Polynomial Complexity Research Articles

PTTS: Zero-knowledge proof-based private token transfer system on Ethereum blockchain and its network flow based balance range privacy attack analysis

Blockchains are decentralized and immutable databases that are shared among the nodes of the network. Although blockchains have attracted a great scale of attention in the recent years by disrupting the traditional financial systems, the transaction privacy is still a challenging issue that needs to be addressed and analysed. We propose a Private Token Transfer System (PTTS) for the Ethereum public blockchain in the first part of this paper. For the proposed framework, zero-knowledge based protocol has been designed using Zokrates and integrated into our private token smart contract. With the help of web user interface designed, the end users can interact with the smart contract without any third-party setup. In the second part of the paper, we provide security and privacy analysis including the replay attack and the balance range privacy attack which has been modeled as a network flow problem. It is shown that in case some balance ranges are deliberately leaked out to particular organizations or adversarial entities, it is possible to extract meaningful information about the user balances by employing minimum cost flow network algorithms that have polynomial complexity. The experimental study reports the Ethereum gas consumption and proof generation times for the proposed framework. It also reports network solution times and goodness rates for a subset of addresses under the balance range privacy attack with respect to number of addresses, number of transactions and ratio of leaked transfer transaction amounts.

Superlinear convergence of an interior point algorithm on linear semi-definite feasibility problems

In the literature, besides the assumption of strict complementarity, superlinear convergence of implementable polynomial-time interior point algorithms using known search directions, namely, the HKM direction, its dual or the NT direction, to solve semi-definite programs (SDPs) is shown by (i) assuming that the given SDP is nondegenerate and making modifications to these algorithms [Kojima et al. Local convergence of predictor–corrector infeasible-interior-point algorithms for SDPs and SDLCPs, Math. Program. Ser. A 80 (1998), pp. 129–160], or (ii) considering special classes of SDPs, such as the class of linear semi-definite feasibility problems (LSDFPs) and requiring the initial iterate to the algorithm to satisfy certain conditions [Sim, Superlinear convergence of an infeasible predictor–corrector path-following interior point algorithm for a semidefinite linear complementarity problem using the Helmberg–Kojima–Monteiro direction, SIAM J. Optim. 21 (2011), pp. 102–126], [Sim, Interior point method on semi-definite linear complementarity problems using the Nesterov-Todd (NT) search direction: Polynomial complexity and local convergence, Comput. Optim. Appl. 74 (2019), pp. 583–621]. Otherwise, these algorithms are not easy to implement even though they are shown to have polynomial iteration complexities and superlinear convergence [Luo et al. Superlinear convergence of a symmetric primal–dual path following algorithm for semidefinite programming, SIAM J. Optim. 8 (1998), pp. 59–81]. The conditions in [Sim, Superlinear convergence of an infeasible predictor–corrector path-following interior point algorithm for a semidefinite linear complementarity problem using the Helmberg–Kojima–Monteiro direction, SIAM J. Optim. 21 (2011), pp. 102–126], [Sim, Interior point method on semi-definite linear complementarity problems using the Nesterov-Todd (NT) search direction: Polynomial complexity and local convergence, Comput. Optim. Appl. 74 (2019), pp. 583–621] that the initial iterate to the algorithm is required to satisfy to have superlinear convergence when solving LSDFPs however are not practical. In this paper, we propose a practical initial iterate to an implementable infeasible interior point algorithm that guarantees superlinear convergence when the algorithm is used to solve the homogeneous feasibility model of an LSDFP.

An algorithmic construction of union-intersection-bounded families

In this paper, we present lower bounds and algorithmic constructions of union-intersection-bounded families of sets. The lower bound is established using the Lovász Local Lemma. This bound matches the best known bound for the size of union-intersection-bounded families of sets. We then use the variable framework for the Lovász Local Lemma, to discuss an algorithm that outputs explicit constructions that attain the lower bound. The algorithm has polynomial complexity in the number of points in the family.

DELTA: Memory-Efficient Training via Dynamic Fine-Grained Recomputation and Swapping

To accommodate the increasingly large-scale models within limited-capacity GPU memory, various coarse-grained techniques, such as recomputation and swapping, have been proposed to optimize memory usage. However, these methods have encountered limitations, either in terms of inefficient memory reduction or diminished training performance. In response to this, our paper introduces DELTA, an innovative approach for memory-efficient large-scale model training that combines fine-grained memory optimization and prefetching technology to reduce memory usage while maintaining high training throughput concurrently. Initially, we formulate the problem of memory-throughput joint optimization as an easy-solving 0/1 Knapsack problem. Leveraging this formalization, we use an improving polynomial complexity heuristic algorithm to address the problem effectively. Furthermore, we introduce a novel bidirectional prefetching technology into dynamic memory management, which significantly accelerates the model training when compared to relying solely on recomputation or swapping. Finally, DELTA offers users an automated training execution library, eliminating the need for manual configuration or specialized expertise. Experimental results demonstrate the effectiveness of DELTA in reducing GPU memory consumption. Compared to state-of-the-art methods, DELTA achieves substantial memory savings ranging from 40% to 72%, while maintaining comparable convergence performance for various models, including ResNet-50, ResNet-101, and BERT-Large. Notably, DELTA enables the training of GPT2-Large and GPT2-XL with batch sizes increased by 5.5 × and 6 ×, respectively, showcasing its versatility and practicality in enabling large-scale model training on GPU hardware.

Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dots

Most problems within and beyond the scientific domain can be framed into one of the following three levels of complexity of function approximation. Type 1: Approximate an unknown function given input/output data. Type 2: Consider a collection of variables and functions, some of which are unknown, indexed by the nodes and hyperedges of a hypergraph (a generalized graph where edges can connect more than two vertices). Given partial observations of the variables of the hypergraph (satisfying the functional dependencies imposed by its structure), approximate all the unobserved variables and unknown functions. Type 3: Expanding on Type 2, if the hypergraph structure itself is unknown, use partial observations of the variables of the hypergraph to discover its structure and approximate its unknown functions. These hypergraphs offer a natural platform for organizing, communicating, and processing computational knowledge. While most scientific problems can be framed as the data-driven discovery of unknown functions in a computational hypergraph whose structure is known (Type 2), many require the data-driven discovery of the structure (connectivity) of the hypergraph itself (Type 3). We introduce an interpretable Gaussian Process (GP) framework for such (Type 3) problems that does not require randomization of the data, access to or control over its sampling, or sparsity of the unknown functions in a known or learned basis. Its polynomial complexity, which contrasts sharply with the super-exponential complexity of causal inference methods, is enabled by the nonlinear ANOVA capabilities of GPs used as a sensing mechanism.

Conformational Analysis of 1,3-Difluorinated Alkanes.

Fluorine substitution can have a profound impact on molecular conformation. Here, we present a detailed conformational analysis of how the 1,3-difluoropropylene motif (-CHF-CH2-CHF-) determines the conformational profiles of 1,3-difluoropropane, anti- and syn-2,4-difluoropentane, and anti- and syn-3,5-difluoroheptane. It is shown that the 1,3-difluoropropylene motif strongly influences alkane chain conformation, with a significant dependence on the polarity of the medium. The conformational effect of 1,3-fluorination is magnified upon chain extension, which contrasts with vicinal difluorination. Experimental evidence was obtained from NMR analysis, where polynomial complexity scaling simulation algorithms were necessary to enable J-coupling extraction from the strong second-order spectra, particularly for the large 16-spin systems of the difluorinated heptanes. These results improve our understanding of the conformational control toolkit for aliphatic chains, yield simple rules for conformation population analysis, and demonstrate quantum mechanical time-domain NMR simulations for liquid state systems with large numbers of strongly coupled spins.

Optimal Piecewise Polynomial Approximation for Minimum Computing Cost by Using Constrained Least Squares.

In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed.

Monotone classes beyond VNP

In this work, we study the natural monotone analogues of various equivalent definitions of VPSPACE: a well studied class (Poizat 2008, Koiran & Perifel 2009, Malod 2011, Mahajan & Rao 2013) that is believed to be larger than VNP. We observe that these monotone analogues are not equivalent unlike their non-monotone counterparts, and propose monotoneVPSPACE (mVPSPACE) to be defined as the monotone analogue of Poizat's definition. With this definition, mVPSPACE turns out to be exponentially stronger than mVNP and also satisfies several desirable closure properties that the other analogues may not.Our initial goal was to understand the monotone complexity of transparent polynomials, a concept that was recently introduced by Hrubeš & Yehudayoff (2021). In that context, we show that transparent polynomials with large support are hard for the monotone analogues of all the known definitions of VPSPACE, except for the one due to Poizat.

Classical simulation of non-Gaussian fermionic circuits

We propose efficient algorithms for classically simulating fermionic linear optics operations applied to non-Gaussian initial states. By gadget constructions, this provides algorithms for fermionic linear optics with non-Gaussian operations. We argue that this problem is analogous to that of simulating Clifford circuits with non-stabilizer initial states: Algorithms for the latter problem immediately translate to the fermionic setting. Our construction is based on an extension of the covariance matrix formalism which permits to efficiently track relative phases in superpositions of Gaussian states. It yields simulation algorithms with polynomial complexity in the number of fermions, the desired accuracy, and certain quantities capturing the degree of non-Gaussianity of the initial state. We study one such quantity, the fermionic Gaussian extent, and show that it is multiplicative on tensor products when the so-called fermionic Gaussian fidelity is. We establish this property for the tensor product of two arbitrary pure states of four fermions with positive parity.

Efficiency of neural-network state representations of one-dimensional quantum spin systems

Neural-network state representations of quantum many-body systems are attracting great attention and more rigorous quantitative analysis of their expressibility and complexity is warranted. Our analysis of the restricted Boltzmann machine (RBM) state representation of one-dimensional (1D) quantum spin systems provides new insight into their computational complexity. We define a class of long-range-fast-decay (LRFD) RBM states with quantifiable upper bounds on truncation errors and provide numerical evidence for a large class of 1D quantum systems that may be approximated by LRFD RBMs of at most polynomial complexities. These results lead us to conjecture that the ground states of a wide range of quantum systems may be exactly represented by LRFD RBMs or a variant of them, even in cases where other state representations become less efficient. At last, we provide the relations between multiple typical state manifolds. Our work proposes a paradigm for doing complexity analysis for generic long-range RBMs which naturally yields a further classification of this manifold. This paradigm and our characterization of their nonlocal structures may pave the way for understanding the natural measure of complexity for quantum many-body states described by RBMs and are generalizable for higher-dimensional systems and deep neural-network quantum states. Published by the American Physical Society 2024

Modeling and Performance Analysis of a Notification-Based Method for Processing Video Queries on the Fly

With the rapid growth of video data, the search for content and events in videos is becoming increasingly relevant, and many challenges arise. Various approaches have been proposed to deal with many issues. However, many open questions are still related to computational cost and latency, especially for real-time applications. Considering the need for new and efficient solutions, the so-called NOP (Notification Oriented Paradigm) could be a suitable alternative. NOP introduced a new way of thinking and developing software in which small collaborative entities perform fact execution and logical decision processing based on precise notifications. Following these concepts and practical tools, this paper proposes a new querying processing method based on NOP, focusing on search and matching in a continuous flow context. Experiments on a labeled dataset demonstrated the suitability of the proposed method for low-latency processing with polynomial complexity. The results are better than the state of the art, which works at exponential cost.

Polynomial computational complexity of matrix elements of finite-rank-generated single-particle operators in products of finite bosonic states

It is known that computing the permanent of the matrix 1+A, where A is a finite-rank matrix, requires a number of operations polynomial in the matrix size. Motivated by the boson-sampling proposal of restricted quantum computation, I extend this result to a generalization of the matrix permanent: An expectation value in a product of a large number of identical bosonic states with a bounded number of bosons. This result complements earlier studies on the computational complexity in boson sampling and related setups. The proposed technique based on the Gaussian averaging is equally applicable to bosonic and fermionic systems. This also allows us to improve an earlier polynomial complexity estimate for the fermionic version of the same problem.

Sufficient control of complex networks

In this paper, we propose to study sufficient control of complex networks, which is to control a sufficiently large portion of the network, where only the quantity of controllable nodes matters. To the best of our knowledge, this is the first time that such a problem is investigated. We prove that the sufficient controllability problem can be converted into a minimum-cost flow problem, for which an algorithm with polynomial complexity can be devised. Further, we study the problem of minimum-cost sufficient control, which is to drive a sufficiently large subset of the network nodes to any predefined state with the minimum cost using a given number of controllers. The problem is NP-hard. We propose an “extended L0-norm-constraint-based Projected Gradient Method” (eLPGM) algorithm, which achieves suboptimal solutions for the problems at small or medium sizes. To tackle the large-scale problems, we propose to convert the control problem into a graph problem and devise an efficient low-complexity “Evenly Divided Control Paths” (EDCP) algorithm to tackle the graph problem. Simulation results on both synthetic and real-life networks are provided, demonstrating the satisfactory performance of the proposed methods.

Complete Neural Networks for Complete Euclidean Graphs

Neural networks for point clouds, which respect their natural invariance to permutation and rigid motion, have enjoyed recent success in modeling geometric phenomena, from molecular dynamics to recommender systems. Yet, to date, no architecture with polynomial complexity is known to be complete, that is, able to distinguish between any pair of non-isomorphic point clouds. We fill this theoretical gap by showing that point clouds can be completely determined, up to permutation and rigid motion, by applying the 3-WL graph isomorphism test to the point cloud's centralized Gram matrix. Moreover, we formulate an Euclidean variant of the 2-WL test and show that it is also sufficient to achieve completeness. We then show how our complete Euclidean WL tests can be simulated by an Euclidean graph neural network of moderate size and demonstrate their separation capability on highly symmetrical point clouds.

Learning to Pivot as a Smart Expert

Linear programming has been practically solved mainly by simplex and interior point methods. Compared with the weakly polynomial complexity obtained by the interior point methods, the existence of strongly polynomial bounds for the length of the pivot path generated by the simplex methods remains a mystery. In this paper, we propose two novel pivot experts that leverage both global and local information of the linear programming instances for the primal simplex method and show their excellent performance numerically. The experts can be regarded as a benchmark to evaluate the performance of classical pivot rules, although they are hard to directly implement. To tackle this challenge, we employ a graph convolutional neural network model, trained via imitation learning, to mimic the behavior of the pivot expert. Our pivot rule, learned empirically, displays a significant advantage over conventional methods in various linear programming problems, as demonstrated through a series of rigorous experiments.

A polynomial algorithm for some instances of NP-complete problems

In this paper, given a fixed reference point and a fixed intersection of finitely many equal radii balls, we consider the problem of finding a point in the said set which is the most distant, under Euclidean distance, to the said reference point. This proble is NP-complete in the general setting. We give sufficient conditions for the existence of an algorithm of polynomial complexity which can solve the problem, in a particular setting. Our algorithm requires that any point in the said intersection to be no closer to the given reference point than the radius of the intersecting balls. Checking this requirement is a convex optimization problem hence one can decide if running the proposed algorithm enjoys the presented theoretical guarantees. We also consider the problem where a fixed initial reference point and a fixed polytope are given and we want to find the farthest point in the polytope to the given reference point. For this problem we give sufficient conditions in which the solution can be found by solving a linear program. Both these problems are known to be NP-complete in the general setup, i.e. the existence of an algorithm which solves any of the above problems without restrictions on the given reference point and search set is undecided so far. Mathematics Subject Classification (2010): 90-08. Received 21 December 2021; Accepted 01 August 2023

Algorithms for the Reconstruction of Genomic Structures with Proofs of Their Low Polynomial Complexity and High Exactness

The mathematical side of applied problems in multiple subject areas (biology, pattern recognition, etc.) is reduced to the problem of discrete optimization in the following mathematical method. We were provided a network and graphs in its leaves, for which we needed to find a rearrangement of graphs by non-leaf nodes, in which the given functional reached its minimum. Such a problem, even in the simplest case, is NP-hard, which means unavoidable restrictions on the network, on graphs, or on the functional. In this publication, this problem is addressed in the case of all graphs being so-called “structures”, meaning directed-loaded graphs consisting of paths and cycles, and the functional as the sum (over all edges in the network) of distances between structures at the endpoints of every edge. The distance itself is equal to the minimal length of sequence from the fixed list of operations, the composition of which transforms the structure at one endpoint of the edge into the structure at its other endpoint. The list of operations (and their costs) on such a graph is fixed. Under these conditions, the given discrete optimization problem is called the reconstruction problem. This paper presents novel algorithms for solving the reconstruction problem, along with full proofs of their low error and low polynomial complexity. For example, for the network, the problem is solved with a zero error algorithm that has a linear polynomial computational complexity; and for the tree the problem is solved using an algorithm with a multiplicative error of at most two, which has a second order polynomial computational complexity.

Isoperimetric inequalities for real‐valued functions with applications to monotonicity testing

AbstractWe generalize the celebrated isoperimetric inequality of Khot, Minzer, and Safra (SICOMP 2018) for Boolean functions to the case of real‐valued functions . Our main tool in the proof of the generalized inequality is a new Boolean decomposition that represents every real‐valued function over an arbitrary partially ordered domain as a collection of Boolean functions over the same domain, roughly capturing the distance of to monotonicity and the structure of violations of to monotonicity. We apply our generalized isoperimetric inequality to improve algorithms for testing monotonicity and approximating the distance to monotonicity for real‐valued functions. Our tester for monotonicity has query complexity , where is the size of the image of the input function. (The best previously known tester makes queries, as shown by Chakrabarty and Seshadhri (STOC 2013).) Our tester is nonadaptive and has 1‐sided error. We prove a matching lower bound for nonadaptive, 1‐sided error testers for monotonicity. We also show that the distance to monotonicity of real‐valued functions that are ‐far from monotone can be approximated nonadaptively within a factor of with query complexity polynomial in and the dimension . This query complexity is nearly optimal for nonadaptive algorithms even for the special case of Boolean functions (The best previously known distance approximation algorithm for real‐valued functions, by Fattal and Ron (TALG 2010) achieves ‐approximation.).

Efficient GPU implementation of randomized SVD and its applications

Matrix decompositions are ubiquitous in machine learning, including applications in dimensionality reduction, data compression and deep learning algorithms. Typical solutions for matrix decompositions have polynomial complexity which significantly increases their computational cost and time. In this work, we leverage efficient processing operations that can be run in parallel on modern Graphical Processing Units (GPUs), predominant computing architecture used e.g. in deep learning, to reduce the computational burden of computing matrix decompositions. More specifically, we reformulate the randomized decomposition problem to incorporate fast matrix multiplication operations (BLAS-3) as building blocks. We show that this formulation, combined with fast random number generators, allows to fully exploit the potential of parallel processing implemented in GPUs. Our extensive evaluation confirms the superiority of this approach over the competing methods and we release the results of this research as a part of the official CUDA implementation.11https://docs.nvidia.com/cuda/cusolver/index.html.

Optimizing Lithium-Ion Battery Modeling: A Comparative Analysis of PSO and GWO Algorithms

In recent years, the modeling and simulation of lithium-ion batteries have garnered attention due to the rising demand for reliable energy storage. Accurate charge cycle predictions are fundamental for optimizing battery performance and lifespan. This study compares particle swarm optimization (PSO) and grey wolf optimization (GWO) algorithms in modeling a commercial lithium-ion battery, emphasizing the voltage behavior and the current delivered to the battery. Bio-inspired optimization tunes parameters to reduce the root mean square error (RMSE) between simulated and experimental outputs. The model, implemented in MATLAB/Simulink, integrates electrochemical parameters and estimates battery behavior under varied conditions. The assessment of terminal voltage revealed notable enhancements in the model through both the PSO and GWO algorithms compared to the non-optimized model. The GWO-optimized model demonstrated superior performance, with a reduced RMSE of 0.1700 (25 °C; 3.6 C, 455 s) and 0.1705 (25 °C; 3.6 C, 10,654 s) compared to the PSO-optimized model, achieving a 42% average RMSE reduction. Battery current was identified as a key factor influencing the model analysis, with optimized models, particularly the GWO model, exhibiting enhanced predictive capabilities and slightly lower RMSE values than the PSO model. This offers practical implications for battery integration into energy systems. Analyzing the execution time with different population values for PSO and GWO provides insights into computational complexity. PSO exhibited greater-than-linear dynamics, suggesting a polynomial complexity of O(nk), while GWO implied a potential polynomial complexity within the range of O(nk) or O(2n) based on execution times from populations of 10 to 1000.