Polynomial Complexity Research Articles

A NEW METHOD FOR SEARCHING A HAMILTON CYCLE ON A GRAPH

In discrete mathematics, there are many problems that belong to the NP class of complexity. The solution of these problems has both theoretical and practical value. One of them is the problem of finding a Hamiltonian cycle on a graph. The aim of the work is to develop a new method and algorithm for solving this problem, which are preferable to the existing methods by time and quality of the solution. The paper analyzes the problem and existing methods of solving it, and identifies the disadvantages of these methods. It is shown that all known methods for solving this problem are based on the implementation of enumeration of solutions or on intuitive heuristics. The first of them are characterized by a non-polynomial amount of time for solving, and the second does not provide the optimal solution. The reason for this state is the inability to formulate conditions that determine the optimal solution of the problem. In this case, the only possible way to solve the problem is still to enumerate over the options, and to reduce the cost of finding a solution, you must resort to reducing the space for iterating over the options. The paper presents new principles for finding a solution and offers a new method for solving the problem. On the basis of the new method, a polynomial algorithm for solving the problem is developed. The search for a Hamiltonian cycle in a graph is reduced to the search for a closed path in a new graph of shortest paths. Dijkstra's algorithm is used to construct the shortest paths graph. The space of enumeration of solutions to the problem consists of solutions that are constructed from each vertex of the graph in the shortest paths graph. Testing of the developed program showed the efficiency of the developed method and algorithm for solving the problem. The proposed solution method significantly reduces the search space and allows us to find the optimal solution with polynomial complexity, both in full and incomplete graphs. The considered method is suitable for parallel implementation, which gives an additional gain in time and allows you to fully use the parallel capabilities of modern multi-core processors.

Cooperative Caching for Multiple Bitrate Videos in Small Cell Edges

Caching popular videos at mobile edge servers (MESs) has been confirmed as a promising method to improve mobile users (MUs) perceived quality of experience (QoE) and to alleviate the server load. However, with the multiple bitrate encoding techniques prevalently employed in modern streaming services, caching deployment is challenging for the following three facts: (1) cooperative caching should be explored for MUs located at overlapped coverage areas of MESs; (2) there exists tradeoff consideration for caching either high bitrate videos or high diversity videos; and (3) the relationship between MU perceived QoE and MU received bitrate, known as QoE function, varies in different services. Aiming to maximize the MU perceived QoE, we formulate the multiple bitrate video caching problem, and prove this problem is NP-hard for any given positive and strictly increasing QoE function. We then propose a polynomial complexity algorithm based on a general QoE function, which can achieve an approximate ratio arbitrarily close to 1/2. Specifically, for a linear QoE function, we explore useful property of optimal solutions, based on which more efficient algorithms are proposed. We demonstrate the effectiveness of our solutions via both theoretical analysis and extensive simulations.

A band selection approach based on Lévy sine cosine algorithm and alternative distribution for hyperspectral image

ABSTRACT Hyperspectral image (HSI) has become one of the most important remote sensing techniques for object interpretation by its abundant band information. As the data dimension increases, band selection technique is utilized to achieve the highest possible classification accuracy with fewer bands. Essentially, it is considered as an NP-hard problem, which is a non-deterministic problem within polynomial complexity and difficult to achieve a satisfactory solution using traditional search methods. Sine cosine algorithm (SCA) is a recently developed swarm intelligence algorithm based on the calculation of sine and cosine functions. To determine the parameters setting in SCA, Lévy flight technique is employed to improve the exploitation phase of the algorithm. In the paper, a new band selection method on the basis of SCA with a Lévy flight is proposed, and an alternative distribution is utilized to decrease the band dimension of HSI. In addition, crossover operation is conducted to enhance the optimal bits of each individual, and an evaluation criterion for assessing the performance of band selection is defined, allowing the classification accuracy and selected number of bands to be as balanced as possible. Experimental results demonstrate that the proposed band selection method is superior to other state-of-the-art approaches in terms of band subsets that achieve higher classification accuracy.

MmWave-NOMA-Based Low-Latency and High-Reliable Communications for Enhancement of V2X Services

The enhanced vehicle-to-everything (eV2X) systems have stricter constraints of latency and reliability than before. To meet the low latency and high reliability (LLHR) demands, the investigation of semi-persistent scheduling (SPS) is of great significance. For the first time, we integrate millimeter wave (mmWave) and non-orthogonal multiple access (NOMA) into SPS strategy. The proposed strategy includes the beam division and user clustering process, the power allocation process, and the vehicular user equipment (VUE) – resource block (RB) matching process. Firstly, the joint-optimization problem in terms of user scheduling indicators and power allocation factors is formulated to minimize the SPS period. This problem is then proved to be non-deterministic polynomial hard (NP-hard) by being reduced into a vertex coloring problem. To solve it with polynomial complexity, we divide it into two subproblems and calculate them by iteration. For one subproblem, the LLHR power control algorithm is proposed to solve the non-convex reliability-optimization problem, which can provide an evaluation indicator for the user scheduling. For the other subproblem, the beam division and VUE clustering algorithm is designed to reduce the complexity of the VUE-RB matching. After that, the matching problem with peer effects is solved by the proposed union-based matching algorithm. The analytical results on stability, convergence and computational complexity are presented, and simulation results show that the proposed SPS strategy outperforms the existing schemes.

Reducing the Standstill Spacing and Duration of Safe Braking Control in Vehicle Platoons With Delays

Braking control, especially in an emergency, is a key technology that is needed to ensure safety in vehicle platoons. However, delays in vehicle platoons can severely affect braking control. This paper proposes an optimized braking control to reduce the standstill spacing and braking duration during delays so that a platoon will stop within a short time frame with a reduced length, thus improving road utilization while ensuring both inter-vehicle and in-vehicle safety. However, two challenges need to be addressed. First, due to the delay in car-following interactions and the nonlinearity of the control law, an analysis model is needed to quantize the duration and distance during an emergency braking with delays. Second, the optimization of these control parameters is an NP-hard problem. Therefore, delay differential equations are introduced to model the braking process, and a crossing criterion is introduced to establish the relationship between the control law and the braking process. The propositions of standstill spacing and braking duration are then derived based on the Runge-Kutta method. According to these criteria, a particle swarm optimization (PSO) based on a lexicographic method with a penalty function is introduced to provide a solution framework with polynomial complexity. Simulation results verify the accuracy of the braking modeling process. Moreover, the results verify the performance of the proposed algorithm and provide a reference for platoon control.

A Study on Heuristic Task Scheduling Optimizing Task Deadline Violations in Heterogeneous Computational Environments

In this paper, we focus on the problem of optimizing deadline violations for executing tasks in various heterogeneous computational environments. To address the problem, we formulated it as a binary nonlinear programming (BNP) model, which maximize the number of completed tasks and optimize the resource utilization of servers. To solve the BNP model in a polynomial complexity, we propose a heuristic task scheduling method, which iteratively schedules a task to the first core such that the accumulated slack time of all scheduled tasks is minimum, until the core cannot finish any task, and executes tasks with the earliest deadline first in each core to execute as many task as possible in a core. Experiment results based on a real world trace show that our method has upto 100% less task violations, and has the best performance in resource efficiency optimization in overall, compared with eight classical and state-of-the-art heuristic methods.

Задача выбора вычислительных процессов, обеспечивающих защиту информации, для серверов распределенной системы и алгоритмы ее решения

Purpose: providing information security on servers of various destinations of an automated system based on the formulation of an optimization task setting for selecting auxiliary processes for information protection, developing and researching algorithms for solving this problem. Method: To solve the problem, the authors propose two exact algorithms for incomplete search with exponential computational complexity, based on the ideas of the Balash method. One algorithm starts with a solution consisting of all ones, and the second algorithm starts with a solution consisting of all zeros. Also proposed are two approximate algorithms with polynomial complexity, based on the ideas of the “greedy” algorithm, one algorithm starts the search from a zero solution, the other from all-one solution. Result: Mathematical model and algorithms for solving the problem of selecting processes to protect information in conditions of limited computing resources of servers. The process selection model is a Boolean programming problem with a non-linear quality indicator and linear constraints. The indicator provides an estimate of the damage prevented when using the selected processes, taking into account the probability or possibility of various attacks on servers, the value of stored data, and the probability of protection from attacks using processes. During the experiments, recommendations were developed for choosing one of two algorithms (search starting from the zero solution and search starting from all-one solution) among pairs of exact and approximate algorithms depending on the availability of resources in order to reduce the time to solve the problem.

A multiple objective optimization model for aircraft arrival and departure scheduling on multiple runways.

This study proposes a multi-objective mixed integer linear programming (MOMILP) model for assigning a set of flights to different runways and determining their actual arrival and departure times. The proposed model envisages unique operation model of each runway (i.e., takeoff, landing, or mixed takeoff and landing). Further, interference in two flights between adjacent runways are also fully considered in this model. The work aims at reveal the optimal relationship between traffic stream characteristics, operation mode of each runway and flight scheduling to simultaneously minimizing flight delays and maximizing runway utilization. Since the problem of interest has a non-deterministic polynomial (NP-hard) complexity, a heuristic-based non-dominated sorting genetic algorithm (NSGA-II) is also presented to find Pareto-optimal solutions in a reasonable amount of time, where coding structure and heuristic algorithm for producing initial population are defined. Finally, a real-world example is provided to compare the difference in quality between the proposed and traditional models, and reveal changes in trends between delay time of flights and idle time of the runways, which can verify the correctness of the model.

RECOMMENDER SYSTEMS: PREDICTING WITH MACHINE LEARNING BASED ON POPULATION-BASED ALGORITHM

The article considers the formulation of the forecasting problem as well as such problems of recommender systems as data sparsity, cold start, scalability, synonymy, fraud, diversity, white crows. Combining the results of collaborative and content filtering gives us two possibilities. On the one hand, to weigh the results according to the content data. On the other hand, to shift these weights towards collaborative filtering as soon as data about a particular user appears. In turn, this improves the accuracy of the recommendations. The authors propose a hybrid model of a recommender system. Such a system includes the characteristics of collaborative and content filtering both. Also, the population-based algorithm for filtering and the architecture of a recommendation system based on it are described in the article. The algorithm consists of the following steps: study the search space; synthesis of solutions, i.e. points of this space; request quality assessment decisions or “fitness”; using it to make “natural selection”. Here we see the learning process about which areas of the search space contain the best solutions. The population of user “characteristics” encoded in the population-based algorithm supports a variety of input data in a hybrid model. The authors propose a coding structure for decisions in a population-based algorithm using the example of a recommender movie viewing system. Drift analysis evaluates the polynomial complexity of the algorithm. The authors demonstrate the results of experimental studies on an array of benchmarks. We also present an assessment of filtration efficiency based on a hybrid model and a population-based algorithm in comparison with the traditional method of collaborative filtering using the Pearson correlation coefficient. We can see that the prediction accuracy of the population-based algorithm is higher than that of the Pearson algorithm.

Petri Net Based Data-Flow Error Detection and Correction Strategy for Business Processes

Conceptual modeling, which includes both control-flow and data-flow modeling, has posed great challenges for conventional business process management systems. To support systematic data-flow modeling and analysis, in this paper, we propose a novel and effective Petri net-based approach. We first introduce a new type of Petri net, called WFIO-net, the firing rule of which is formally defined by extending classical Petri net with read and write semantics, to model both the control-flow and data-flow information. Then, we discuss about three possible data-flow errors over a WFIO-net, i.e., the missing, redundant and conflicting data errors. To detect such data-flow errors, we develop a polynomial complexity algorithm based on the so-called Activity Data Incidence Matrix (ADIM) of a WFIO-net. Following by that, we propose three effective correction strategies to resolve the detected data-flow anomalies. Finally, we present a property loan approval business process case study for our approach, and the study results demonstrate that the proposed detection and correction approaches are indeed very effective and can be applied to real-life settings.

Computation Offloading Scheduling for Periodic Tasks in Mobile Edge Computing

Motivated by various delay sensitive applications, we address the problem of coordinating the offloading decisions of wireless devices that periodically generate computationally intensive tasks. We consider autonomous devices that aim at minimizing their own cost by choosing when to perform their tasks and whether or not to offload their tasks to an edge cloud through one of the multiple wireless links. We develop a game theoretical model of the problem, prove the existence of pure strategy Nash equilibria and propose a polynomial complexity algorithm for computing an equilibrium. Furthermore, we characterize the structure of the equilibria, and by providing an upper bound on the price of anarchy of the game we establish an asymptotically tight bound on the approximation ratio of the proposed algorithm. Our simulation results show that the proposed algorithm achieves significant performance gain compared to uncoordinated computation offloading at a computational complexity that is on average linear in the number of devices.

Integrated scheduling and link adaptation for heterogeneous networks: design and performance analysis

ABSTRACT This paper presents novel integrated scheduling and link adaptation (ISLA) schemes for links which have a common spectrum and possibly different rates and reliability constraints. We consider the problem of weighted sum average rate maximisation of wireless links. In an orthogonal transmission system, a link is selected using instantaneous signal-to-noise (SNR) of all the links, and its rate and power are set in a jointly optimised manner. To demonstrate the effectiveness of the proposed approach, we analyse ISLA schemes with continuous rate adaptation using constant or adaptive transmission powers. ISLA schemes are designed in similar settings for practical discrete rate-adaptive systems using adaptive modulation and coding. To this end, a design framework is presented, which relies on a partitioning of space of links’ SNRs into regions, similar to structured vector quantisation. Within this framework, a particular ISLA design with high performance and polynomial complexity is suggested. The proposed schemes are designed before the start of the transmission, based on closed-form solutions, and can easily be applied to multiple access or broadcast networks. We also analyse several schemes for comparison, which employ modified Round-Robin, opportunistic scheduling or online generalised scheduling. Numerical results demonstrate how the proposed ISLA schemes outperform the benchmark schemes and effectively meet various user requirements.1

Time Decision for Multi-Input and Multi-Output Networked Control Systems

In a networked control system with multiple inputs and multiple outputs, the send time is the time instant at which a sensor sends the sampling signal to the controllers, and the perform time is the time instant at which an actuator starts performing the control commands. The relationship between the send time and perform time is nonlinear from the traditional perspective. This article formulates such a nonlinear timing relationship as a linear form based on a special algebraic structure. An algorithm is developed to find all legal send times that allow the control commands to be performed at the given time. It is pointed out that the problem for finding all legal send times is nondeterministic polynomial (NP)-hard. In addition, the method for finding the latest legal send time is presented, which has a polynomial complexity. This article also analyzes the robustness of networked control systems with perturbations, which decides the send time such that the actuators can perform the control commands at the specified time no matter how the time delays vary in bounded intervals. Numerical examples are used to illustrate the presented algorithm.

Optimal Joint Channel Estimation and Data Detection for Massive SIMO Wireless Systems: A Polynomial Complexity Solution

By exploiting large antenna arrays, massive MIMO (multiple input multiple output) systems can greatly increase spectral and energy efficiency over traditional MIMO systems. However, increasing the number of antennas at the base station (BS) makes the uplink joint channel estimation and data detection (JED) challenging in massive MIMO systems. In this paper, we consider the JED problem for massive SIMO (single input multiple output) wireless systems, which is a special case of wireless systems with large antenna arrays. We propose exact Generalized Likelihood Ratio Test (GLRT) optimal JED algorithms with low expected complexity, for both constant-modulus and nonconstant-modulus constellations. We show that, despite the large number of unknown channel coefficients, the expected computational complexity of these algorithms is polynomial in channel coherence time (T) and the number of receive antennas (N), even when the number of receive antennas grows polynomially in the channel coherence time (N=O(T <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</sup> ) suffices to guarantee an expected computational complexity cubic in T and linear in N). Simulation results show that the GLRT-optimal JED algorithms achieve significant performance gains (up to 5 dB improvement in energy efficiency) with low computational complexity.

Implementing termination analysis on quantum programming

Termination analysis is an essential part in programming. Especially quantum programming concerning measurement, entanglement and even superposition are the foundations of bizarre behaviours in quantum programs. In this paper, we analyse and extend the theoretical theorems on termination analysis proposed by Ying et al. into computational theorems and algorithms. The new algorithm without the Jordan decomposition process has a significant acceleration with polynomial complexity both on terminating and almost-surely terminating programs. Moreover, the least upper bound of termination programs steps is studied and utilized to output the substituted matrix representation of quantum programs. We also implement four groups of experiments to illustrate the advantages of the new algorithm in case of processing a simplified quantum walk example comparing with the original counterpart.

On the well-posedness, equivalency and low-complexity translation techniques of discrete-time hybrid automaton and piecewise affine systems

The main contribution of this paper is to present the systematic and low-complexity translation techniques between a class of hybrid systems referred to as automaton-based DHA and piecewise affine (PWA) systems. As an starting point the general modeling framework of the automaton-based DHA is represented which models the controlled and uncontrolled switching phenomena between linear continuous dynamics including discrete and continuous states, inputs and outputs. The basic theoretical definitions on the state trajectories of the proposed DHA with forward and backward evolutions which yield forward and backward piecewise affine (FPWA and BPWA) systems are given. Next, the well-posedness and equivalency properties are proposed and the sufficient conditions under which the well-posedness property is achieved with the automaton-based DHA and PWA systems are given. It is shown that the graphical structure of the proposed automaton-based DHA makes it possible to obtain analytically the equivalent PWA system with a polynomial complexity in contrast to the existing numerical translation techniques via decomposed structure of the DHA with an exponential complexity. Examples are presented to confirm the effectiveness of the proposed translation techniques.

Joint User Pairing, Mode Selection, and Power Control for D2D-Capable Cellular Networks Enhanced by Nonorthogonal Multiple Access

Nonorthogonal multiple access (NOMA) and device-to-device (D2D) are two promising technologies that have great potential in improving user connectivity. In this paper, we incorporate NOMA into the D2D-capable cellular networks and propose a new NOMA-aided D2D access scheme. In the proposed scheme, the D2D users (DUEs) can operate in four spectrum-sharing modes, which are the extension of the traditional underlay mode. To fully exploit the advantages of the NOMA-and-D2D integrated framework, we formulate a connectivity-maximization problem by jointly considering user pairing, mode selection, and power control under the constraints of the decoding thresholds of cellular users and DUEs. Based on the graph theory, we devise an efficient algorithm with polynomial complexity to solve the formulated problem optimally. We first analytically obtain the optimal transmission power and spectrum-sharing mode for every possible user pair through a graphical method. Based on the power control and mode selection policies, we transform the user pairing problem into a min-cost max-flow problem which can be tackled by the Ford-Fulkerson algorithm. Finally, simulation results indicate that the NOMA-aided D2D access scheme outperforms the traditional underlay mode, and the proposed algorithm yields a large performance gain in comparison with other schemes in terms of user connectivity and power consumption.

On hard instances of non-commutative permanent

Recent developments on the complexity of the non-commutative determinant and permanent (Arvind and Srinivasan, 2010; Chien et al., 2011; Bläser, 2013; Gentry, 2014) have settled the complexity of non-commutative determinant with respect to the structure of the underlying algebra. Continuing the research further, we look to obtain more insights on hard instances of non-commutative permanent.We show that any Algebraic Branching Program (ABP) computing the Cayley permanent of a collection of disjoint directed two-cycles with distinct variables as edge labels requires exponential size on an average, where the average is taken over all possible permutations of variables. For graphs where every connected component contains at most six vertices, we show that evaluating the Cayley permanent over any algebra containing 2 × 2 matrices is #P complete.Further, we obtain efficient algorithms for computing the Cayley permanent/ determinant on graphs with bounded component size, when vertices within each component are not far apart from each other in the Cayley ordering. This gives a tight upper and lower bounds for size of ABPs computing the permanent of disjointtwo-cycles. Finally, we exhibit more families of non-commutative polynomial evaluation problems that are complete for #P.Our results demonstrate that apart from the structure of underlying algebras, relative ordering of the variables plays a crucial role in determining the complexity of non-commutative polynomials.

Another look at degree lower bounds for polynomial calculus

Polynomial complexity is an algebraic proof system inspired by Gröbner bases which was introduced by Clegg, Edmonds and Impagliazzo. Alekhnovich and Razborov devised a method for proving degree lower bounds for polynomial calculus. We present an alternative account of their method, which also encompasses a generalization due to Galesi and Lauria.

Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism

A function f :{ −1,1} n → { −1,1} is a k -junta if it depends on at most k of its variables. We consider the problem of tolerant testing of k -juntas, where the testing algorithm must accept any function that is ε- close to some k -junta and reject any function that is ε′-far from every k ′-junta for some ε′ = O (ε) and k ′ = O ( k ). Our first result is an algorithm that solves this problem with query complexity polynomial in k and 1/ε. This result is obtained via a new polynomial-time approximation algorithm for submodular function minimization (SFM) under large cardinality constraints, which holds even when only given an approximate oracle access to the function. Our second result considers the case where k ′ = k . We show how to obtain a smooth tradeoff between the amount of tolerance and the query complexity in this setting. Specifically, we design an algorithm that, given ρ ∈ (0,1), accepts any function that is ε ρ/16-close to some k -junta and rejects any function that is ε-far from every k -junta. The query complexity of the algorithm is O ( k log k /ε ρ (1-ρ) k . Finally, we show how to apply the second result to the problem of tolerant isomorphism testing between two unknown Boolean functions f and g . We give an algorithm for this problem whose query complexity only depends on the (unknown) smallest k such that either f or g is close to being a k -junta.