arXiv subjects

Pranjal Dutta

Publications and source records attributed to Pranjal Dutta.

13 recordsLinked to original sources

arXiv2026

Exponential lower bound via exponential sums

Valiant's famous VP vs. VNP conjecture states that the symbolic permanent polynomial does not have polynomial-size algebraic circuits. However, the best upper bound on the size of the circuits computing the permanent is exponential. Informally, VNP is an exponential sum of VP-circuits. In this paper we study whether, in general, exponential sums (of algebraic circuits) require exponential-size algebraic circuits. We show that the famous Shub-Smale $\tau$-conjecture indeed implies such an exponential lower bound for an exponential sum. Our main tools come from parameterized complexity. Along the way, we also prove an exponential fpt (fixed-parameter tractable) lower bound for the parameterized algebraic complexity class VW$_{nb}^0$[P], assuming the same conjecture. VW$_{nb}^0$[P] can be thought of as the weighted sums of (unbounded-degree) circuits, where only $\pm 1$ constants are cost-free. To the best of our knowledge, this is the first time the Shub-Smale $\tau$-conjecture has been applied to prove explicit exponential lower bounds. Furthermore, we prove that when this class is fpt, then a variant of the counting hierarchy, namely the linear counting hierarchy collapses. Moreover, if a certain type of parameterized exponential sums is fpt, then integers, as well as polynomials with coefficients being definable in the linear counting hierarchy have subpolynomial $\tau$-complexity. Finally, we characterize a related class VW[F], in terms of permanents, where we consider an exponential sum of algebraic formulas instead of circuits. We show that when we sum over cycle covers that have one long cycle and all other cycles have constant length, then the resulting family of polynomials is complete for VW[F] on certain types of graphs.

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arXiv2025

Recent Advances in Debordering Methods

Border complexity captures functions that can be approximated by low-complexity ones. Debordering is the task of proving an upper bound on some non-border complexity measure in terms of a border complexity measure, thus getting rid of limits. Debordering lies at the heart of foundational complexity theory questions relating Valiant's determinant versus permanent conjecture (1979) and its geometric complexity theory (GCT) variant proposed by Mulmuley and Sohoni (2001). The debordering of matrix multiplication tensors by Bini (1980) played a pivotal role in the development of efficient matrix multiplication algorithms. Consequently, debordering finds applications in both establishing computational complexity lower bounds and facilitating algorithm design. Recent years have seen notable progress in debordering various restricted border complexity measures. In this survey, we highlight these advances and discuss techniques underlying them.

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arXiv2025

Geometric complexity theory for product-plus-power

According to Kumar's recent surprising result (ToCT'20), a small border Waring rank implies that the polynomial can be approximated as a sum of a constant and a small product of linear polynomials. We prove the converse of Kumar's result and establish a tight connection between border Waring rank and the model of computation in Kumar's result. In this way, we obtain a new formulation of border Waring rank, up to a factor of the degree. We connect this new formulation to the orbit closure problem of the product-plus-power polynomial. We study this orbit closure from two directions: 1. We deborder this orbit closure and some related orbit closures, i.e., prove all points in the orbit closure have small non-border algebraic branching programs. 2. We fully implement the geometric complexity theory approach against the power sum by generalizing the ideas of Ikenmeyer-Kandasamy (STOC'20) to this new orbit closure. In this way, we obtain new multiplicity obstructions that are constructed from just the symmetries of the polynomials.

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arXiv2025

Algebraic metacomplexity and representation theory

In the algebraic metacomplexity framework we prove that the decomposition of metapolynomials into their isotypic components can be implemented efficiently, namely with only a quasipolynomial blowup in the circuit size. We use this to resolve an open question posed by Grochow, Kumar, Saks & Saraf (2017). Our result means that many existing algebraic complexity lower bound proofs can be efficiently converted into isotypic lower bound proofs via highest weight metapolynomials, a notion studied in geometric complexity theory. In the context of algebraic natural proofs, it means that without loss of generality algebraic natural proofs can be assumed to be isotypic. Our proof is built on the Poincaré-Birkhoff-Witt theorem for Lie algebras and on Gelfand-Tsetlin theory, for which we give the necessary comprehensive background.

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arXiv2024

The Complexity of Order-Finding for ROABPs

We study the \emph{order-finding problem} for Read-once Oblivious Algebraic Branching Programs (ROABPs). Given a polynomial $f$ and a parameter $w$, the goal is to find an order $σ$ in which $f$ has an ROABP of \emph{width} $w$. We show that this problem is NP-hard in the worst case, even when the input is a constant degree polynomial that is given in its dense representation. We provide a reduction from CutWidth to prove these results. Owing to the exactness of our reduction, all the known results for the hardness of approximation of Cutwidth also transfer directly to the order-finding problem. Additionally, we also show that any constant-approximation algorithm for the order-finding problem would imply a polynomial time approximation scheme (PTAS) for it. On the algorithmic front, we design algorithms that solve the order-finding problem for generic ROABPs in polynomial time, when the width $w$ is polynomial in the individual degree $d$ of the polynomial $f$. That is, our algorithm is efficient for most/random ROABPs, and requires more time only on a lower-dimensional subspace (or subvariety) of ROABPs. Even when the individual degree is constant, our algorithm runs in time $n^{O(\log w)}$ for most/random ROABPs. This stands in strong contrast to the case of (Boolean) ROBPs, where only heuristic order-finding algorithms are known.

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arXiv2024

Derandomizing Multivariate Polynomial Factoring for Low Degree Factors

For a polynomial $f$ from a class $\mathcal{C}$ of polynomials, we show that the problem to compute all the constant degree irreducible factors of $f$ reduces in polynomial time to polynomial identity tests (PIT) for class $\mathcal{C}$ and divisibility tests of $f$ by constant degree polynomials. We apply the result to several classes $\mathcal{C}$ and obtain the constant degree factors in 1. polynomial time, for $\mathcal{C}$ being polynomials that have only constant degree factors, 2. quasipolynomial time, for $\mathcal{C}$ being sparse polynomials, 3. subexponential time, for $\mathcal{C}$ being polynomials that have constant-depth circuits. Result 2 and 3 were already shown by Kumar, Ramanathan, and Saptharishi with a different proof and their time complexities necessarily depend on black-box PITs for a related bigger class $\mathcal{C}'$. Our complexities vary on whether the input is given as a blackbox or whitebox. We also show that the problem to compute the sparse factors of polynomial from a class $\mathcal{C}$ reduces in polynomial time to PIT for class $\mathcal{C}$, divisibility tests of $f$ by sparse polynomials, and irreducibility preserving bivariate projections for sparse polynomials. For $\mathcal{C}$ being sparse polynomials, it follows that it suffices to derandomize irreducibility preserving bivariate projections for sparse polynomials in order to compute all the sparse irreducible factors efficiently. When we consider factors of sparse polynomials that are sums of univariate polynomials, a subclass of sparse polynomials, we obtain a polynomial time algorithm. This was already shown by Volkovich with a different proof.

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arXiv2024

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^{\varphi(\Delta)/2} \, s^{\varphi(\Delta)/2}}, \end{equation*} where $\Delta$ denotes the exponent of the group $G$ (that is, the maximum order of an element in $G$) and $\varphi$ 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 $\epsilon$-far from every $s$-sparse Boolean function, with query complexity $poly\left((2s)^{\varphi(\Delta)},1/\epsilon \right)$. In addition, we generalize the classical notion of Boolean degree to arbitrary Abelian groups and establish an $\Omega(\sqrt{s})$ lower bound for adaptive sparsity testing.

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arXiv2024

Fixed-parameter debordering of Waring rank

Border complexity measures are defined via limits (or topological closures), so that any function which can approximated arbitrarily closely by low complexity functions itself has low border complexity. Debordering is the task of proving an upper bound on some non-border complexity measure in terms of a border complexity measure, thus getting rid of limits. Debordering is at the heart of understanding the difference between Valiant's determinant vs permanent conjecture, and Mulmuley and Sohoni's variation which uses border determinantal complexity. The debordering of matrix multiplication tensors by Bini played a pivotal role in the development of efficient matrix multiplication algorithms. Consequently, debordering finds applications in both establishing computational complexity lower bounds and facilitating algorithm design. Currently, very few debordering results are known. In this work, we study the question of debordering the border Waring rank of polynomials. Waring and border Waring rank are very well studied measures in the context of invariant theory, algebraic geometry, and matrix multiplication algorithms. For the first time, we obtain a Waring rank upper bound that is exponential in the border Waring rank and only linear in the degree. All previous known results were exponential in the degree. For polynomials with constant border Waring rank, our results imply an upper bound on the Waring rank linear in degree, which previously was only known for polynomials with border Waring rank at most 5.

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arXiv2023

Homogeneous Algebraic Complexity Theory and Algebraic Formulas

We study algebraic complexity classes and their complete polynomials under \emph{homogeneous linear} projections, not just under the usual affine linear projections that were originally introduced by Valiant in 1979. These reductions are weaker yet more natural from a geometric complexity theory (GCT) standpoint, because the corresponding orbit closure formulations do not require the padding of polynomials. We give the \emph{first} complete polynomials for VF, the class of sequences of polynomials that admit small algebraic formulas, under homogeneous linear projections: The sum of the entries of the non-commutative elementary symmetric polynomial in 3 by 3 matrices of homogeneous linear forms. Even simpler variants of the elementary symmetric polynomial are hard for the topological closure of a large subclass of VF: the sum of the entries of the non-commutative elementary symmetric polynomial in 2 by 2 matrices of homogeneous linear forms, and homogeneous variants of the continuant polynomial (Bringmann, Ikenmeyer, Zuiddam, JACM '18). This requires a careful study of circuits with arity-3 product gates.

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arXiv2023

Deterministic identity testing paradigms for bounded top-fanin depth-4 circuits

Polynomial Identity Testing (PIT) is a fundamental computational problem. The famous depth-$4$ reduction result by Agrawal and Vinay (FOCS 2008) has made PIT for depth-$4$ circuits an enticing pursuit. A restricted depth-4 circuit computing a $n$-variate degree-$d$ polynomial of the form $\sum_{i = 1}^{k} \prod_{j} g_{ij}$, where $°g_{ij} \leq δ$ is called $Σ^{[k]}ΠΣΠ^{[δ]}$ circuit. On further restricting $g_{ij}$ to be sum of univariates we obtain $Σ^{[k]}ΠΣ\wedge$ circuits. The largely open, special-cases of $Σ^{[k]}ΠΣΠ^{[δ]}$ for constant $k$ and $δ$, and $Σ^{[k]}ΠΣ\wedge$ have been a source of many great ideas in the last two decades. For eg. depth-$3$ ideas of Dvir and Shpilka (STOC 2005), Kayal and Saxena (CCC 2006), and Saxena and Seshadhri (FOCS 2010 and STOC 2011). Further, depth-$4$ ideas of Beecken, Mittmann and Saxena (ICALP 2011), Saha, Saxena and Saptharishi (Comput.Compl. 2013), Forbes (FOCS 2015), and Kumar and Saraf (CCC 2016). Additionally, geometric Sylvester-Gallai ideas of Kayal and Saraf (FOCS 2009), Shpilka (STOC 2019), and Peleg and Shpilka (CCC 2020, STOC 2021). Very recently, a subexponential-time blackbox PIT algorithm for constant-depth circuits was obtained via lower bound breakthrough of Limaye, Srinivasan, Tavenas (FOCS 2021). We solve two of the basic underlying open problems in this work. We give the first polynomial-time PIT for $Σ^{[k]}ΠΣ\wedge$. We also give the first quasipolynomial time blackbox PIT for both $Σ^{[k]}ΠΣ\wedge$ and $Σ^{[k]}ΠΣΠ^{[δ]}$. A key technical ingredient in all the three algorithms is how the logarithmic derivative, and its power-series, modify the top $Π$-gate to $\wedge$.

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arXiv2022

Efficient reductions and algorithms for variants of Subset Sum

Given $(a_1, \dots, a_n, t) \in \mathbb{Z}_{\geq 0}^{n + 1}$, the Subset Sum problem ($\mathsf{SSUM}$) is to decide whether there exists $S \subseteq [n]$ such that $\sum_{i \in S} a_i = t$. There is a close variant of the $\mathsf{SSUM}$, called $\mathsf{Subset~Product}$. Given positive integers $a_1, ..., a_n$ and a target integer $t$, the $\mathsf{Subset~Product}$ problem asks to determine whether there exists a subset $S \subseteq [n]$ such that $\prod_{i \in S} a_i=t$. There is a pseudopolynomial time dynamic programming algorithm, due to Bellman (1957) which solves the $\mathsf{SSUM}$ and $\mathsf{Subset~Product}$ in $O(nt)$ time and $O(t)$ space. In the first part, we present {\em search} algorithms for variants of the Subset Sum problem. Our algorithms are parameterized by $k$, which is a given upper bound on the number of realisable sets (i.e.,~number of solutions, summing exactly $t$). We show that $\mathsf{SSUM}$ with a unique solution is already NP-hard, under randomized reduction. This makes the regime of parametrized algorithms, in terms of $k$, very interesting. Subsequently, we present an $\tilde{O}(k\cdot (n+t))$ time deterministic algorithm, which finds the hamming weight of all the realisable sets for a subset sum instance. We also give a poly$(knt)$-time and $O(\log(knt))$-space deterministic algorithm that finds all the realisable sets for a subset sum instance. In the latter part, we present a simple and elegant randomized $\tilde{O}(n + t)$ time algorithm for $\mathsf{Subset~Product}$. Moreover, we also present a poly$(nt)$ time and $O(\log^2 (nt))$ space deterministic algorithm for the same. We study these problems in the unbounded setting as well. Our algorithms use multivariate FFT, power series and number-theoretic techniques, introduced by Jin and Wu (SOSA'19) and Kane (2010).

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arXiv2017

A note on faithful coupling of Markov chains

One often needs to turn a coupling $(X_i, Y_i)_{i\geq 0}$ of a Markov chain into a sticky coupling where once $X_T = Y_T$ at some $T$, then from then on, at each subsequent time step $T'\geq T$, we shall have $X_{T'} = Y_{T'}$. However, not all of what are considered couplings in literature, even Markovian couplings, can be turned into sticky couplings, as proved by Rosenthal through a counter example. Rosenthal then proposed a strengthening of the Markovian coupling notion, termed as faithful coupling, from which a sticky coupling can indeed be obtained. We identify the reason why a sticky coupling could not be obtained in the counter example of Rosenthal, which motivates us to define a type of coupling which can obviously be turned into a sticky coupling. We show then that the new type of coupling that we define, and the faithful coupling as defined by Rosenthal, are actually identical. Our note may be seen as a demonstration of the naturalness of the notion of faithful coupling.

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arXiv2017

Discovering the roots: Uniform closure results for algebraic classes under factoring

Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots $r$ is small but the multiplicities are exponentially large. Our method sets up a linear system in $r$ unknowns and iteratively builds the roots as formal power series. For an algebraic circuit $f(x_1,\ldots,x_n)$ of size $s$ we prove that each factor has size at most a polynomial in: $s$ and the degree of the squarefree part of $f$. Consequently, if $f_1$ is a $2^{Ω(n)}$-hard polynomial then any nonzero multiple $\prod_{i} f_i^{e_i}$ is equally hard for arbitrary positive $e_i$'s, assuming that $\sum_i \text{deg}(f_i)$ is at most $2^{O(n)}$. It is an old open question whether the class of poly($n$)-sized formulas (resp. algebraic branching programs) is closed under factoring. We show that given a polynomial $f$ of degree $n^{O(1)}$ and formula (resp. ABP) size $n^{O(\log n)}$ we can find a similar size formula (resp. ABP) factor in randomized poly($n^{\log n}$)-time. Consequently, if determinant requires $n^{Ω(\log n)}$ size formula, then the same can be said about any of its nonzero multiples. As part of our proofs, we identify a new property of multivariate polynomial factorization. We show that under a random linear transformation $τ$, $f(τ\overline{x})$ completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. This with allRootsNI are the techniques that help us make progress towards the old open problems, supplementing the large body of classical results and concepts in algebraic circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \& Burgisser, FOCS 2001).

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