of unsatisfiability of Boolean formulas using resolution over parities is an outstanding problem that has received a lot of attention after its introduction by Raz and Tzamaret [Ann. Pure Appl. Log.'08]. Very recently, Efremenko, Garlík and Itsykson [ECCC'23] proved the first exponential lower bounds on the size of ResLin proofs that were additionally restricted to be bottom-regular. We show that there are formulas for which such regular ResLin proofs of unsatisfiability continue to have exponential size even though there exists short proofs of their unsatisfiability in ordinary, non-regular resolution. This is the first super-polynomial separation between the power of general ResLin and and that of regular ResLin for any natural notion of regularity. Our argument, while building upon the work of Efremenko et al., uses additional ideas from the literature on lifting theorems.

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arXiv2022

Lifting to Parity Decision Trees Via Stifling

We show that the deterministic decision tree complexity of a (partial) function or relation $f$ lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation $f \circ g$ as long as the gadget $g$ satisfies a property that we call stifling. We observe that several simple gadgets of constant size, like Indexing on 3 input bits, Inner Product on 4 input bits, Majority on 3 input bits and random functions, satisfy this property. It can be shown that existing randomized communication lifting theorems ([Göös, Pitassi, Watson. SICOMP'20], [Chattopadhyay et al. SICOMP'21]) imply PDT-size lifting. However there are two shortcomings of this approach: first they lift randomized decision tree complexity of $f$, which could be exponentially smaller than its deterministic counterpart when either $f$ is a partial function or even a total search problem. Second, the size of the gadgets in such lifting theorems are as large as logarithmic in the size of the input to $f$. Reducing the gadget size to a constant is an important open problem at the frontier of current research. Our result shows that even a random constant-size gadget does enable lifting to PDT size. Further, it also yields the first systematic way of turning lower bounds on the width of tree-like resolution proofs of the unsatisfiability of constant-width CNF formulas to lower bounds on the size of tree-like proofs in the resolution with parity system, i.e., $\textit{Res}$($\oplus$), of the unsatisfiability of closely related constant-width CNF formulas.

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arXiv2021

Query-to-Communication Lifting Using Low-Discrepancy Gadgets

Lifting theorems are theorems that relate the query complexity of a function $f:\{0,1\}^{n}\to\{0,1\}$ to the communication complexity of the composed function $f \circ g^{n}$, for some "gadget" $g:\{0,1\}^{b}\times\{0,1\}^{b}\to\{0,1\}$. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in the recent years. In addition, such theorems can be viewed as a strong generalization of a direct-sum theorem for the gadget $g$. We prove a new lifting theorem that works for all gadgets $g$ that have logarithmic length and exponentially-small discrepancy, for both deterministic and randomized communication complexity. Thus, we significantly increase the range of gadgets for which such lifting theorems hold. Our result has two main motivations: First, allowing a larger variety of gadgets may support more applications. In particular, our work is the first to prove a randomized lifting theorem for logarithmic-size gadgets, thus improving some applications of the theorem. Second, our result can be seen as a strong generalization of a direct-sum theorem for functions with low discrepancy.

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arXiv2021

Monotone Complexity of Spanning Tree Polynomial Re-visited

We prove two results that shed new light on the monotone complexity of the spanning tree polynomial, a classic polynomial in algebraic complexity and beyond. First, we show that the spanning tree polynomials having $n$ variables and defined over constant-degree expander graphs, have monotone arithmetic complexity $2^{Ω(n)}$. This yields the first strongly exponential lower bound on the monotone arithmetic circuit complexity for a polynomial in VP. Before this result, strongly exponential size monotone lower bounds were known only for explicit polynomials in VNP (Gashkov-Sergeev'12, Raz-Yehudayoff'11, Srinivasan'20, Cavalar-Kumar-Rossman'20, Hrubes-Yehudayoff'21). Recently, Hrubes'20 initiated a program to prove lower bounds against general arithmetic circuits by proving $ε$-sensitive lower bounds for monotone arithmetic circuits for a specific range of values for $ε\in (0,1)$. We consider the spanning tree polynomial $ST_{n}$ defined over the complete graph on $n$ vertices and show that the polynomials $F_{n-1,n} - ε\cdot ST_{n}$ and $F_{n-1,n} + ε\cdot ST_{n}$ defined over $n^2$ variables, have monotone circuit complexity $2^{Ω(n)}$ if $ε\geq 2^{-Ω(n)}$ and $F_{n-1,n} = \prod_{i=2}^n (x_{i,1} +\cdots + x_{i,n})$ is the complete set-multilinear polynomial. This provides the first $ε$-sensitive exponential lower bound for a family of polynomials inside VP. En-route, we consider a problem in 2-party, best partition communication complexity of deciding whether two sets of oriented edges distributed among Alice and Bob form a spanning tree or not. We prove that there exists a fixed distribution, under which the problem has low discrepancy with respect to every nearly-balanced partition. This result could be of interest beyond algebraic complexity.

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arXiv2020

Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on $n$ input bits, each of which has approximate Fourier sparsity at most $O(n^3)$ and randomized parity decision tree complexity $Θ(n)$. This improves upon the recent work of Chattopadhyay, Mande and Sherif (JACM '20) both qualitatively (in terms of designing a large number of examples) and quantitatively (improving the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).

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arXiv2020

Computing the multilinear factors of lacunary polynomials without heights

We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in $\ell^n$ where $\ell$ is the lacunary size of the input polynomial and $n$ its number of variables, that is in particular polynomial in the logarithm of its degree. We also provide a randomized algorithm for the same problem of complexity polynomial in $\ell$ and $n$. Over other fields of characteristic zero and finite fields of large characteristic, our algorithms compute the multilinear factors having at least three monomials of multivariate polynomials. Lower bounds are provided to explain the limitations of our algorithm. As a by-product, we also design polynomial-time deterministic polynomial identity tests for families of polynomials which were not known to admit any. Our results are based on so-called Gap Theorem which reduce high-degree factorization to repeated low-degree factorizations. While previous algorithms used Gap Theorems expressed in terms of the heights of the coefficients, our Gap Theorems only depend on the exponents of the polynomials. This makes our algorithms more elementary and general, and faster in most cases.

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arXiv2019

Quantum Query-to-Communication Simulation Needs a Logarithmic Overhead

Buhrman, Cleve and Wigderson (STOC'98) observed that for every Boolean function $f : \{-1, 1\}^n \to \{-1, 1\}$ and $\bullet : \{-1, 1\}^2 \to \{-1, 1\}$ the two-party bounded-error quantum communication complexity of $(f \circ \bullet)$ is $O(Q(f) \log n)$, where $Q(f)$ is the bounded-error quantum query complexity of $f$. Note that the bounded-error randomized communication complexity of $(f \circ \bullet)$ is bounded by $O(R(f))$, where $R(f)$ denotes the bounded-error randomized query complexity of $f$. Thus, the BCW simulation has an extra $O(\log n)$ factor appearing that is absent in classical simulation. A natural question is if this factor can be avoided. Høyer and de Wolf (STACS'02) showed that for the Set-Disjointness function, this can be reduced to $c^{\log^* n}$ for some constant $c$, and subsequently Aaronson and Ambainis (FOCS'03) showed that this factor can be made a constant. That is, the quantum communication complexity of the Set-Disjointness function (which is $\mathsf{NOR}_n \circ \wedge$) is $O(Q(\mathsf{NOR}_n))$. Perhaps somewhat surprisingly, we show that when $ \bullet = \oplus$, then the extra $\log n$ factor in the BCW simulation is unavoidable. In other words, we exhibit a total function $F : \{-1, 1\}^n \to \{-1, 1\}$ such that $Q^{cc}(F \circ \oplus) = Θ(Q(F) \log n)$. To the best of our knowledge, it was not even known prior to this work whether there existed a total function $F$ and 2-bit function $\bullet$, such that $Q^{cc}(F \circ \bullet) = ω(Q(F))$.

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arXiv2019

Lower Bounds for Linear Decision Lists

We demonstrate a lower bound technique for linear decision lists, which are decision lists where the queries are arbitrary linear threshold functions. We use this technique to prove an explicit lower bound by showing that any linear decision list computing the function $MAJ \circ XOR$ requires size $2^{0.18 n}$. This completely answers an open question of Tur{á}n and Vatan [FoCM'97]. We also show that the spectral classes $PL_1, PL_\infty$, and the polynomial threshold function classes $\widehat{PT}_1, PT_1$, are incomparable to linear decision lists.

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arXiv2017

Weights at the Bottom Matter When the Top is Heavy

Proving super-polynomial lower bounds against depth-2 threshold circuits of the form THR of THR is a well-known open problem that represents a frontier of our understanding in boolean circuit complexity. By contrast, exponential lower bounds on the size of THR of MAJ circuits were shown by Razborov and Sherstov (SIAM J.Comput., 2010) even for computing functions in depth-3 AC^0. Yet, no separation among the two depth-2 threshold circuit classes were known. In fact, it is not clear a priori that they ought to be different. In particular, Goldmann, Hastad and Razborov (Computational Complexity, 1992) showed that the class MAJ of MAJ is identical to the class MAJ of THR. In this work, we provide an exponential separation between THR of MAJ and THR of THR. We achieve this by showing a function f that is computed by linear size THR of THR circuits and yet has exponentially large sign rank. This, by a well-known result, implies that f requires exponentially large THR of MAJ circuits to be computed. Our result suggests that the sign rank method alone is unlikely to prove strong lower bounds against THR of THR circuits. The main technical ingredient of our work is to prove a strong sign rank lower bound for an XOR function. This requires novel use of approximation theoretic tools.

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arXiv2017

Simulation Theorems via Pseudorandom Properties

We generalize the deterministic simulation theorem of Raz and McKenzie [RM99], to any gadget which satisfies certain hitting property. We prove that inner-product and gap-Hamming satisfy this property, and as a corollary we obtain deterministic simulation theorem for these gadgets, where the gadget's input-size is logarithmic in the input-size of the outer function. This answers an open question posed by Göös, Pitassi and Watson [GPW15]. Our result also implies the previous results for the Indexing gadget, with better parameters than was previously known. A preliminary version of the results obtained in this work appeared in [CKL+17].

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arXiv2017

Dual polynomials and communication complexity of $\textsf{XOR}$ functions

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 $D_f(i) \neq D_f(i+2)$, 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 $Ω(k/ \log(n/k))$. 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 $O(n^{1/2-ε})$, for any constant $ε> 0$.

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arXiv2016

Tight Network Topology Dependent Bounds on Rounds of Communication

We prove tight network topology dependent bounds on the round complexity of computing well studied $k$-party functions such as set disjointness and element distinctness. Unlike the usual case in the CONGEST model in distributed computing, we fix the function and then vary the underlying network topology. This complements the recent such results on total communication that have received some attention. We also present some applications to distributed graph computation problems. Our main contribution is a proof technique that allows us to reduce the problem on a general graph topology to a relevant two-party communication complexity problem. However, unlike many previous works that also used the same high level strategy, we do not reason about a two-party communication problem that is induced by a cut in the graph. To `stitch' back the various lower bounds from the two party communication problems, we use the notion of timed graph that has seen prior use in network coding. Our reductions use some tools from Steiner tree packing and multi-commodity flow problems that have a delay constraint.

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arXiv2016

Tribes Is Hard in the Message Passing Model

We consider the point-to-point message passing model of communication in which there are $k$ processors with individual private inputs, each $n$-bit long. Each processor is located at the node of an underlying undirected graph and has access to private random coins. An edge of the graph is a private channel of communication between its endpoints. The processors have to compute a given function of all their inputs by communicating along these channels. While this model has been widely used in distributed computing, strong lower bounds on the amount of communication needed to compute simple functions have just begun to appear. In this work, we prove a tight lower bound of $Ω(kn)$ on the communication needed for computing the Tribes function, when the underlying graph is a star of $k+1$ nodes that has $k$ leaves with inputs and a center with no input. Lower bound on this topology easily implies comparable bounds for others. Our lower bounds are obtained by building upon the recent information theoretic techniques of Braverman et.al (FOCS'13) and combining it with the earlier work of Jayram, Kumar and Sivakumar (STOC'03). This approach yields information complexity bounds that is of independent interest.

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arXiv2015

The Range of Topological Effects on Communication

We continue the study of communication cost of computing functions when inputs are distributed among $k$ processors, each of which is located at one vertex of a network/graph called a terminal. Every other node of the network also has a processor, with no input. The communication is point-to-point and the cost is the total number of bits exchanged by the protocol, in the worst case, on all edges. Chattopadhyay, Radhakrishnan and Rudra (FOCS'14) recently initiated a study of the effect of topology of the network on the total communication cost using tools from $L_1$ embeddings. Their techniques provided tight bounds for simple functions like Element-Distinctness (ED), which depend on the 1-median of the graph. This work addresses two other kinds of natural functions. We show that for a large class of natural functions like Set-Disjointness the communication cost is essentially $n$ times the cost of the optimal Steiner tree connecting the terminals. Further, we show for natural composed functions like $\text{ED} \circ \text{XOR}$ and $\text{XOR} \circ \text{ED}$, the naive protocols suggested by their definition is optimal for general networks. Interestingly, the bounds for these functions depend on more involved topological parameters that are a combination of Steiner tree and 1-median costs. To obtain our results, we use some new tools in addition to ones used in Chattopadhyay et. al. These include (i) viewing the communication constraints via a linear program; (ii) using tools from the theory of tree embeddings to prove topology sensitive direct sum results that handle the case of composed functions and (iii) representing the communication constraints of certain problems as a family of collection of multiway cuts, where each multiway cut simulates the hardness of computing the function on the star topology.

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