Non-polynomial Time Research Articles

An End-Host-Importance-Aware Secure Service-Enabled Hybrid SDN Deployment

Security is critical to networks, but TCP/IP-based legacy networks are difficult to advance new security functions due to the use of costly inflexible hardware devices and error-prone network configurations. Recent literature explores the paradigm of consolidating security services with the forwarding functionality using Software-defined Networking (SDN). Existing full SDN deployment, replacing all legacy network devices with SDN devices, is cost-prohibitive. Whereas the hybrid SDN that only upgrades partial legacy devices to SDN switches is considered practical. However, the challenge is to minimize threats and deployment expenses simultaneously under heterogeneous end-host businesses that have various importance. In this paper, we study the challenge and propose the (EASON) problem. We mathematically formulate the EASON problem as an integer programming problem, prove its non-polynomial time complexity, and propose a heuristic algorithm called BonSéc. We conduct rigorous simulations on real-world topologies and traces. Experimental results show that BonSéc achieves comparable security and cost performances to the optimal solution on small topologies. Meanwhile, it is scalable on larger topologies.

Analysis of Polynomial Time and Non-Polynomial Time of Algorithms

Abstract: The P vs NP problem is one of the most significant open problems in computer science and mathematics. This problem asks whether every problem that can be solved in polynomial time can also be verified in polynomial time. The purpose of this research paper is to explore the P vs NP problem and its relevance in the analysis of algorithms. We will discuss the techniques used to design and analyze algorithms, such as divide-and-conquer, dynamic programming, and greedy algorithms, and their relation to the P vs NP problem. We will also examine some examples of polynomial-time algorithms and NP-hard problems and analyze their time and space complexity, addition to which we will analyze the NP-Complete Problem and finally looking the current scenario of the P & NP and stating its significance.

Review and analysis of existing methods of scheduling classes

One of the most important steps in organizing the quality of the educational process in educational institutions and the effectiveness of the use of scientific and pedagogical potential is the task of forming a high-quality lesson schedule. A qualitatively designed lesson schedule should ensure the uniform loading of student groups and teaching staff. The problem of university course planning (UCTP) is a specific problem of course planning at a university, with simultaneous use of resources such as students, faculty, and classrooms. These problems are considered non-polynomial time (NP) and Combinatorial Optimization (COP) problems, which means that they can be solved by optimization algorithms to obtain the desired table. Several methods have been used in universities to solve table problems, and most of them use optimization methods. This article discusses six methods for solving problems with the schedule of classes in universities: Sequential methods, сluster methods, сonstraint-based methods, meta-heuristic methods, generalized search, hybrid evolutionary algorithms, multi-criteria approaches, case-based reasoning techniques, hyper-heuristics, adaptive approaches. The purpose of this article is a comprehensive review of optimization approaches to solving the problems of scheduling in universities. In addition, when solving such problems, the accepted algorithms for Meta-heuristic optimization are shown. In addition, methods have been developed to analyze and measure the performance of a solution by specifying frequently used control data sets.

Optimization Techniques in University Timetabling Problem: Constraints, Methodologies, Benchmarks, and Open Issues

University timetabling problems are a yearly challenging task and are faced repeatedly each semester. The problems are considered non-polynomial time (NP) and combinatorial optimization problems (COP), which means that they can be solved through optimization algorithms to produce the aspired optimal timetable. Several techniques have been used to solve university timetabling problems, and most of them use optimization techniques. This paper provides a comprehensive review of the most recent studies dealing with concepts, methodologies, optimization, benchmarks, and open issues of university timetabling problems. The comprehensive review starts by presenting the essence of university timetabling as NP-COP, defining and clarifying the two formed classes of university timetabling: University Course Timetabling and University Examination Timetabling, illustrating the adopted algorithms for solving such a problem, elaborating the university timetabling constraints to be considered achieving the optimal timetable, and explaining how to analyze and measure the performance of the optimization algorithms by demonstrating the commonly used benchmark datasets for the evaluation. It is noted that meta-heuristic methodologies are widely used in the literature. Additionally, recently, multi-objective optimization has been increasingly used in solving such a problem that can identify robust university timetabling solutions. Finally, trends and future directions in university timetabling problems are provided. This paper provides good information for students, researchers, and specialists interested in this area of research. The challenges and possibilities for future research prospects are also explored.

Distribution of Multi MmWave UAV Mounted RIS Using Budget Constraint Multi-Player MAB

Millimeter wave (mmWave), reconfigurable intelligent surface (RIS), and unmanned aerial vehicles (UAVs) are considered vital technologies of future six-generation (6G) communication networks. In this paper, various UAV mounted RIS are distributed to support mmWave coverage over several hotspots where numerous users exist in harsh blockage environment. UAVs should be spread among the hotspots to maximize their average achievable data rates while minimizing their hovering and flying energy consumptions. To efficiently address this non-polynomial time (NP) problem, it will be formulated as a centralized budget constraint multi-player multi-armed bandit (BCMP-MAB) game. In this formulation, UAVs will act as the players, the hotspots as the arms, and the achievable sum rates of the hotspots as the profit of the MAB game. This formulated MAB problem is different from the traditional one due to the added constraints of the limited budget of UAVs batteries as well as collision avoidance among UAVs, i.e., a hotspot should be covered by only one UAV at a time. Numerical analysis of different scenarios confirm the superior performance of the proposed BCMP-MAB algorithm over other benchmark schemes in terms of average sum rate and energy efficiency with comparable computational complexity and convergence rate.

L0-Regularized Learning for High-Dimensional Additive Hazards Regression

Sparse learning in high-dimensional survival analysis is of great practical importance, as exemplified by modern applications in credit risk analysis and high-throughput genomic data analysis. In this article, we consider the L0-regularized learning for simultaneous variable selection and estimation under the framework of additive hazards models and utilize the idea of primal dual active sets to develop an algorithm targeted at solving the traditionally nonpolynomial time optimization problem. Under interpretable conditions, comprehensive statistical properties, including model selection consistency, oracle inequalities under various estimation losses, and the oracle property, are established for the global optimizer of the proposed approach. Moreover, our theoretical analysis for the algorithmic solution reveals that the proposed L0-regularized learning can be more efficient than other regularization methods in that it requests a smaller sample size as well as a lower minimum signal strength to identify the significant features. The effectiveness of the proposed method is evidenced by simulation studies and real-data analysis. Summary of Contribution: Feature selection is a fundamental statistical learning technique under high dimensions and routinely encountered in various areas, including operations research and computing. This paper focuses on the L0-regularized learning for feature selection in high-dimensional additive hazards regression. The matching algorithm for solving the nonconvex L0-constrained problem is scalable and enjoys comprehensive theoretical properties.

Optimization of open flow controller placement in software defined networks

The world is entering into the era of Big Data where computer networks are an essential part. However, the current network architecture is not very convenient to configure such leap. Software defined network (SDN) is a new network architecture which argues the separation of control and data planes of the network devices by centralizing the former in high level, centralised devices and efficient supervisors, called controllers. This paper proposes a mathematical model that helps optimizing the locations of the controllers within the network while minimizing the overall cost under realistic constrains. Our method includes finding the minimum cost of placing the controllers; these costs are the network latency, controller processing power and link bandwidth. Different types of network topologies have been adopted to consider the data profile of the controllers, links of controllers and locations of switches. The results showed that as the size of input data increased, the time to find the optimal solution also increased in a non-polynomial time. In addition, the cost of solution is increased linearly with the input size. Furthermore, when increasing allocating possible locations of the controllers, for the same number of switches, the cost was found to be less.

Round-Optimal Secure Multi-party Computation

Secure multi-party computation (MPC) is a central cryptographic task that allows a set of mutually distrustful parties to jointly compute some function of their private inputs where security should hold in the presence of a malicious adversary that can corrupt any number of parties. Despite extensive research, the precise round complexity of this “standard-bearer” cryptographic primitive is unknown. Recently, Garg, Mukherjee, Pandey and Polychroniadou, in EUROCRYPT 2016 demonstrated that the round complexity of any MPC protocol relying on black-box proofs of security in the plain model must be at least four. Following this work, independently Ananth, Choudhuri and Jain, CRYPTO 2017 and Brakerski, Halevi, and Polychroniadou, TCC 2017 made progress towards solving this question and constructed four-round protocols based on non-polynomial time assumptions. More recently, Ciampi, Ostrovsky, Siniscalchi and Visconti in TCC 2017 closed the gap for two-party protocols by constructing a four-round protocol from polynomial-time assumptions. In another work, Ciampi, Ostrovsky, Siniscalchi and Visconti TCC 2017 showed how to design a four-round multi-party protocol for the specific case of multi-party coin-tossing.

POLYNOMIAL ALGORITHM FOR FINDING A HAMILTONIAN CYCLE ON A GRAPH

The subject of research is the solution of the problem of finding a Hamiltonian cycle on a graph, which in discrete mathematics belongs to the NP complexity class and still retains interest. The aim of this work is to develop a new algorithm for solving this problem, which guarantees finding the optimal solution with polynomial time. In [1], an analysis of the current state of the problem was performed, the shortcomings of existing methods of solving were noted, and new principles and method of finding solution were presented. Known methods for solving the problem are based on the implementation of a search of possible solutions or on intuitive heuristics. Enumeration methods of solutions are unacceptable in terms of costs, since they characterized by non-polynomial time. Heuristic methods are satisfactory in time, but they do not guarantee finding the optimal solution. The popularity of enumeration methods is explained by the linear simplicity of searching in a pre-known set of acceptable solutions to the problem. But the factorial dependence of the power of the set of solutions (n-1)! from the number of vertices of the graph n makes it impossible to use such methods for large-size problems in practice. The desire to significantly reduce time costs leads to attempts to reasonably reduce the enumeration set or to the development of various heuristics, which actually indicates that it is impossible to formulate conditions for finding the optimal solution to the problem as a whole. This article describes the conditions that determine the optimal solution of the problem and a polynomial algorithm for solving the problem that embodies the described method. Finding the optimal solution to the problem is reduced to finding a closed path in a new graph of shortest paths. The shortest paths graph built on the original graph of the problem by using Dijkstra's algorithm. The enumeration set for determining the optimal solution of the problem consists of solutions that are constructed from each vertex of the source graph in the shortest paths graph. The size of this set is estimated as n(n-1). The developed parallel algorithm guarantees finding the optimal solution, significantly reduces the initial search space, and allows you to find a solution with polynomial complexity. Testing of the program showed the efficiency of the developed method and algorithm for solving the problem.

On the Planning and Design Problem of Fog Computing Networks

This paper proposes an exact model for the planning and design problem of fog networks. More precisely, a mathematical model is proposed to simultaneously determine the optimal location, the capacity and the number of fog node(s) as well as the interconnection between the installed fog nodes and the cloud. The goal of the model is to minimize the delay in the network and the amount of traffic sent to the cloud data center. To address this multi-objective optimization problem, three optimization techniques are used: the weighted sum, the hierarchical and the trade-off methods. The weighted sum method aggregates all the lone objective functions into a single objective by applying a weighted vector. The hierarchical method takes a sequential approach by tightly constraining the more important objective function. The trade-off method solves a single objective function and translates all other objective functions into constraints. These methods are then compared in terms of average delay, amount of traffic sent to the cloud and amount of CPU time required to find optimal solution(s). Since we are dealing with a multi-objective optimization problem and that multiple optimal solutions can be found, the fuzzy-based mechanism and the hypervolume indicator have been used. Computational results show that as the problem size increases, the delay and the traffic also increase in a linear form; whereas, the solution time increases in non-polynomial time. The weighted sum method was able to achieve the best trade-off results for the delay and the traffic, whereas the hierarchical method was able to return minimum delay but with worse traffic going to the cloud. As the model considers realistic edge device traffic parameters, constraints, and various topology aspects, it can be helpful for the planning and deployment of fog networks and how they operate within a cloud infrastructure.

Leveraging the power of quantum computing for breaking RSA encryption

ABSTRACT Encryption is the process of securing confidential data that bars a third party’s access to the information.RSA encryption utilises the property of complexity classes wherein the problem of prime integer factorization lies inside the Non-Polynomial time (NP-Hard) class, which makes it impervious to classical computers. Since it is so hard to break even for a computer, it becomes important to do encryption for all the secure transactions. Although it lies outside the capabilities of traditional computing, the recent developments in the field of quantum computing can be utilised to break RSA Encryption. The approach involves mapping of qubits used in a quantum machine to a constraint satisfaction problem (CSP) and then using them to check for factors. This consists of the use of a Multiplicative Boolean circuit in which the qubits utilised by the machine replaces the variables. These Qubits are then mapped as per the gates involved, and the factorization problem is thus transformed into a CSP problem, through which, the factors can be easily found. Once known, these factors can be used to calculate the public and private keys effectively breaking the encryption security. We provide a novel approach to highlight the importance of developing Post-Quantum cryptography techniques for providing a secure channel of communication.

BetheSF: Efficient computation of the exact tagged-particle propagator in single-file systems via the Bethe eigenspectrum

Single-file diffusion is a paradigm for strongly correlated classical stochastic many-body dynamics and has widespread applications in soft condensed matter and biophysics. However, exact results for single-file systems are sparse and limited to the simplest scenarios. We present an algorithm for computing the non-Markovian time-dependent conditional probability density function of a tagged-particle in a single-file of N particles diffusing in a confining external potential. The algorithm implements an eigenexpansion of the full interacting many-body problem obtained by means of the coordinate Bethe ansatz. While formally exact, the Bethe eigenspectrum involves the generation and evaluation of permutations, which becomes unfeasible for single-files with an increasing number of particles N. Here we exploit the underlying exchange symmetries between the particles to the left and to the right of the tagged-particle and show that it is possible to reduce the complexity of the algorithm from the worst case scenario O(N!) down to O(N). A C++ code to calculate the non-Markovian probability density function using this algorithm is provided. Solutions for simple model potentials are readily implemented including single-file diffusion in a flat and a ‘tilted’ box, as well as in a parabolic potential. Notably, the program allows for implementations of solutions in arbitrary external potentials under the condition that the user can supply solutions to the respective single-particle eigenspectra. Program summaryProgram Title: BetheSFCPC Library link to program files:http://dx.doi.org/10.17632/3bs74vf72n.1Licensing provisions: MITProgramming language: C++ (C++17 support required)Supplementary material: makefile, README, SingleFileBluePrint.hppNature of problem: Diffusive single-files are mathematical models of effectively one-dimensional strongly correlated many-body systems. While the dynamics of the full system is Markovian, the diffusion of a tracer-particle in a single-file is an example of non-Markovian and anomalous diffusion. The many-body Fokker–Planck equation governing the system’s dynamics can be solved using the coordinate Bethe ansatz. A naïve implementation of such a solution runs in non-polynomial time since it requires the generation of permutations of the elements of a multiset.Solution method: In this paper we show how, exploiting the exchange symmetries of the system, it is possible to reduce the complexity of the algorithm to evaluate the solution, using a permutation-generation algorithm, from O(N!) in the worst case scenario to O(N) in the best case scenario, which corresponds to tagging the first or the last particle, where N stands for the number of particles in the single-file.Additional comments including restrictions and unusual features: The code may overflow for large single-files N≥170. All the benchmarks ran on the following CPU: Intel Xeon E3-1270 v2 3.50 GHz 4 cores. The compiler used is g++ 7.3.1 (SUSE Linux) with the optimization-O3 turned on. The code to produce all the data in the figures is included in the files: figure2.cpp, figure3.cpp, figure4.cpp, figure5a.cpp and figure5b.cpp

An Evolutionary Algorithm for Short Addition Chains

The encryption efficiency of the Rivest-Shamir-Adleman cryptosystem is based on decreasing the number of multiplications in the modular exponentiation (ME) operation. An addition chain (AC) is one of the strategies used to reduce the time consumed by ME through generating a shortest/short chain. Due to the non-polynomial time required for generating a shortest AC, several algorithms have been proposed to find a short AC in a faster time. In this paper, we use the evolutionary algorithm (EA) to find a short AC for a natural number. We discuss and present the role of every component of the EA, including the population, mutation operator, and survivor selection. Then we study, practically, the effectiveness of the proposed method in terms of the length of chain it generates by comparing it with three kinds of algorithms: (1) exact, (2) non-exact deterministic, and (3) non-exact non-deterministic. The experiment is conducted on all natural numbers that have 10, 11, 12, 13, and 14 bits. The results demonstrate that the proposed algorithm has good performance compared to the other three types of algorithms.

QoS-Aware Web Service Selection with Internal Complementarity

Service composition is a key enabling technology in service-oriented computing for developing versatile applications by integrating various existing interoperable services. Although actively studied, most existing works on service composition neglect the existence of complementarity among candidate services within a service class, so-called internal complementarity. In fact, complementary candidate services within a service class can be composed to form a composite candidate service which may yield better service utility than that provided by any existing candidate service within that service class. This work focuses on web service composition where internal complementarity is more likely to happen. Specifically, we aim at addressing the problem of QoS-aware web service selection with internal complementarity (WSS-IC). We first transform this problem into a multi-choice multi-dimensional knapsack problem (MMKP) and prove such a transformation per se has non-polynomial time complexity in the worse case.Then, we perform complexity analysis to demonstrate that existing approaches to MMKPs are not computationally feasible to resolve QoS-aware WSS-IC. This fact motivates us to propose an iteratively improving framework for deriving the solution iteration by iteration while taking into account both solution structure and QoS constraints. At each iteration, the current solution gets improved by solving a disjunctively constrained knapsack problem. To verify the effectiveness of the proposed framework, two heuristic approaches are implemented under this framework. Experimental results demonstrate that our approaches outperform the compared methods in terms of both solution quality and computation time.

Parallel time–space reduction by unbiased filtering for solving the 0/1-Knapsack problem

The 0/1-Knapsack problem (0/1-KP) is one of popular NP-hard problems since its optimal solutions are meaningful to data-science computing in real-world applications (i.e., maximum profit of product planning-and-loading, minimum encryption of reliable information-transmission, etc.). Theoretically, the optimal solutions of the 0/1-KP can be solved by the dynamic programming (DP) but in non-polynomial time, which is not reasonable for big data n and massive limited-capacity C. Practically, the hybrid swarm-optimization could improve the 0/1-KP solutions in polynomial time but best fitness-solutions may not be optimal. Therefore, this paper proposes the parallel time–space reduction for solving the 0/1-KP in polynomial time. Our innovation and contribution focus on achieving 1. the fast processing as the FS (feature sorting) and the optimal solutions close to the DP by our unbiased-filtering and 2. the optimized time-complexity O(n/p log2p + C’/p) by our parallel time–space reduction. In this new approach, we integrate the FS method (for the efficient initial solutions) and the optimal DP (on the remaining critical-objects n’ and knapsack-capacity C’), incorporated with the proposed unbiased-filtering. Finally, performance is evaluated by simulation study on a variety of problem-sizes. In experiment, most of our solutions are optimal as comparable to the optimal solutions of the DP.

Heuristic modeling for sustainable procurement and logistics in a supply chain using big data

Drastic climate change has enforced business organizations to manage their carbon emissions. Procurement and transportation is one of the supply chain business operations where carbon emissions are huge. This paper proposes an environmentally sustainable procurement and logistics model for a supply chain. The proposed models are of MINLP (Mixed Integer Non Linear Program) and MILP (Mixed Integer Linear Program) form requiring a variety of the real time parameters from buyer and supplier side such as costs, capacities, lead-times and emissions. Based on real time data, the models provide an optimal sustainable procurement and transportation decision. It is also shown that large sized problems possessing essential 3V's of big data, i.e., volume, variety and velocity consume non-polynomial time and cannot be solved optimally. Therefore, a heuristic (H-1) is also proposed to solve the large sized problems involving big data. T-test significance is also conducted between optimal and heuristic solutions obtained using 42 randomly generated data instances possessing essential characteristics of big data. Encouraging results in terms of solution quality and computational time are obtained.

A polynomial time repeated cuts algorithm for the time cost tradeoff problem: The linear and convex crashing cost deadline problem

The time cost tradeoff problem in project management, TCTP, is to achieve a given deadline on the project completion time by expediting the normal durations of activities at the cheapest cost possible. The linear TCTP, in which the expediting costs of each activity are linear, as a function of the number of time periods reduced, can be solved using linear programming. We present here an algorithm that solves the linear or convex costs TCTP that runs in polynomial time and calls only for a minimum s,t-cut routine at each iteration. This repeated cuts algorithm is related to the non-polynomial time algorithm by Phillips and Dessouky (1977), the PD-algorithm. The PD-algorithm reduces the project duration, at each iteration, by one time unit, at a minimum cost. The choice of the activities to expedite, in order to reduce the project duration by one unit, is determined by a solution to a minimum s,t-cut in a respective graph.We present here previously unknown properties of the PD-algorithm, and a new concept of cut-decomposition. These properties are used in devising the repeated cuts algorithm based on scaling. The repeated cuts algorithm solves in polynomial time, the linear as well as the convex TCTP. The algorithm solves the TCTP problem in polynomial time even when the durations and/or the target deadline are not necessarily integers.

Stable Beneficial Group Activity Formation

Computational models are one of the very powerful tools for expressing everyday situations that are derived from human interactions. In this paper, an investigation of the problem of forming beneficial groups based on the members' preferences and the coordinator's own strategy is presented . It is assumed that a coordinator has a good intention behind trimming members' preferences to meet the ultimate goal of forming the group. His strategy is justified and evaluated by Nash stability. There are two variations of the problem: the Anonymous Stable Beneficial Group Activity Formation and the General Stable Beneficial Group Activity Formation. The computational complexity of solving both variations has been analyzed. Finding stable groups needs non-polynomial time algorithm to be solved. A polynomial time solution is presented and enhanced with examples

Parameterized complexity of basic decision problems for tree automata

There are many decision problems in automata theory (including membership, emptiness, inclusion and universality problems) that are NP-hard for some classes of tree automata (TA). The study of their parameterized complexity allows us to find new bounds of their nonpolynomial time algorithmic behaviours. We present results of such a study for classical TA, rigid tree automata, TA with global equality and disequality and t-DAG automata. As parameters we consider the number of states, the cardinality of the signature, the size of the term or the t-dag and the size of the automaton.

On the Power of Randomization in Algorithmic Mechanism Design

In many settings the power of truthful mechanisms is severely bounded. In this paper we use randomization to overcome this problem in the multi-unit auction setting. In particular, we construct a fully polynomial-time approximation scheme (FPTAS) for multi-unit auctions that is truthful in expectation, whereas there is evidence that no polynomial-time truthful deterministic mechanism provides an approximation ratio better than 2. We leverage the FPTAS to show for the first time that truthful in expectation polynomial-time mechanisms are provably stronger than polynomial-time universally truthful mechanisms. Specifically, we show that there is a setting, related to multi-unit auctions, in which (1) there is a nonpolynomial time truthful mechanism that always outputs the optimal solution, and that (2) no universally truthful randomized mechanism can provide an approximation ratio better than 2 in polynomial time, but (3) an FPTAS that is truthful in expectation exists.