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Swagato Sanyal

Publications and source records attributed to Swagato Sanyal.

At least 19 recordsLinked to original sources
arXiv2026

Complexity of learning matchings and half graphs via edge queries

The problem of learning or reconstructing an unknown graph from a known family via partial-information queries arises as a mathematical model in various contexts. The most basic type of access to the graph is via \emph{edge queries}, where an algorithm may query the presence/absence of an edge between a pair of vertices of its choosing, at unit cost. While more powerful query models have been extensively studied in the context of graph reconstruction, the basic model of edge queries seems to have not attracted as much attention. In this paper we study the edge query complexity of learning a hidden bipartite graph, or equivalently its bipartite adjacency matrix, in the classical as well as quantum settings. We focus on learning matchings and half graphs, which are graphs whose bipartite adjacency matrices are a row/column permutation of the identity matrix and the lower triangular matrix with all entries on and below the principal diagonal being 1, respectively. - For matchings of size $n$, we show a tight deterministic bound of $n(n-1)/2$ and an asymptotically tight randomized bound of $Θ(n^2)$. A quantum bound of $Θ(n^{1.5})$ was shown in a recent work of van Apeldoorn et al.~[ICALP'21]. - For half graphs whose bipartite adjacency matrix is a column-permutation of the $n \times n$ lower triangular matrix, we give tight $Θ(n \log n)$ bounds in both deterministic and randomized settings, and an $Ω(n)$ quantum lower bound. \item - For general half graphs, we observe that the problem is equivalent to a natural generalization of the famous nuts-and-bolts problem, leading to a tight $Θ(n \log n)$ randomized bound. - We also present a simple quicksort-style method that instantiates to a $O(n \log^2 n)$ randomized algorithm and a tight $O(n \log n)$ quantum algorithm.

cs.CC↗
arXiv2026

Quantum Search With Generalized Wildcards

In the search with wildcards problem [Ambainis, Montanaro, Quantum Inf.~Comput.'14], one's goal is to learn an unknown bit-string $x \in \{-1,1\}^n$. An algorithm may, at unit cost, test equality of any subset of the hidden string with a string of its choice. Ambainis and Montanaro showed a quantum algorithm of cost $O(\sqrt{n} \log n)$ and a near-matching lower bound of $Ω(\sqrt{n})$. Belovs [Comput.~Comp.'15] subsequently showed a tight $O(\sqrt{n})$ upper bound. We consider a natural generalization of this problem, parametrized by a subset $\cal{Q} \subseteq 2^{[n]}$, where an algorithm may test whether $x_S = b$ for an arbitrary $S \in \cal{Q}$ and $b \in \{-1,1\}^S$ of its choice, at unit cost. We show near-tight bounds when $\cal{Q}$ is any of the following collections: bounded-size sets, contiguous blocks, prefixes, and only the full set. All of these results are derived using a framework that we develop. Using symmetries of the task at hand we show that the quantum query complexity of learning $x$ is characterized, up to a constant factor, by an optimization program, which is succinctly described as follows: `maximize over all odd functions $f : \{-1,1\}^n \to \mathbb{R}$ the ratio of the maximum value of $f$ to the maximum (over $T \in \cal{Q}$) standard deviation of $f$ on a subcube whose free variables are exactly $T$.' To the best of our knowledge, ours is the first work to use the primal version of the negative-weight adversary bound (which is a maximization program typically used to show lower bounds) to show new quantum query upper bounds without explicitly resorting to SDP duality.

quant-ph↗
arXiv2026

Structure of sparse Boolean functions over Abelian groups, and its application to testing

We study Fourier-sparse Boolean functions over general finite Abelian groups. A Boolean function $f : G \to \{-1,+1\}$ is $s$-sparse if it has at most $s$ non-zero Fourier coefficients. We introduce a general notion of granularity of Fourier coefficients and prove that every non-zero coefficient of an $s$-sparse Boolean function has magnitude at least \begin{equation*} \frac{1}{2^{φ(Δ)/2} \, s^{φ(Δ)/2}}, \end{equation*} where $Δ$ denotes the exponent of the group $G$ (that is, the maximum order of an element in $G$) and $φ$ is the Euler's totient function. This generalizes the celebrated result of Gopalan et al. (SICOMP 2011) for $\mathbb{Z}_2^n$, extending it to all finite Abelian groups via new techniques from group theory and algebraic number theory. Using our new structural results on the Fourier coefficients of sparse functions, we design an efficient sparsity testing algorithm for Boolean functions. The tester distinguishes whether a given function is $s$-sparse or $ε$-far from every $s$-sparse Boolean function, with query complexity $poly\left((2s)^{φ(Δ)},1/ε\right)$. In addition, we generalize the classical notion of Boolean degree to arbitrary Abelian groups and establish an $Ω(\sqrt{s})$ lower bound for adaptive sparsity testing.

cs.CC↗
arXiv2025

Voter Participation Control in Online Polls

News outlets, surveyors, and other organizations often conduct polls on social networks to gain insights into public opinion. Such a poll is typically started by someone on a social network who sends it to her friends. If a person participates in the poll, the poll information gets published on her wall, which in turn enables her friends to participate, and the process continues. Eventually, a subset of the population participates in the poll, and the pollster learns the outcome of that poll. We initiate the study of a new but natural type of election control in such online elections. We study how difficult/easy it is to sway the outcome of such polls in one's favor/against (aka constructive vs destructive) by any malicious influencer who nudges/bribes people for seemingly harmless actions like non-participation. These questions are important from the standpoint of studying the power of resistance of online voting against malicious behavior. The destructive version is also important to quantify the robustness of the winner of an online voting. We show that both problems are computationally intractable even if the election is over only two candidates and the influencer has an infinite amount of money to spend (that is, every voter can be persuaded to not participate). We strengthen this result by proving that the computational task remains substantially challenging even if the underlying network is a tree. Finally, we show that there is a polynomial-time algorithm for the constructive version of the problem when we have O(1) candidates, and the treewidth of the underlying graph is O(1); the algorithm for the destructive version does not even need to assume O(1) number of candidates. Hence, we observe that the destructive version is computationally easier than the constructive version.

cs.MA↗
arXiv2024

On the communication complexity of finding a king in a tournament

A tournament is a complete directed graph. A king in a tournament is a vertex v such that every other vertex is reachable from v via a path of length at most 2. It is well known that every tournament has at least one king, one of which is a maximum out-degree vertex. The tasks of finding a king, a maximum out-degree vertex and a source in a tournament has been relatively well studied in the context of query complexity. We study the communication complexity of these tasks, where the edges are partitioned between two players. The following are our main results for n-vertex tournaments: 1) The deterministic communication complexity of finding whether a source exists is tilde{Theta}(log^2 n). 2) The deterministic and randomized communication complexities of finding a king are Theta(n). The quantum communication complexity is tilde{Theta}(sqrt{n}). 3) The deterministic, randomized and quantum communication complexities of finding a maximum out-degree vertex are Theta(n log n), tilde{Theta}(n) and tilde{Theta}(sqrt{n}), respectively. Our upper bounds hold for all partitions of edges, and the lower bounds for a specific partition of the edges. To show the first bullet above, we show, perhaps surprisingly, that finding a source in a tournament is equivalent to the well-studied Clique vs. Independent Set (CIS) problem on undirected graphs. Our bounds for finding a source then follow from known bounds on the complexity of the CIS problem. In view of this equivalence, we can view the task of finding a king in a tournament to be a natural generalization of CIS. One of our lower bounds uses a fooling-set based argument, and all our other lower bounds follow from carefully-constructed reductions from Set-Disjointness.

cs.CC↗
arXiv2024

Randomized query composition and product distributions

Let R_eps denote randomized query complexity for error probability eps, and R:=R_{1/3}. In this work we investigate whether a perfect composition theorem R(f o g^n)=Omega(R(f).R(g)) holds for a relation f in {0,1}^n * S and a total inner function g:{0,1}^m \to {0, 1}. Let D^(prod) denote the maximum distributional query complexity with respect to any product (over variables) distribution. In this work we show the composition theorem R(f o g^n)=Omega(R(f).D^{prod}(g)) up to logarithmic factors. In light of the minimax theorem which states that R(g) is the maximum distributional complexity of g over any distribution, our result makes progress towards answering the composition question. We prove our result by means of a complexity measure R^(prod)_(eps) that we define for total Boolean functions. We show it to be equivalent (up to logarithmic factors) to the sabotage complexity measure RS() defined by Ben-David and Kothari (ICALP 2019): RS(g) = Theta(R^(prod)_(1/3)(g)) (up to log factors). We ask if our bound RS(g) = Omega(D^(prod)(g)) (up to log factors) is tight. We answer this question in the negative, by showing that for the NAND tree function, sabotage complexity is polynomially larger than D^(prod). Our proof yields an alternative and different derivation of the tight lower bound on the bounded error randomized query complexity of the NAND tree function (originally proved by Santha in 1985), which may be of independent interest. Our result gives an explicit polynomial separation between R and D^(prod) which, to our knowledge, was not known prior to our work.

cs.CC↗
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↗
arXiv2023

Feature-based Individual Fairness in k-Clustering

Ensuring fairness in machine learning algorithms is a challenging and essential task. We consider the problem of clustering a set of points while satisfying fairness constraints. While there have been several attempts to capture group fairness in the $k$-clustering problem, fairness at an individual level is relatively less explored. We introduce a new notion of individual fairness in $k$-clustering based on features not necessarily used for clustering. We show that this problem is NP-hard and does not admit a constant factor approximation. Therefore, we design a randomized algorithm that guarantees approximation both in terms of minimizing the clustering distance objective and individual fairness under natural restrictions on the distance metric and fairness constraints. Finally, our experimental results against six competing baselines validate that our algorithm produces individually fairer clusters than the fairest baseline by 12.5% on average while also being less costly in terms of the clustering objective than the best baseline by 34.5% on average.

cs.LG↗
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.

cs.CC↗
arXiv2022

Decision Tree Complexity versus Block Sensitivity and Degree

Relations between the decision tree complexity and various other complexity measures of Boolean functions is a thriving topic of research in computational complexity. It is known that decision tree complexity is bounded above by the cube of block sensitivity, and the cube of polynomial degree. However, the widest separation between decision tree complexity and each of block sensitivity and degree that is witnessed by known Boolean functions is quadratic. In this work, we investigate the tightness of the existing cubic upper bounds. We improve the cubic upper bounds for many interesting classes of Boolean functions. We show that for graph properties and for functions with a constant number of alternations, both of the cubic upper bounds can be improved to quadratic. We define a class of Boolean functions, which we call the zebra functions, that comprises Boolean functions where each monotone path from 0^n to 1^n has an equal number of alternations. This class contains the symmetric and monotone functions as its subclasses. We show that for any zebra function, decision tree complexity is at most the square of block sensitivity, and certificate complexity is at most the square of degree. Finally, we show using a lifting theorem of communication complexity by G{ö}{ö}s, Pitassi and Watson that the task of proving an improved upper bound on the decision tree complexity for all functions is in a sense equivalent to the potentially easier task of proving a similar upper bound on communication complexity for each bi-partition of the input variables, for all functions. In particular, this implies that to bound the decision tree complexity it suffices to bound smaller measures like parity decision tree complexity, subcube decision tree complexity and decision tree rank, that are defined in terms of models that can be efficiently simulated by communication protocols.

cs.CC↗
arXiv2022

Sampling-Based Winner Prediction in District-Based Elections

In a district-based election, we apply a voting rule $r$ to decide the winners in each district, and a candidate who wins in a maximum number of districts is the winner of the election. We present efficient sampling-based algorithms to predict the winner of such district-based election systems in this paper. When $r$ is plurality and the margin of victory is known to be at least $\varepsilon$ fraction of the total population, we present an algorithm to predict the winner. The sample complexity of our algorithm is $\mathcal{O}\left(\frac{1}{\varepsilon^4}\log \frac{1}{\varepsilon}\log\frac{1}δ\right)$. We complement this result by proving that any algorithm, from a natural class of algorithms, for predicting the winner in a district-based election when $r$ is plurality, must sample at least $Ω\left(\frac{1}{\varepsilon^4}\log\frac{1}δ\right)$ votes. We then extend this result to any voting rule $r$. Loosely speaking, we show that we can predict the winner of a district-based election with an extra overhead of $\mathcal{O}\left(\frac{1}{\varepsilon^2}\log\frac{1}δ\right)$ over the sample complexity of predicting the single-district winner under $r$. We further extend our algorithm for the case when the margin of victory is unknown, but we have only two candidates. We then consider the median voting rule when the set of preferences in each district is single-peaked. We show that the winner of a district-based election can be predicted with $\mathcal{O}\left(\frac{1}{\varepsilon^4}\log\frac{1}{\varepsilon}\log\frac{1}δ\right)$ samples even when the harmonious order in different districts can be different and even unknown. Finally, we also show some results for estimating the margin of victory of a district-based election within both additive and multiplicative error bounds.

cs.AI↗
arXiv2022

One-way communication complexity and non-adaptive decision trees

We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the Inner Product on a constant number of inputs. Let $IP$ denote Inner Product on $2b$ bits. - If $f$ is a total Boolean function that depends on all of its inputs, the bounded-error one-way quantum communication complexity of $f \circ IP$ equals $Ω(n(b-1))$. - If $f$ is a partial Boolean function, the deterministic one-way communication complexity of $f \circ IP$ is at least $Ω(b \cdot D_{dt}^{\rightarrow}(f))$, where $D_{dt}^{\rightarrow}(f)$ denotes the non-adaptive decision tree complexity of $f$. Montanaro and Osborne [arXiv'09] observed that the deterministic one-way communication complexity of $f \circ XOR_2$ equals the non-adaptive parity decision tree complexity of $f$. In contrast, we show the following with the gadget $AND_2$. - There exists a function for which even the quantum non-adaptive AND decision tree complexity of $f$ is exponentially large in the deterministic one-way communication complexity of $f \circ AND_2$. - For symmetric functions $f$, the non-adaptive AND decision tree complexity of $f$ is at most quadratic in the (even two-way) communication complexity of $f \circ AND_2$. In view of the first point, a lower bound on non-adaptive AND decision tree complexity of $f$ does not lift to a lower bound on one-way communication complexity of $f \circ AND_2$. In our final result we show that for all $f$, the deterministic one-way communication complexity of $F = f \circ AND_2$ is at most $(rank(M_{F}))(1 - Ω(1))$, where $M_{F}$ denotes the communication matrix of $F$. This shows that the rank upper bound on one-way communication complexity (which can be tight in general) is not tight for AND-composed functions.

cs.CC↗
arXiv2021

Tight Chang's-lemma-type bounds for Boolean functions

Chang's lemma (Duke Mathematical Journal, 2002) is a classical result with applications across several areas in mathematics and computer science. For a Boolean function $f$ that takes values in {-1,1} let $r(f)$ denote its Fourier rank. For each positive threshold $t$, Chang's lemma provides a lower bound on $wt(f):=\Pr[f(x)=-1]$ in terms of the dimension of the span of its characters with Fourier coefficients of magnitude at least $1/t$. We examine the tightness of Chang's lemma w.r.t. the following three natural settings of the threshold: - the Fourier sparsity of $f$, denoted $k(f)$, - the Fourier max-supp-entropy of $f$, denoted $k'(f)$, defined to be $\max \{1/|\hat{f}(S)| : \hat{f}(S) \neq 0\}$, - the Fourier max-rank-entropy of $f$, denoted $k''(f)$, defined to be the minimum $t$ such that characters whose Fourier coefficients are at least $1/t$ in absolute value span a space of dimension $r(f)$. We prove new lower bounds on $wt(f)$ in terms of these measures. One of our lower bounds subsumes and refines the previously best known upper bound on $r(f)$ in terms of $k(f)$ by Sanyal (ToC, 2019). Another lower bound is based on our improvement of a bound by Chattopadhyay, Hatami, Lovett and Tal (ITCS, 2019) on the sum of the absolute values of the level-$1$ Fourier coefficients. We also show that Chang's lemma for the these choices of the threshold is asymptotically outperformed by our bounds for most settings of the parameters involved. Next, we show that our bounds are tight for a wide range of the parameters involved, by constructing functions (which are modifications of the Addressing function) witnessing their tightness. Finally we construct Boolean functions $f$ for which - our lower bounds asymptotically match $wt(f)$, and - for any choice of the threshold $t$, the lower bound obtained from Chang's lemma is asymptotically smaller than $wt(f)$.

cs.CC↗
arXiv2020

On parity decision trees for Fourier-sparse Boolean functions

We study parity decision trees for Boolean functions. The motivation of our study is the log-rank conjecture for XOR functions and its connection to Fourier analysis and parity decision tree complexity. Let f be a Boolean function with Fourier support S and Fourier sparsity k. 1) We prove via the probabilistic method that there exists a parity decision tree of depth O(sqrt k) that computes f. This matches the best known upper bound on the parity decision tree complexity of Boolean functions (Tsang, Wong, Xie, and Zhang, FOCS 2013). Moreover, while previous constructions (Tsang et al., FOCS 2013, Shpilka, Tal, and Volk, Comput. Complex. 2017) build the trees by carefully choosing the parities to be queried in each step, our proof shows that a naive sampling of the parities suffices. 2) We generalize the above result by showing that if the Fourier spectra of Boolean functions satisfy a natural "folding property", then the above proof can be adapted to establish existence of a tree of complexity polynomially smaller than O(sqrt k). We make a conjecture in this regard which, if true, implies that the communication complexity of an XOR function is bounded above by the fourth root of the rank of its communication matrix, improving upon the previously known upper bound of square root of rank (Tsang et al., FOCS 2013, Lovett, J. ACM. 2016). 3) It can be shown by elementary techniques that for any Boolean function f and all pairs (alpha, beta) of parities in S, there exists another pair (gamma, delta) of parities in S such that alpha + beta = gamma + delta. We show, among other results, that there must exist several gamma in F_2^n such that there are at least three pairs (alpha_1, alpha_2) of parities in S with alpha_1 + alpha_2 = gamma.

cs.CC↗
arXiv2018

A composition theorem for randomized query complexity via max conflict complexity

Let $R_ε(\cdot)$ stand for the bounded-error randomized query complexity with error $ε> 0$. For any relation $f \subseteq \{0,1\}^n \times S$ and partial Boolean function $g \subseteq \{0,1\}^m \times \{0,1\}$, we show that $R_{1/3}(f \circ g^n) \in Ω(R_{4/9}(f) \cdot \sqrt{R_{1/3}(g)})$, where $f \circ g^n \subseteq (\{0,1\}^m)^n \times S$ is the composition of $f$ and $g$. We give an example of a relation $f$ and partial Boolean function $g$ for which this lower bound is tight. We prove our composition theorem by introducing a new complexity measure, the max conflict complexity $\bar χ(g)$ of a partial Boolean function $g$. We show $\bar χ(g) \in Ω(\sqrt{R_{1/3}(g)})$ for any (partial) function $g$ and $R_{1/3}(f \circ g^n) \in Ω(R_{4/9}(f) \cdot \bar χ(g))$; these two bounds imply our composition result. We further show that $\bar χ(g)$ is always at least as large as the sabotage complexity of $g$, introduced by Ben-David and Kothari.

cs.CC↗
arXiv2018

A Composition Theorem via Conflict Complexity

Let $\R(\cdot)$ stand for the bounded-error randomized query complexity. We show that for any relation $f \subseteq \{0,1\}^n \times \mathcal{S}$ and partial Boolean function $g \subseteq \{0,1\}^n \times \{0,1\}$, $\R_{1/3}(f \circ g^n) = Ω(\R_{4/9}(f) \cdot \sqrt{\R_{1/3}(g)})$. Independently of us, Gavinsky, Lee and Santha \cite{newcomp} proved this result. By an example demonstrated in their work, this bound is optimal. We prove our result by introducing a novel complexity measure called the \emph{conflict complexity} of a partial Boolean function $g$, denoted by $χ(g)$, which may be of independent interest. We show that $χ(g) = Ω(\sqrt{\R(g)})$ and $\R(f \circ g^n) = Ω(\R(f) \cdot χ(g))$.

cs.CC↗
arXiv2017

Quadratically Tight Relations for Randomized Query Complexity

Let $f:\{0,1\}^n \rightarrow \{0,1\}$ be a Boolean function. The certificate complexity $C(f)$ is a complexity measure that is quadratically tight for the zero-error randomized query complexity $R_0(f)$: $C(f) \leq R_0(f) \leq C(f)^2$. In this paper we study a new complexity measure that we call expectational certificate complexity $EC(f)$, which is also a quadratically tight bound on $R_0(f)$: $EC(f) \leq R_0(f) = O(EC(f)^2)$. We prove that $EC(f) \leq C(f) \leq EC(f)^2$ and show that there is a quadratic separation between the two, thus $EC(f)$ gives a tighter upper bound for $R_0(f)$. The measure is also related to the fractional certificate complexity $FC(f)$ as follows: $FC(f) \leq EC(f) = O(FC(f)^{3/2})$. This also connects to an open question by Aaronson whether $FC(f)$ is a quadratically tight bound for $R_0(f)$, as $EC(f)$ is in fact a relaxation of $FC(f)$. In the second part of the work, we upper bound the distributed query complexity $D^μ_ε(f)$ for product distributions $μ$ by the square of the query corruption bound ($\mathrm{corr}_ε(f)$) which improves upon a result of Harsha, Jain and Radhakrishnan [2015]. A similar statement for communication complexity is open.

cs.CC↗
arXiv2017

A Composition Theorem for Randomized Query Complexity

Let the randomized query complexity of a relation for error probability $ε$ be denoted by $R_ε(\cdot)$. We prove that for any relation $f \subseteq \{0,1\}^n \times \mathcal{R}$ and Boolean function $g:\{0,1\}^m \rightarrow \{0,1\}$, $R_{1/3}(f\circ g^n) = Ω(R_{4/9}(f)\cdot R_{1/2-1/n^4}(g))$, where $f \circ g^n$ is the relation obtained by composing $f$ and $g$. We also show that $R_{1/3}\left(f \circ \left(g^\oplus_{O(\log n)}\right)^n\right)=Ω(\log n \cdot R_{4/9}(f) \cdot R_{1/3}(g))$, where $g^\oplus_{O(\log n)}$ is the function obtained by composing the xor function on $O(\log n)$ bits and $g^t$.

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