Algorithm Of Polynomial Complexity Research Articles

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

TWENTY SIMILARITY FUNCTIONS OF TWO FINITE SEQUENCES

The article discusses the various numerical functions that determine the degree of "similarity" of the two given final sequences. These similarity measures are based on the concept we define of embedding in a sequence. A special case of such an attachment is the usual sub-subsequence. Other cases further require equality of distances between adjacent sub-sequence symbols in both sequences. This is generalization of the concept of a sequence segment (substring) in which these distances are unit. In addition, equality of distances from the beginning of the sequences to the first embedding symbol or from the last embedding symbol to the end of the sequences may be required. Except these last two cases, the attachment can be in a sequence several times. The literature uses functions such as the number of common attachments or the number of attachment occurrence pairs in a sequence. In addition to them, we enter three more functions: the sum of the lengths of total investments, the sum of the minima of the number of occurrences of a common embedding in both sequences and the similarity function based on the largest number of symbols of the common embedding. Intotal, 20 numerical functions are considered, for 17 of which algorithms (including new ones) of polynomial complexity are proposed, for two more functions, algorithms have exponential complexity with reduced a measure of degree. The Conclusion gives a brief comparative description of these investments and functions.

THE SHUFFLE FROG LEAPING ALGORITHM FOR THE PRODUCTION LOCATION PROBLEM

The production location problem is one of the most well-known discrete optimization problems. There are many options for setting this task. As a rule, all these variants of the problem of location of production belong to the class of NP-hard problems, that is, for its exact solution of such a problem, algorithms of polynomial complexity are currently unknown. So far, no effective methods have been developed for calculating the lower bounds for this problem, which make it possible to assess the achievement of the optimum. Exact algorithms for this problem are reduced to a complete enumeration of options. In this regard, the use of exact algorithms for solving the problem of production location often turns out to be inappropriate and impossible due to the large time costs. Therefore, the development and study of heuristic optimization methods is of considerable interest. One of the promising areas is the development of algorithms based on well-known metaheuristic approaches that are successfully used to solve many discrete optimization problems. In this paper, we show that one of the classes of production location problems is reduced to the problem of covering a complete graph with vertex-disjoint stars. It is proved that in this formulation the problem can be considered as an optimization problem on an oriented fragmentary structure. This allows you to create hybrid algorithms for finding suboptimal solutions to problems of this class based on a combination of well-known metaheuristics and a fragmentary algorithm. The mixed jumping frogs algorithm was chosen as a metaheuristic. It is shown that this algorithm can be used to find suboptimal solutions on a set of permutations. On the other hand, in the presence of an oriented fragmentary structure, the discrete optimization problem can be reduced to an optimization problem on a set of permutations. Thus, a simple and rather effective method for finding suboptimal solutions to the production location problem has been obtained. The method can be easily transferred to other classes of discrete optimization problems, which can be considered as problems on an oriented fragmentary structure.

On the Quality of Dispatching of Mondrian-Type Arrays in Grid Systems

Grid systems of centralized architecture, with multisite dispatching, characterized by the ability to execute a multiprocessor application on several parallel systems simultaneously, are modeled by a resource quadrant. The user's request for maintenance by the Grid system dispatcher is modeled by a resource rectangle with horizontal and vertical dimensions, respectively, equal to the number of time and processor resource units required to fulfill the request. Due to the exponential complexity of the optimal distribution of computational and time resources, heuristic algorithms of polynomial complexity based on the operations of dynamic integration of resource rectangles in the environment of resource rectangles are of practical value. The quality of dispatching is evaluated by a non-Euclidean heuristic measure that takes into account the area and shape of the occupied resource area. The quality of dispatching arrays of exact form with applications requiring approximately the same work, understood as the product of the number of required processors at runtime, is analyzed. In this paper, the quality of six polynomial-level algorithms is evaluated when dispatching arrays with applications of approximately the same area equal to the product of the number of required processors at the time of execution of the application. The adaptability of the analyzed algorithms is demonstrated on seven test arrays induced by Mondrian squares. Such arrays contain circular, hyperbolic and parabolic type applications. It is shown that the smallest value of the heuristic measure of H-level algorithms in length is 0.76, whereas in V-level algorithms in height, the smallest value of the heuristic measure is 0.83. At the same time, the H-level algorithm in length with a minimum deviation has the lowest value of the maximum of the heuristic measure of 0.5 + 0.59 on the test arrays of resource rectangles under consideration. When servicing arrays of exact forms with applications of approximately the same measure in Grid systems, it is recommended to use the considered polynomial H-level algorithm in length with minimal deviation.

Efficient exact algorithms for continuous bi‐objective performance‐energy optimization of applications with linear energy and monotonically increasing performance profiles on heterogeneous high performance computing platforms

AbstractPerformance and energy are the two most important objectives for optimization on heterogeneous high performance computing platforms. This work studies a mathematical problem motivated by the bi‐objective optimization of data‐parallel applications on such platforms for performance and energy. First, we formulate the problem and present an exact algorithm of polynomial complexity solving the problem where all the application profiles of objective type one are continuous and strictly increasing, and all the application profiles of objective type two are linear increasing. We then apply the algorithm to develop solutions for two related optimization problems of parallel applications on heterogeneous hybrid platforms, one for performance and dynamic energy and the other for performance and total energy. Our proposed solution methods are then employed to solve the two bi‐objective optimization problems for two data‐parallel applications, matrix multiplication and gene sequencing, on a hybrid platform employing five heterogeneous processors, namely, two different Intel multicore CPUs, an Nvidia K40c GPU, an Nvidia P100 PCIe GPU, and an Intel Xeon Phi.

Using solvable classes of systems for scheduling of real-time computations with minimized jitter

An approach to scheduling a computational process in distributed real-time systems with jitter minimization is considered. The approach is based on the concept of a solvable class of systems for which there exist optimal scheduling algorithms of polynomial complexity.

Resilience Indices for Power/Cyberphysical Systems

An engineered system is designed to deliver certain performance related to its quality-of-service, and while doing so, it must also maintain stable operation. Resilience of a system is its ability to continue to offer system performance stably, while withstanding any adverse events. Motivated by this concept, we propose to measure the resilience level of a power system by quantifying its stability level as measured by: transient stability margin (TSM), critical clearance time (CCT), relay margin (RM), and load security margin (LSM), as well as its performance level as measured by: load loss (LL) and recovery/repair time (RT) while being exposed to adverse events. For comparability, we also propose a normalization for each of the 6 measures to a number in the unit interval [0, 1], which is scale-invariant, and further probabilistically average each of those across all possible sequences of faults (of a specified length) against their occurrence probabilities to arrive at a set of 6 unit-interval valued indices. New polynomial complexity algorithms (in the number of generators) are proposed for estimating TSM (in form of volume of region of stability) and CCT; new quadratic program formulation for precise computation of RM is developed and implemented; also, new security and stability informed notions of LSM and LL are introduced and implemented by extending continuation power flow. Such quantification of resilience levels provides a numerical measure to compare the relative abilities of different power grids to withstand the impact of sequences of adverse events. The proposed approach is illustrated by computing and comparing the resilience of three similar power system topologies differing only in the location of generators. The framework is further validated by implementing it on the IEEE 30-bus test system.

Factorization of Boolean Polynomials: Parallel Algorithms and Experimental Evaluation

Polynomial factorization is a classical algorithmic algebra problem with a wide range of applications. Of particular interest is factorization over finite fields, among which fields of order two are probably the most important ones when representing Boolean functions by Zhegalkin polynomials. In particular, factorization of Boolean polynomials corresponds to conjunctive decomposition of Boolean functions given in algebraic normal form. In addition, factorization enables decomposition of functions given in full disjunctive normal form (DNF) and positive DNF, as well as Cartesian decomposition of relational data. These applications demonstrate the importance of developing fast factorization algorithms. In this paper, we consider some recently proposed factorization algorithms of polynomial complexity and describe a parallel MIMD implementation that takes advantage of both task-level and data-level parallelism. We conduct some experiments on logic synthesis benchmarks and synthetic (random) polynomials to demonstrate significant factorization speedup. In conclusion, we discuss results of testing a parallel implementation of the algorithm on a massively parallel multicore architecture (REDEFINE).

Critical pipeline of the acyclic wave processor

This paper is devoted to studying the performance of an acyclic wave processor consisting of heterogeneous processor elements having various delays. Data channels have sufficiently large buffers. The main result is the proof of the conjecture that for an arbitrary amount of input data, an acyclic wave processor contains a pipeline whose operating time is equal to the data processing time using the entire wave processor. This pipeline is called critical. For partially testing the conjecture, a multi-threaded application simulating the work of a two-dimensional pipeline has been developed. Earlier, the author proposed an algorithm for calculating the input processing time, the complexity of which depends not only on the number of stages but also on the amount of data. The proven conjecture eliminates this drawback, it allows building algorithms for calculating data processing time, the complexity of which does not depend on the amount of data. In particular, for any class of acyclic wave processors that have some algorithm of polynomial complexity for listing maximum pipelines, it is easy to construct a polynomial algorithm for calculating the data processing time for an arbitrary volume.

Joint FSO Fronthaul and Millimeter-Wave Access Link Optimization in Cloud Small Cell Networks: A Statistical-QoS Aware Approach

We investigate resource optimization for the downlink cloud small cell network, where the baseband unit pool communicates with the buffer-aided small remote radio heads (SRRHs) through free space optical fronthaul, and SRRHs transmit to the user equipments (UEs) by using time division multiplexing-based millimeter wave access links. Our objective is to maximize the supportable aggregate data arrival rate in the network by exploiting the inter-dependence of fronthaul and access links. Toward this objective, we consider maximum acceptable end-to-end queue-length bound violation probability constraints, load-balancing constraints in the access link, fronthaul link selection constraints, and transmit power budget constraints of fronthaul and access links. Since the joint fronthaul and access link optimization is a non-convex and combinatorial problem, we develop an iterative solution by decomposing the original optimization problem into two sub-problems. The first sub-problem optimally obtains fronthaul and access link power allocation and fronthaul link selection by using Lagrangian dual decomposition and canonical one-to-one matching techniques. By employing the Lagrangian dual decomposition and alternating optimization techniques, the second sub-problem obtains near optimal data arrival rate for each UE, UE-SRRH associations, fronthaul rate allocation among the transmitted data for the UEs, and the transmission duration scheduling in millimeter wave access link. An algorithm of polynomial complexity is developed in order to determine the supportable aggregate data arrival rate by considering the statistical quality-of-service requirements, and its convergence is proved. The simulation results depict that the proposed scheme significantly improves the aggregate data arrival rate over several benchmark schemes.

Power-Efficient Resource Allocation in C-RANs With SINR Constraints and Deadlines

In this paper, we address the problem of power-efficient resource management in Cloud Radio Access Networks (C-RANs). Specifically, we consider the case where Remote Radio Heads (RRHs) perform data transmission, and signal processing is executed in a virtually centralized Base-Band Units (BBUs) pool. Users request to transmit at different time instants; they demand minimum signal-to-noise-plus-interference ratio (SINR) guarantees, and their requests must be accommodated within a given deadline. These constraints pose significant challenges to the management of C-RANs and, as we will show, considerably impact the allocation of processing and radio resources in the network. Accordingly, we analyze the power consumption of the C-RAN system, and we formulate the power consumption minimization problem as a weighted joint scheduling of processing and power allocation problem for C-RANs with minimum SINR and finite horizon constraints. The problem is a Mixed Integer Non-Linear Program (MINLP), and we propose an optimal offline solution based on Dynamic Programming (DP). We show that the optimal solution is of exponential complexity, thus we propose a sub-optimal greedy online algorithm of polynomial complexity. We assess the performance of the two proposed solutions through extensive numerical results. Our solution aims to reach an appropriate trade-off between minimizing the power consumption and maximizing the percentage of satisfied users. We show that it results in power consumption that is only marginally higher than the optimum, at significantly lower complexity.

Sparsity in max-plus algebra and systems

We study sparsity in the max-plus algebraic setting. We seek both exact and approximate solutions of the max-plus linear equation with minimum cardinality of support. In the former case, the sparsest solution problem is shown to be equivalent to the minimum set cover problem and, thus, NP-complete. In the latter one, the approximation is quantified by the $\ell_{1}$ residual error norm, which is shown to have supermodular properties under some convex constraints, called lateness constraints. Thus, greedy approximation algorithms of polynomial complexity can be employed for both problems with guaranteed bounds of approximation. We also study the sparse recovery problem and present conditions, under which, the sparsest exact solution solves it. Through multi-machine interactive processes, we describe how the present framework could be applied to two practical discrete event systems problems: resource optimization and structure-seeking system identification. We also show how sparsity is related to the pruning problem. Finally, we present a numerical example of the structure-seeking system identification problem and we study the performance of the greedy algorithm via simulations.

Dimension Reduction and Feedback Stabilization for Max-Plus Linear Systems and Applications in VLSI Array Processors

This paper investigates the dimension reduction and feedback stabilization of max-plus linear systems and applies them to control and optimize the very large scale integration (VLSI) array processors. We introduce the weakly similar relation between max-plus matrices and the pseudoequivalent relation between autonomous max-plus linear systems, and point out that two systems are pseudoequivalent if and only if their state matrices are weakly similar. The reduced system is defined by using the pseudoequivalence, whose dimension is determined by the row rank of the original state matrix. We focus on obtaining a reduced system which maintains the stability and retains the steady-state period. An algorithm of polynomial complexity is developed to find such a reduced system. The reduced system is then used to design a state feedback controller to stabilize a max-plus linear system. Finally, we use the VLSI array processors as an example to demonstrate how the presented methods work in practical applications.

On the Possibility of Network Scheduling With Polynomial Complexity and Delay

Considering the collection of all networks with independent set interference model, Shah, Tse, and Tsitsiklis showed that there exist scheduling algorithms with polynomial complexity and delay, only if the maximum independent set problem can be solved in polynomial time (equivalently, P=NP). In this paper, we extend this result to arbitrary collections of networks and present a clear-cut criterion for the existence of polynomial complexity and delay scheduling algorithms relative to a given collection of networks with arbitrary interference models, not confined to independent set interference or SINR models, and not necessarily encompassing all network topologies. This amounts to the equivalence of polynomial scheduling and effective approximation of maximum weighted actions .

Deep Haar scattering networks

An orthogonal Haar scattering transform is a deep network computed with a hierarchy of additions, subtractions and absolute values over pairs of coefficients. Unsupervised learning optimizes Haar pairs to obtain sparse representations of training data with an algorithm of polynomial complexity. For signals defined on a graph, a Haar scattering is computed by cascading orthogonal Haar wavelet transforms on the graph, with Haar wavelets having connected supports. It defines a representation which is invariant to local displacements of signal values on the graph. When the graph connectivity is unknown, unsupervised Haar learning can provide a consistent estimation of connected wavelet supports. Classification results are given on image data bases, defined on regular grids or graphs, with a connectivity which may be known or unknown.

A Framework for Structural Input/Output and Control Configuration Selection in Large-Scale Systems

This paper addresses problems on the structural design of large-scale control systems. An efficient and unified framework is proposed to select the minimum number of manipulated/measured variables to achieve structural controllability/observability of the system, and to select the minimum number of feedback interconnections between measured and manipulated variables such that the closed-loop system has no structural fixed modes. Global solutions are computed using polynomial complexity algorithms in the number of the state variables of the system. Finally, graph-theoretic characterizations are proposed, which allow a characterization of all possible solutions.

Detection of embeddings in binary Markov chains

AbstractThe paper is concerned with problems in steganography on the detection of embeddings and statistical estimation of positions at which message bits are embedded. Binary stationary Markov chains with known or unknown matrices of transition probabilities are used as mathematical models of cover sequences (container hles). Based on the runs statistics and the likelihood ratio statistic, statistical tests are constructed for detecting the presence of embeddings. For a family of contiguous alternatives, the asymptotic power of statistical tests based on the runs statistics is found. An algorithm of polynomial complexity is developed for the statistical estimation of positions with embedded bits. Results of computer experiments are presented.

On the Design and Analysis of Fair Contact Plans in Predictable Delay-Tolerant Networks

Delay-tolerant networks (DTNs) have become a promising architecture for wireless sensor systems in challenged communication environments where traditional solutions based on persistent connectivity either fail or show serious weaknesses. As a result, different routing schemes have been investigated that take into account the time-evolving nature of the network topology. Among them, contact graph routing has been proposed for space environments with predictable connectivity. In order to evaluate routing decisions, DTN nodes need to know the contact plan in advance, which comprises all communication links among nodes that will be available in the future. Since not all potential contacts can belong to the contact plan, its design requires analyzing conflicting contacts in order to select those that meet an overall goal. In this paper, we consider the design of contact plans that can maximize fairness requirements while still maximizing the overall capacity as well. To this end, we propose to formalize the problem by means of an optimization model and evaluate its performance in terms of different fairness metrics. Since this model can be computationally intractable for a large number of contacts, we also propose to tackle it as a matching problem, resulting in algorithms of polynomial complexity, and compare these results with those of the original model. We show that fairness can be properly modeled to design contact plans and that efficient algorithms do exist to compute these plans quite accurately while also improving overall network routing metrics for a proposed case study.

On Quality of Monitoring for Multichannel Wireless Infrastructure Networks

Passive monitoring utilizing distributed wireless sniffers is an effective technique to monitor activities in wireless infrastructure networks for fault diagnosis, resource management, and critical path analysis. In this paper, we introduce a quality of monitoring (QoM) metric defined by the expected number of active users monitored, and investigate the problem of maximizing QoM by judiciously assigning sniffers to channels based on the knowledge of user activities in a multichannel wireless network. Two types of capture models are considered. The user-centric model assumes the frame-level capturing capability of sniffers such that the activities of different users can be distinguished while the sniffer-centric model only utilizes the binary channel information (active or not) at a sniffer. For the user-centric model, we show that the implied optimization problem is NP-hard, but a constant approximation ratio can be attained via polynomial complexity algorithms. For the sniffer-centric model, we devise stochastic inference schemes to transform the problem into the user-centric domain, where we are able to apply our polynomial approximation algorithms. The effectiveness of our proposed schemes and algorithms is further evaluated using both synthetic data as well as real-world traces from an operational WLAN.

Joint scheduling and power control for multicasting in cellular wireless networks

We consider a cellular wireless network. Our aim is to develop adaptive-power scheduling algorithms that enable area base stations to coordinate, on a time-division basis, the transmission of multicast packets to identified client nodes, for scenarios under which a prescribed code rate is employed. Such mechanisms can be implemented by 4G long term evolution (LTE) systems using multimedia broadcast multicast services (MBMS), meshed WiFi networks, or mobile backbone based ad hoc wireless networks. We show that the joint scheduling and power control problem can be represented as a mixed-integer linear programming model, which is NP-hard. Consequently, we present three heuristic algorithms of polynomial complexity for solving the problem in a practical manner. For small network layouts, our centralized heuristic algorithm is shown to achieve a throughput rate performance that is close to that attained by the optimal scheme. All three of our heuristic algorithms are shown to yield excellent throughput rate performance behavior. The use of power adaptations, when compared with fixed transmit power operations, leads to enhanced throughput rate performance while also lowering communications energy consumption levels. Comparisons with LTE MBMS over single frequency network based schemes have shown our algorithms to offer enhanced spectral efficiency performance, while providing high client coverage.