Computational complexity continuum within Ising formulation of NP problems

A promising approach to achieve computational supremacy over the classical von Neumann architecture explores classical and quantum hardware as Ising machines. The minimisation of the Ising Hamiltonian is known to be NP-hard problem yet not all problem instances are equivalently hard to optimise. Given that the operational principles of Ising machines are suited to the structure of some problems but not others, we propose to identify computationally simple instances with an ‘optimisation simplicity criterion’. Neuromorphic architectures based on optical, photonic, and electronic systems can naturally operate to optimise instances satisfying this criterion, which are therefore often chosen to illustrate the computational advantages of new Ising machines. As an example, we show that the Ising model on the Möbius ladder graph is ‘easy’ for Ising machines. By rewiring the Möbius ladder graph to random 3-regular graphs, we probe an intermediate computational complexity between P and NP-hard classes with several numerical methods. Significant fractions of polynomially simple instances are further found for a wide range of small size models from spin glasses to maximum cut problems. A compelling approach for distinguishing easy and hard instances within the same NP-hard class of problems can be a starting point in developing a standardised procedure for the performance evaluation of emerging physical simulators and physics-inspired algorithms.

A Competitive Volleyball Algorithm to Solve Global Optimization with the Engineering Problems

In real life, many engineering problems are nonlineble NP problems, in order to solve some of these problems, we put forward a competitive volleyball algorithm.The algorithm proposed in this paper is a meta-heuristic technique based on swarm optimization. It is inspired by the competition between volleyball teams in a league and the improvement in players’ overall abilities in order to win the Most Valuable Player award. Several specific terms relating to competition, such as pre-match reinforcement, single round robin mechanism, optimal strategy constitute the structure of the algorithm. The sensitivity of several parameters of the algorithm is analyzed and tested for three types of benchmark functions: unimodal, high-dimensional multimodal and low-dimensional multimodal functions. Through the use of these three types of test functions, the performance of this algorithm is compared with nine classical metaheuristic algorithms: Genetic Algorithm (GA), Differential Evolution (DE), Harmony Search (HS), Artificial Bee Colony (ABC), Particle Swarm Optimization (PSO), Sine Cosine Algorithm (SCA), Soccer League Competition (SLC), League Championship Algorithm (LCA) and Volleyball Premier League (VPL). CVA has been used to solve three real-world engineering problems. The results show that the performance of the CVA is behaviorally promising and better than the other classical metaheuristic algorithms.

P vs NP: P is Equal to NP: Desired Proof

Computations and computational complexity are fundamental for mathematics and all computer science, including web load time, cryptography (cryptocurrency mining), cybersecurity, artificial intelligence, game theory, multimedia processing, computational physics, biology (for instance, in protein structure prediction), chemistry, and the P vs. NP problem that has been singled out as one of the most challenging open problems in computer science and has great importance as this would essentially solve all the algorithmic problems that we have today if the problem is solved, but the existing complexity is deprecated and does not solve complex computations of tasks that appear in the new digital age as efficiently as it needs. Therefore, we need to realize a new complexity to solve these tasks more rapidly and easily. This paper presents proof of the equality of P and NP complexity classes when the NP problem is not harder to compute than to verify in polynomial time if we forget recursion that takes exponential running time and goes to regress only (every problem in NP can be solved in exponential time, and so it is recursive, this is a key concept that exists, but recursion does not solve the NP problems efficiently). The paper’s goal is to prove the existence of an algorithm solving the NP task in polynomial running time. We get the desired reduction of the exponential problem to the polynomial problem that takes O(log n) complexity.

Quantum verification of NP problems with single photons and linear optics

Quantum computing is seeking to realize hardware-optimized algorithms for application-related computational tasks. NP (nondeterministic-polynomial-time) is a complexity class containing many important but intractable problems like the satisfiability of potentially conflict constraints (SAT). According to the well-founded exponential time hypothesis, verifying an SAT instance of size n requires generally the complete solution in an O(n)-bit proof. In contrast, quantum verification algorithms, which encode the solution into quantum bits rather than classical bit strings, can perform the verification task with quadratically reduced information about the solution in $$\tilde O(\sqrt n )$$ O ̃ ( n ) qubits. Here we realize the quantum verification machine of SAT with single photons and linear optics. By using tunable optical setups, we efficiently verify satisfiable and unsatisfiable SAT instances and achieve a clear completeness-soundness gap even in the presence of experimental imperfections. The protocol requires only unentangled photons, linear operations on multiple modes and at most two-photon joint measurements. These features make the protocol suitable for photonic realization and scalable to large problem sizes with the advances in high-dimensional quantum information manipulation and large scale linear-optical systems. Our results open an essentially new route toward quantum advantages and extend the computational capability of optical quantum computing.

Meerkat Clan Algorithm for Solving N-Queen Problems

The swarm intelligence and evolutionary methods are commonly utilized by researchers in solving the difficult combinatorial and Non-Deterministic Polynomial (NP) problems. The N-Queen problem can be defined as a combinatorial problem that became intractable for the large ‘n’ values and, thereby, it is placed in the NP class of problems. In the present study, a solution is suggested for the N-Queen problem, on the basis of the Meerkat Clan Algorithm (MCA). The problem of n-Queen can be mainly defined as one of the generalized 8-Queen problem forms, for which the aim is placing 8 queens in a way that none of the queens has the ability of killing the others with the use of the standard moves of the chess queen. The Meerkat Clan environment is a directed graph, called the search space, produced for the efficient search of valid n-queens’ placement, in a way that they do not cause harm to one another. This paper also presents the development of an intelligent heuristic function which is helpful to find the solution with high speed and effectiveness. This study includes a detailed discussion of the problem background, problem complexity, Meerkat Clan Algorithm, and comparisons of the problem solution with the Practical Swarm Optimization (PSO) and Genetic Algorithm (GA. It is an entirely review-based work which implemented the suggested designs and architectures of the methods and a fair amount of experimental results.

Research and modeling of operational planning of cargo operations in the sea port

Статья продолжает прежние исследования автора в области построения модели оперативного планирования грузовых операций в морском порту. Затрагивается широкий круг вопросов (стратификация, семиотика, искусственный интеллект). Основное внимание уделено моделированию на верхней «математической» страте - объёмно-календарному планированию на основе многоэтапного нелинейного и динамического программирований (НП), эвристик и экстраполирования. Оптимальный план должен поступать на нижнюю страту, где рассматриваются каргопланы, грузовые технологические схемы, а средства моделирования – системы искусственного интеллекта. В основе моделей верхней страты - диспач-демередж, сталийное время, норма одновременной обработки. Предложены три группы моделей. Во-первых, n-этапные сепарабельные задачи НП; показан алгоритм решения двухэтапной задачи НП динамическим программированием с понижением размерности и множителем Лагранжа; обсуждаются проблемы решения при n>2. Во-вторых, экстраполяционные модели; они включают в себя в качестве «ядер циклов» модели из 1-й группы; обсуждаются способы применения эвристик. В-третьих, несепарабельные задачи НП, которые учитывают процедуры вхождения судов в норму одновременной обработки. Обозначено направление исследований и проектирования модели планирования. The article continues the author's previous research in the field of building a model for operational planning of cargo operations in a seaport. It covers a wide range of issues (stratification, semiotics, artificial intelligence). The main attention is paid to modeling on the upper "mathematical" stratum-volume-calendar planning based on multi-stage nonlinear and dynamic programming (NP), heuristics and extrapolation. The optimal plan should be sent to the lower stratum, where cargoplans, cargo technological schemes are considered, and modeling tools – artificial intelligence systems. The upper stratum models are based on dispatch-demurrage, steel time, and the rate of simultaneous processing. Three groups of models are proposed. First, n-stage separable NP problems; an algorithm for solving a two-stage NP problem by dynamic programming with reduced dimension and a Lagrange multiplier is shown; solution problems for n>2 are discussed. Second, extrapolation of the model; they include models from group 1 as "cycle cores"; ways to apply heuristics are discussed. Third, non-separable NP tasks that take into account the procedures for vessels entering the simultaneous processing norm. The direction of research and design of the planning model is indicated.

aspBEEF: Explaining Predictions Through Optimal Clustering

In this paper we introduce aspBEEF, a tool for generating explanations for the outcome of an arbitrary machine learning classifier. This is done using Grover’s et al. framework known as Balanced English Explanations of Forecasts (BEEF) that generates explanations in terms of in terms of finite intervals over the values of the input features. Since the problem of obtaining an optimal BEEF explanation has been proved to be NP-complete, BEEF existing implementation computes an approximation. In this work we use instead an encoding into the Answer Set Programming paradigm, specialized in solving NP problems, to guarantee that the computed solutions are optimal.

Complexity continuum within Ising formulation of NP problems

A promising approach to achieve computational supremacy over the classical von Neumann architecture explores classical and quantum hardware as Ising machines. The minimisation of the Ising Hamiltonian is known to be NP-hard problem for certain interaction matrix classes, yet not all problem instances are equivalently hard to optimise. We propose to identify computationally simple instances with an `optimisation simplicity criterion'. Such optimisation simplicity can be found for a wide range of models from spin glasses to k-regular maximum cut problems. Many optical, photonic, and electronic systems are neuromorphic architectures that can naturally operate to optimise problems satisfying this criterion and, therefore, such problems are often chosen to illustrate the computational advantages of new Ising machines. We further probe an intermediate complexity for sparse and dense models by analysing circulant coupling matrices, that can be `rewired' to introduce greater complexity. A compelling approach for distinguishing easy and hard instances within the same NP-hard class of problems can be a starting point in developing a standardised procedure for the performance evaluation of emerging physical simulators and physics-inspired algorithms.

Trustworthiness and a Zero Leakage OTMP-P2L Scheme Based on NP Problems for Edge Security Access

Resource constraints have prevented comprehensive cryptography and multifactor authentication in numerous Internet of Things (IoT) connectivity scenarios. Existing IoT systems generally adopt lightweight security protocols that lead to compromise and privacy leakage. Edge computing enables better access control and privacy protection, furthermore, blockchain architecture has achieved a trusted store of value by open-source and distributed consensus mechanisms. To embrace these new paradigms, we propose a scheme that employs one-time association multitasking proofs for peer to local authentication (OTMP-P2L). The scheme chooses relevant nondeterministic polynomial (NP) problem tasks, and manages localized trust and anonymity by using smart devices such as phones and pads, thereby enabling IoT devices to autonomously perform consensus validation with an enhanced message authentication code. This nested code is a one-time zero-knowledge proof that comprises multiple logic verification arguments. To increase diversity and reduce the workload of each one, these arguments are chained by a method that establishes some of the inputs of the following task from the output of previous tasks. We implemented a smart lock system and confirmed that the scheme outperforms IoT authentication methods. The result demonstrates superior flexibility through dynamic difficulty strategies and succinct non-interactive peer-to-peer (P2P) verification.