arXiv subjects

Prateek Dwivedi

Publications and source records attributed to Prateek Dwivedi.

14 recordsLinked to original sources

arXiv2026

Tight Lower Bounds for Algebraic Communication and Applications

Communication complexity studies how much information must be exchanged to solve a problem whose input is split among several parties. The classical setting deals with Boolean inputs split between two parties. We study an algebraic variant, where the inputs are vectors over a field $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$. Alice and Bob have inputs $X\in \mathbb{F}^n$ and $Y\in \mathbb{F}^n$, respectively. We consider two kinds of tasks: the polynomial evaluation problem (compute the value of a polynomial $g\in \mathbb{F}[X,Y]$), and the set-recognition problem (decide whether (X,Y) is in $S$, for $S\subseteq \mathbb{F}^{n} \times \mathbb{F}^n$). In both settings, Alice and Bob send evaluations of polynomials depending only on their own inputs. In the set-recognition problem, a referee receives the messages and may apply polynomial tests to the messages received so far; the outcomes of these tests determine acceptance or rejection. The protocols may be deterministic or probabilistic. We study: - Upper bounds and reductions: We give non-trivial upper bounds for a range of natural polynomial evaluation and set-recognition problems and prove reductions between different problems, which help organize the landscape of the model. - A lower bound framework and tight lower bounds: Our main technical contribution is a general framework for proving lower bounds for algebraic set-recognition problems. We prove several probabilistic lower bounds for natural problems, giving tight or near-tight characterizations of their algebraic communication. - Applications of the framework: Finally, we give two applications of our framework: proving lower bounds for a class of left-to-right algebraic algorithms (algebraic scanners) and a more general algebraic computational setting inspired by the BSS model.

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arXiv2026

A primer on the closure of algebraic complexity classes under factoring

Polynomial factorisation is a fundamental problem in computational algebra. Over the past half century, a variety of algorithmic techniques have been developed to tackle different variants of this problem. In parallel, algebraic complexity theory classifies polynomials into complexity classes based on their computational hardness. This raises a natural question: Are these complexity classes closed under factorisation? In this survey, we revisit pivotal techniques in polynomial factorisation: Hensel lifting, Newton iteration, and Lagrange inversion. These techniques have played an essential role in resolving key factoring questions in algebraic complexity for more than half a century. We examine and organise the known results through the lens of these techniques, discussing their underlying mathematical equivalence while reflecting on how their applications vary depending on the problem context. We focus on prominent algebraic complexity classes, including $\text{VP}$ (circuits of polynomial size and degree), its closure $\overline{\text{VP}}$, the class $\text{VNP}$ (verifier circuits of polynomial size and degree), $\text{VBP}$ (polynomial-size branching programs), $\text{VF}$ (polynomial-size formulas), and $\text{VP}_{\text{nb}}$ (circuits of polynomial size and exponential degree). We also discuss bounded-depth circuits and sparse polynomials. Along the way, we highlight several unresolved open problems.

cs.CC↗

arXiv2026

Lower Bounds in Algebraic Complexity via Symmetry and Homomorphism Polynomials

Valiant's conjecture asserts that the circuit complexity classes VP and VNP are distinct, meaning that the permanent does not admit polynomial-size algebraic circuits. As it is the case in many branches of complexity theory, the unconditional separation of these complexity classes seems elusive. In stark contrast, the symmetric analogue of Valiant's conjecture has been proven by Dawar and Wilsenach (2020): the permanent does not admit symmetric algebraic circuits of polynomial size, while the determinant does. Symmetric algebraic circuits are both a powerful computational model and amenable to proving unconditional lower bounds. In this paper, we develop a symmetric algebraic complexity theory by introducing symmetric analogues of the complexity classes VP, VBP, and VF called symVP, symVBP, and symVF. They comprise polynomials that admit symmetric algebraic circuits, skew circuits, and formulas, respectively, of polynomial orbit size. Having defined these classes, we show unconditionally that $\mathsf{symVF} \subsetneq \mathsf{symVBP} \subsetneq \mathsf{symVP}$. To that end, we characterise the polynomials in symVF and symVBP as those that can be written as linear combinations of homomorphism polynomials for patterns of bounded treedepth and pathwidth, respectively. This extends a previous characterisation by Dawar, Pago, and Seppelt (2026) of symVP. Finally, we show that symVBP and symVP contain homomorphism polynomials which are VBP- and VP-complete, respectively. We give general graph-theoretic criteria for homomorphism polynomials and their linear combinations to be VBP-, VP-, or VNP-complete. These conditional lower bounds drastically enlarge the realm of natural polynomials known to be complete for VNP, VP, or VBP. Under the assumption VFPT $\neq$ VW[1], we precisely identify the homomorphism polynomials that lie in VP as those whose patterns have bounded treewidth.

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arXiv2025

On Closure Properties of Read-Once Oblivious Algebraic Branching Programs

We investigate the closure properties of read-once oblivious Algebraic Branching Programs (roABPs) under various natural algebraic operations and prove the following. - Non-closure under factoring: There is a sequence of explicit polynomials $(f_n(x_1,\ldots, x_n))_n$ that have $\mathsf{poly}(n)$-sized roABPs such that some irreducible factor of $f_n$ does not have roABPs of superpolynomial size in any order. - Non-closure under powering: There is a sequence of polynomials $(f_n(x_1,\ldots, x_n))_n$ with $\mathsf{poly}(n)$-sized roABPs such that any super-constant power of $f_n$ does not have roABPs of polynomial size in any order (and $f_n^n$ requires exponential size in any order). - Non-closure under symmetric compositions: There are symmetric polynomials $(f_n(e_1,\ldots, e_n))_n$ that have roABPs of polynomial size such that $f_n(x_1,\ldots, x_n)$ do not have roABPs of subexponential size. (Here, $e_1,\ldots, e_n$ denote the elementary symmetric polynomials in $n$ variables.) These results should be viewed in light of known results on models such as algebraic circuits, (general) algebraic branching programs, formulas and constant-depth circuits, all of which are known to be closed under these operations. To prove non-closure under factoring, we construct hard polynomials based on expander graphs using gadgets that lift their hardness from sparse polynomials to roABPs. For symmetric compositions, we show that the circulant polynomial requires roABPs of exponential size in every variable order.

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arXiv2025

Monotone Bounded-Depth Complexity of Homomorphism Polynomials

For every fixed graph $H$, it is known that homomorphism counts from $H$ and colorful $H$-subgraph counts can be determined in $O(n^{t+1})$ time on $n$-vertex input graphs $G$, where $t$ is the treewidth of $H$. On the other hand, a running time of $n^{o(t / \log t)}$ would refute the exponential-time hypothesis. Komarath, Pandey and Rahul (Algorithmica, 2023) studied algebraic variants of these counting problems, i.e., homomorphism and subgraph $\textit{polynomials}$ for fixed graphs $H$. These polynomials are weighted sums over the objects counted above, where each object is weighted by the product of variables corresponding to edges contained in the object. As shown by Komarath et al., the $\textit{monotone}$ circuit complexity of the homomorphism polynomial for $H$ is $Θ(n^{\mathrm{tw}(H)+1})$. In this paper, we characterize the power of monotone $\textit{bounded-depth}$ circuits for homomorphism and colorful subgraph polynomials. This leads us to discover a natural hierarchy of graph parameters $\mathrm{tw}_Δ(H)$, for fixed $Δ\in \mathbb N$, which capture the width of tree-decompositions for $H$ when the underlying tree is required to have depth at most $Δ$. We prove that monotone circuits of product-depth $Δ$ computing the homomorphism polynomial for $H$ require size $Θ(n^{\mathrm{tw}_Δ(H^{\dagger})+1})$, where $H^{\dagger}$ is the graph obtained from $H$ by removing all degree-$1$ vertices. This allows us to derive an optimal depth hierarchy theorem for monotone bounded-depth circuits through graph-theoretic arguments.

cs.CC↗

arXiv2025

Emergence of Order in Chemically Active Droplets: Temporal Dynamics and Collective Behavior

Collective behaviors such as swarming, chemical signaling, and clustering are fundamental to biological microorganisms, enabling hierarchical colony formation, coordinated motion, and enhanced nutrient accessibility crucial for their survival. Over the past few decades, extensive research has been dedicated to unraveling the mechanisms underlying these diverse collective patterns through experimental model systems. Among these, active droplets have emerged as valuable synthetic analogs, effectively replicating key biological attributes and serving as ideal platforms for investigating collective phenomena. This research explores the collective behavior of 4-Cyano-4-pentyl-biphenyl (5CB) oil droplets across varying Péclet ($Pe$) numbers. At high $Pe$, droplets exhibit a pusher mode of propulsion and form dynamic chain-like patterns. Decreasing $Pe$ enhances repulsive interactions among droplets, resulting in the inhibition of clustering. In the low $Pe$ regime, their repulsive interactions predominated by chemical field lead to the emergence of an ordered structure. Furthermore, we illustrate how active droplets efficiently navigate within a soft structured environment. These findings contribute to our comprehension of self-organized phenomena in active matter systems and provide insights for designing strategies for controlled locomotion in intricate fluidic environments.

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arXiv2025

Chemical interactions in active droplets

Interactions among biologically active agents is facilitated by their self-generated chemical and hydrodynamic fields. In order to elucidate the pair-wise interactions between such micro-organisms, we employ active droplets as a model system, capable of self-generating chemical and hydrodynamic fields. We demonstrate that the solute Péclet number ($Pe$), characterizing the relative strength of its convective to diffusive transport, plays a crucial role in determining how the chemical and hydrodynamic fields impact their interactions. Our findings reveal that at low $Pe$, the interaction is predominantly governed by chemo-repulsive effects, leading to droplets avoiding physical contact. Conversely, at elevated $Pe$, hydrodynamic interactions become more influential, leading to physical engagement. However, irrespective of $Pe$, the interaction of a droplet with the chemical trail of another droplet is always governed by chemo-repulsive effects. Furthermore, our results establish that the chemo-repulsive deflection/rebounding of droplets is influenced by the droplets' inherent chemical polarity, as determined by its $Pe$, independent of their approach orientation. Our findings offer a methodology for tuning the outcomes of binary interactions among chemically active droplets, laying the groundwork for potential studies on their collective dynamics.

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arXiv2024

Motility and pair-wise interactions of chemically active droplets in 1-D confinement

Self-propelled droplets serve as ideal model systems to delve deeper into understanding of the motion of biological micro-swimmers by simulating their motility. Biological microorganisms are renowned for showcasing a diverse array of dynamic swimming behaviors when confronted with physical constraints. This study aims to elucidate the impact of physical constraints on swimming characteristics of biological microorganisms. To achieve this, we present observations on the individual and pair-wise behavior of micellar solubilized self-propelled 4-Cyano-4'-pentyl-biphenyl (5CB) oil droplets in a square capillary channel filled with a surfactant trimethyl ammonium bromide (TTAB) aqueous solution. To explore the effect of the underlying Péclet ($Pe$) number of the swimming droplets, the study is also performed in the presence of additives such as high molecular weight polymer Polyethylene oxide (PEO) and molecular solute glycerol. The capillary confinement restricts droplet to predominantly one-dimensional (1D) motion, albeit with noticeable differences in their motion across the three scenarios. Through a characterization of the chemical and hydrodynamic flow fields surrounding the droplets, we illustrate that the modification of the droplets' chemical field due to confinement varies significantly based on the underlying differences in the Péclet number ($Pe$) in these cases. This alteration in the chemical field distribution notably affects the individual droplets' motion. Moreover, these distinct chemical field interactions between the droplets also lead to variations in their pair-wise motion, ranging from behaviors like chasing to scattering.

cond-mat.soft↗

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 $\wedge$.

cs.CC↗

arXiv2022

Mode-Switching of Active Droplets in Macromolecular Solutions

Typical bodily and environmental fluids encountered by biological swimmers consist of dissolved macromolecules such as proteins and polymers, often rendering them non Newtonian. To mimic such scenarios, we investigate the motion of swimming droplets in an ambient medium doped with polymers as macromolecular solutes. Active droplets mimic the essential propulsive characteristics of several biological swimmers and serve as ideal model systems to widen our understanding of their locomotive strategies. Our experiments reveal extreme sensitivity of droplet motion to the presence of macromolecular solutes in the ambient medium. Through in-situ visualization of the self-generated chemical field around the droplet, we report unexpectedly high diffusivity of filled micelles in the presence of high molecular weight polymer solutes or macromolecules. This is attributed to the limitation of Stokes-Einstein relationship in accurately predicting micelle diffusivity due to significant size disparity between micelles and the macromolecular solute. With an increase in polymer concentration, particle image velocimetry reveals a mode-switching, from the conventional pusher mode to a puller mode of propulsion, characterized by a more persistent droplet motion. With a further increase in concentration, a secondary transition from smooth to a jittery mode of propulsion occurs. A robust Peclet number framework is proposed that successfully captures the observed mode-switching of active droplets. Our experiments unveil a novel route to orchestrate complex transitions in active droplet propulsion by doping the ambient medium with suitable choice of macromolecules.

cond-mat.soft↗

arXiv2022

Deforming Active Droplets in Viscoelastic Media

To mimic the motion of biological swimmers in bodily fluids, a novel experimental system of micellar solubilization driven active droplets in a visco-elastic polymeric solution is presented. The visco-elastic nature of the medium, characterized by the Deborah number (De), is tuned by varying the surfactant (fuel) and polymer concentration in the ambient medium. At moderate De, the droplet exhibits a steady deformed shape, markedly different from the spherical shape observed in Newtonian media. A theoretical analysis based on the normal stress balance at the interface is shown to accurately predict the droplet shape. With a further increase in De, time-periodic deformations accompanied by oscillatory transitions in swimming modes are observed. The study unveils the rich complexity in the motion of active droplets in viscoelastic fluids, which has been hitherto unexplored.

cond-mat.soft↗

arXiv2021

Rheotaxis of Active Droplets

Rheotaxis is a well-known phenomenon among microbial organisms and artificial active colloids, wherein the swimmers respond to an imposed flow. We report the first experimental evidence of upstream rheotaxis by spherical active droplets. It is shown that the presence of a nearby wall and the resulting strong flow-gradient at the droplet level is at the root of this phenomenon. Experiments with optical cells of different heights reveal that rheotaxis is observed only for a finite range of shear rates, independent of the bulk flow-rate. We conjecture that the flow induced distortion of an otherwise isotropic distribution of filled/empty micelles around the droplet propels it against the flow. We also show that nematic droplets exhibit elastic stress-induced oscillations during their rheotactic flight. A promising potential of manipulating the rheotactic behavior to trap as well as shuttle droplets between target locations is demonstrated, paving way to potentially significant advancement in bio-medical applications.

cond-mat.soft↗

arXiv2021

Interaction of Active Janus Particles with Passive Tracers

In this study, we investigated the motion of active SiO2-Pt Janus particles in the 2D bath of smaller silica tracers dispersed in varying areal densities. The effect on the organization of the tracer particles around the active JPs was also explored. Our experiments indicate that the interaction between the tracers and the active JPs mainly depend on the nature of collision marked by the duration of contact. For all the concentration regimes, we have shown that the short time collisions do not have significant impact on the motion of active JPs, however, during moderate/long-time collisions tracer(s) can lead to a significant change in active JPs motion and even cause them to rotate. In the concentrated regime, our experiments reveal the emergence of a novel organizational behavior of the passive tracers on the trailing Pt and the leading SiO2 with a strong dependence on the nature of collision.

cond-mat.soft↗

arXiv2020

Solute Induced Jittery Motion of Self-Propelled Droplets

The intriguing role of the presence of solutes in the activity of a self-propelling droplet is investigated. A system of self-propelling micron sized 4-pentyl-4-biphenylcarbonitrile (5CB) droplets in an aqueous solution of tetradecyltrimethylammonium bromide (TTAB) as surfactant is considered. It is shown that addition of glycerol causes the active 5CB droplet to exhibit a transition from smooth to jittery motion. The motion is found to be independent of the droplet size and the nematic state of 5CB. Analogous experiments with Polyacrylamide (PAAm), Polyvinylpyrrolidone (PVP) and Polyvinyl Alcohol (PVA), as solutes confirm that such a transition cannot merely be explained solely based on the viscosity or Peclet number of the system. We propose that the specific nature of physicochemical interactions between the solute and the droplet interface is at the root of this transition. The experiments show that the time-scales associated with the influx and redistribution of surfactants at the interface are altered in the presence of solutes. Glycerol and PVP significantly enhance the rate of solubilization of the 5CB droplets resulting in a quicker re-distribution of the adsorbed TTAB molecules on the interface, causing the droplet to momentarily stop and then restart in an independent direction. On the other hand, low solubilization rates in the presence of PAAm and PVA lead to smooth trajectories. Our hypothesis is supported by the time evolution of droplet size and interfacial velocity measurements in the presence and absence of solute. Overall, our results provide fundamental insights into the complex interactions emerging due to the presence of solutes.

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