Exponential Complexity Research Articles

Smart Routes: A System for Development and Comparison of Algorithms for Solving Vehicle Routing Problems with Realistic Constraints

The problem of route optimization with realistic constraints is becoming extremely relevant in the face of global urban population growth. While we are aware of approaches that theoretically provide an exact optimal solution, their application becomes challenging as the problem size increases because of exponential complexity. We investigate the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW) and compare solutions obtaining by exact solver SCIP [1] with heuristic algorithms such as LKH, 2-OPT, 3-OPT [2], the OR-Tools framework [3], and the deep learning model JAMPR [4]. We demonstrate that for problem of size 50 deep learning and classical heuristic solutions became close to SCIP exact solution but requires less time. Additionally for problems with size 100, SCIP exact methods ∼ 13 times slower that neural and classical heuristics with the same route cost and on ∼ 50% worse for the first feasible solution on the same time. To conduct experiments, we developed the Smart Routes platform for solving route optimization problems, which includes exact, heuristic, and deep learning models, and facilitates convenient integration of custom algorithms and datasets.

Generating mutants of monotone affinity towards stronger protein complexes through adversarial learning

Despite breakthroughs achieved in protein sequence-to-structure and function-to-sequence predictions, the affinity-to-mutation prediction problem remains unsolved. Such a problem is of exponential complexity deemed to find a mutated protein or protein complex having a guaranteed binding-affinity change. Here we introduce an adversarial learning-based mutation method that creates optimal amino acid substitutions and changes the mutant’s affinity change significantly in a preset direction. The key aspect in our method is the adversarial training process that dynamically labels the real side of the protein data and generates fake pseudo-data accordingly to construct a deep learning architecture for guiding the mutation. The method is sufficiently flexible to generate both single- and multipointed mutations at the adversarial learning step to mimic the natural circumstances of protein evolution. Compared with random mutants, our mutated sequences have in silico exhibited more than one order of change in magnitude of binding free energy change towards stronger complexes in the case study of Novavax–angiotensin-converting enzyme-related carboxypeptidase vaccine construct optimization. We also applied the method iteratively each time, using the output as the input sequence of the next iteration, to generate paths and a landscape of mutants with affinity-increasing monotonicity to understand SARS-CoV-2 Omicron’s spike evolution. With these steps taken for effective generation of protein mutants of monotone affinity, our method will provide potential benefits to many other applications including protein bioengineering, drug design, antibody reformulation and therapeutic protein medication.

A Module-Level Configuration Methodology for Programmable Camouflaged Logic

Logic camouflage is a widely adopted technique that mitigates the threat of intellectual property (IP) piracy and overproduction in the integrated circuit (IC) supply chain. Camouflaged logic achieves functional obfuscation through physical-level ambiguity and post-manufacturing programmability. However, discussions on programmability are confined to the level of logic cells/gates, limiting the broader-scale application of logic camouflage. In this work, we propose a novel module-level configuration methodology for programmable camouflaged logic that can be implemented without additional hardware ports and with negligible resources. We prove theoretically that the configuration of the programmable camouflaged logic cells can be achieved through the inputs and netlist of the original module. Further, we propose a novel lightweight ferroelectric FET (FeFET)-based reconfigurable logic gate (rGate) family and apply it to the proposed methodology. With the flexible replacement and the proposed configuration-aware conversion algorithm, this work is characterized by the input-only programming scheme as well as the combination of high output error rate and point-function-like defense. Evaluations show an average of >95% of the alternative rGate location for camouflage, which is sufficient for the security-aware design. We illustrate the exponential complexity in function state traversal and the enhanced defense capability of locked blackbox against Boolean Satisfiability (SAT) attacks compared with key-based methods. We also preserve an evident output Hamming distance and introduce negligible hardware overheads in both gate-level and module-level evaluations under typical benchmarks.

Recursion Algorithm for Highlighting the Maximum Clique of the Graph

The work considers the polynomial algorithm for calculating the maximum clique of the non orientable graph. The algorithm is based on the properties of the subspace of sections and the subspace of cycles belonging to the the space sugrapha of a non — unoriented graph. It is shown that the algorithm for the allocation of the set of maximum clicks of the graph has an exponential computational complexity. Examples of highlighting the maximum clique of the graph are considered.

On the estimation of the core for TU games

The core of a transferable utility (TU) game, if it is not empty, is prescribed by the set of all stable allocations. The exact determination of the core reaches exponential time complexity. Therefore, its exact computation is often avoided as the number of players increases. In this work, we propose an estimator for the core of a TU game based on the statistical theory of set estimation. Concretely, we provide a core reconstruction that is obtained in polynomial time for general dimension. Additionally, convergence rates for the estimation error are derived. Finally, a consistent core-center estimator is established as a geometrical application of this methodology.

Deep Graph Reinforcement Learning for Solving Multicut Problem.

The multicut problem, also known as correlation clustering, is a classic combinatorial optimization problem that aims to optimize graph partitioning given only node (dis)similarities on edges. It serves as an elegant generalization for several graph partitioning problems and has found successful applications in various areas such as data mining and computer vision. However, the multicut problem with an exponentially large number of cycle constraints proves to be NP-hard, and existing solvers either suffer from exponential complexity or often give unsatisfactory solutions due to inflexible heuristics driven by hand-designed mechanisms. In this article, we propose a deep graph reinforcement learning method to solve the multicut problem within a combinatorial decision framework involving sequential edge contractions. The customized subgraph neural network adapts to the dynamically edge-contracted graph environment by extracting bilevel connected features from both contracted and original graphs. Our method can learn to infer feasible multicut solutions end-to-end toward optimization of the multicut objective in a data-driven manner. More specifically, by exploring the decision space adaptively, it implicitly gains heuristic knowledge from topological patterns of instances and thereby generates more targeted heuristics overcoming the short-sightedness inherent in the hand-designed ones. During testing, the learned heuristics iteratively contract graphs to construct high-quality solutions within polynomial time. Extensive experiments on synthetic and real-world multicut instances show the superiority of our method over existing combinatorial solvers, while also maintaining a certain level of out-of-distribution generalization ability.

DATA SECURITY IN BIG DATA: CHALLENGES, STRATEGIES, AND FUTURE TRENDS

In the dynamic landscape of big data analytics, this paper explores the critical dimension of data security, addressing challenges, strategies, and emerging trends. Recognizing the exponential growth and complexity of big data, the study delves into the multifaceted challenges posed by its volume, velocity, and variety. Through an extensive literature review, this paper provides an overview of existing studies on data security in big data environments, highlighting the current gaps in understanding. The paper then evaluates strategies employed to fortify data security, emphasizing encryption techniques, access controls, and authentication mechanisms. Drawing insights from real-world case studies, it showcases successful implementations of these strategies, providing practical examples for organizations navigating the intricacies of big data security. As technology continues to evolve, the paper investigates emerging trends in data security, such as the integration of artificial intelligence and machine learning for advanced threat detection, the potential of blockchain for enhancing data integrity, and the adoption of zero-trust architectures. The analysis includes a discussion on the implications of these technologies on the future of data security in big data environments. The methodology section outlines the research design and data collection methods, setting the foundation for the subsequent results and discussion. The results section summarizes key findings, addressing the effectiveness of current strategies and technologies in mitigating data security risks. The discussion section critically analyses the implications of these findings, identifying potential areas for future research and improvement. In conclusion, this paper provides a comprehensive examination of data security in the era of big data, offering valuable insights for practitioners, researchers, and policymakers. By addressing challenges, evaluating strategies, and anticipating future trends, the study contributes to the ongoing discourse surrounding the intersection of data security and big data analytics.

Trie-PMS8: A trie-tree based robust solution for planted motif search problem

Finding patterns in biological sequences is a crucial and intriguing task. This paper explores the (Ɩ, d) motif search problem, also known as Planted Motif Search (PMS), and discusses its challenging nature as an NP-hard problem. PMS and (Ɩ, d) motif search algorithms are believed to represent the next generation of tools for motif discovery. In this context, PMS deals with n biological sequences and two parameters, Ɩ and d, to identify sequences of Ɩ length that occur in all input strings with, at most, d mismatches. Many existing exact PMS algorithms exhibit exponential time complexity in worst-case scenarios. This paper introduces an innovative algorithm that focuses on improving the efficiency of the sample-driven portion of the process. Specifically, dynamic programming techniques are employed to avoid redundant calculations in frequently used subtrees. Furthermore, this paper presents novel approaches to enhance algorithm performance, such as utilizing a trie tree that significantly reduces the time for the “sort rows by size” step. It has also reduced the spaces that take linked lists on LL-PMS8 (Hasan et al., Jun., 2022) or reduced the number of l-mers. Using trie tree as the main way to speed things up gives a much better result than older versions of PMS methods like LL-PMS8 (Hasan et al., Jun., 2022). Overall time complexity reduced than the previous method is 26.17 % and 16.48 % for real-world and generated datasets (Hasan et al., 2020).

LP formulation of stochastic Bayesian two‐player zero‐sum games with long horizon

AbstractThis article studies two‐player zero‐sum stochastic Bayesian games where each player has its own dynamic state that is unknown to the other. To compute the optimal strategy for the players, prior work uses sequence form to construct linear programming formulation, whose size grows exponentially with respect to the time horizon of the game. The exponential computational complexity restricts us in games with short horizon. While window‐by‐window or receding horizon methods can be used to extend the horizon, how to update the initial parameters in each window is still unknown. Based on the existing results about dual games, this article proposes LP formulations to update the initial parameters in each window. A window‐by‐window method is given with performance analysis. The main results are demonstrated in a security problem of underwater sensor networks.

PAC learning halfspaces in non-interactive local differential privacy model with public unlabeled data

In this paper, we study the problem of PAC learning halfspaces in the non-interactive local differential privacy model (NLDP). To breach the barrier of exponential sample complexity, previous results studied a relaxed setting where the server has access to some additional public but unlabeled data. We continue in this direction. Specifically, we consider the problem under the standard setting instead of the large margin setting studied before. Under different mild assumptions on the underlying data distribution, we propose two approaches that are based on the Massart noise model and self-supervised learning and show that it is possible to achieve sample complexities that are only linear in the dimension and polynomial in other terms for both private and public data, which significantly improve the previous results. Our methods could also be used for other private PAC learning problems.

On the Complexity of Chow and Hurwitz Forms

We consider the bit complexity of computing Chow forms of projective varieties defined over integers and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants and obtain a single exponential complexity upper bound. Earlier computational results for Chow forms were in the arithmetic complexity model; thus, our result represents the first bit complexity bound. We also extend our algorithm to Hurwitz forms in projective space and we explore connections between multiprojective Hurwitz forms and matroid theory. The motivation for our work comes from incidence geometry where intriguing computational algebra problems remain open.

A NOVEL ALGORITHM FOR FINDING ALL REDUCTS IN THE INCOMPLETE DECISION TABLE

Attribute reduction, or attribute selection in the decision table, is a fundamental problem of rough set theory. Currently, many scientists are interested in and developing these issues. Unfortunately, most studies focus mainly on the complete decision table. On incomplete decision tables, researchers have proposed tolerance relations and designed attribute reduction algorithms based on different measures. However, these algorithms only return a reduct and do not preserve information in the decision tables. This paper will propose an efficient method to determine entire reducts of incomplete decision tables according to the relational database approach. In the complex case, this algorithm has exponential computational complexity. However, this algorithm has polynomial computational complexity in the different cases of databases.

Average-case complexity of a branch-and-bound algorithm for Min Dominating Set

The average-case complexity of a branch-and-bound algorithm for Min Dominating Set problem in random graphs in the G(n,p) model is studied. We identify phase transitions between subexponential and exponential average-case complexities, depending on the growth of the probability p with respect to the number n of nodes.

Experimental certification of contextuality, coherence, and dimension in a programmable universal photonic processor.

Quantum superposition of high-dimensional states enables both computational speed-up and security in cryptographic protocols. However, the exponential complexity of tomographic processes makes certification of these properties a challenging task. In this work, we experimentally certify coherence witnesses tailored for quantum systems of increasing dimension using pairwise overlap measurements enabled by a six-mode universal photonic processor fabricated with a femtosecond laser writing technology. In particular, we show the effectiveness of the proposed coherence and dimension witnesses for qudits of dimensions up to 5. We also demonstrate advantage in a quantum interrogation task and show it is fueled by quantum contextuality. Our experimental results testify to the efficiency of this approach for the certification of quantum properties in programmable integrated photonic platforms.

Dynamic probabilistic risk assessment with K-shortest-paths planning for generating discrete dynamic event trees

Conventional risk assessment methods are often not well suited for fast-changing dynamic and complex systems since the results from the analysis may be averages or valid for a short time before the system’s state changes. A response to this problem is dynamic probabilistic risk assessment (DPRA), which considers the ever-changing nature of such systems and how their dynamic behavior affects the likelihood of future accident scenarios. Performing DPRA is difficult for complex systems — i.e., systems with many interconnected subsystems and components, for example, autonomous systems. There is a combinatorial “explosion” when considering how component failures affect one another and the overall system performance, known in the DPRA literature as the “state explosion problem”, causing DPRA methods to have poor computational performance for large-scale systems. Although solutions to state explosion alleviate the average-case performance, most DPRA methods remain computationally expensive, with exponential worst-case time complexities. In this paper, a method for solving DPRA problems with K-Shortest-Paths planning algorithms is proposed. The method, called KPRA, consists in framing a subset of DPRA problems as relaxed versions of the K-Shortest-Paths (KSP) planning problem, allowing these DPRA problems to be solved by a graph search algorithm called K*, which has a theoretical log-linear worst-case complexity. Therefore, in theory, KPRA with K* solves DPRA problems with computational performance better than exponential. KPRA was implemented and applied to a case study of DPRA for an autonomous ship for validation and comparison with conventional DPRA methods. The case study consists of two ships, one of them autonomous, in a crossing encounter with a possible collision risk. The task is to find the most critical situations for the autonomous ship, i.e., the scenarios where its collision risk is the highest. KPRA’s performance in solving the case study is compared with two conventional DPRA methods. The results show that the KPRA implementation in this work can solve the case study, i.e., produce an output equivalent to the other methods, with a polynomial worst-case computational complexity, i.e., more efficiently than the other methods with exponential complexities.

Efficient Mobility Management in Mobile Edge Computing Networks: Joint Handover and Service Migration

Mobile edge computing (MEC) has been envisioned as an essential technology for latency-critical applications by providing computing services in close proximity to mobile users. Bringing MEC to come into practice, how to support user mobility remains challenging. In addition to seeking a thorny trade-off between service latency and migration cost, both interactions in space and time exist in mobility management, which requires collaboration among users and perfect prior knowledge, including user mobility and network information. In this paper, we propose an efficient mobility management framework for MEC networks, in which mobility management is operated centered around users’ performance and cost, while radio access and computing service provision are loosely coupled. With a loosely coupled design, the proposed framework exhibits more flexibility and incurs higher complexity. Focusing on multi-user and multi-cell MEC networks, this joint control problem is formulated to maximize the long-term total utility accounting for the service delay and migration cost. Considering the exponential complexity, a distributed mobility management approach is developed, which combines game theory and user-oriented deep reinforcement learning to deal with the interactions in space and time. Simulation results show the efficiency and scalability of the proposed approach.

Variational Quantum Linear Solver-based Combination Rules in Dempster–Shafer Theory

Dempster–Shafer Theory (DST), as a method of handling uncertain information, is widely used in decision-making and information fusion. But the issue of exponential computational complexity limits its real-time application. Since quantum computers with its natural parallel computing capability can theoretically achieve speedup, an HHL-based algorithm for implementing DST operations on quantum circuits has been proposed. However, the algorithm is impossible to be implemented with high accuracy due to the limited performance of quantum computers in the current noisy intermediate-scale quantum (NISQ) era. In response to that, we utilize an up-to-date scheme Variational Quantum Linear Solver (VQLS) to conduct DST operations by solving the linear system. Besides, VQLS further reveals the structural consistency of DST with quantum computation, which enables faster circuit configuration and computation. In this paper, we first realize the transformation in belief functions by VQLS and then extends it to a methods of implementing combination rules. Simulated under a realistic classification task, the feasibility and accuracy of the proposed VQLS-based method are verified. The new VQLS-based method proposed is able to achieve combination rules at a lower error level with fewer quantum resources, which is more suitable for the NISQ era.

A novel approach and hybrid parallel algorithms for solving the fixed charge transportation problem

This article is dedicated to the efficient resolution of the fixed charge transport problem (FCTP) with the goal of identifying optimal solutions within reduced timeframes. FCTP is a combinatorial and NP-complete problem known for its exponential time complexity relative to problem size. Metaheuristic methods, including genetic algorithms, represent effective techniques for obtaining high-quality FCTP solutions. Consequently, the integration of parallel algorithms emerges as a strategy for expediting problem-solving. The proposed approach, referred to as the parallel genetic algorithm (PGA), entails the application of a genetic algorithm across multiple parallel architectures to tackle the FCTP problem. The primary aim is to explore fresh solutions for the fixed charge transportation problem using genetic algorithms while concurrently optimizing the time required to achieve these solutions through parallelism. The FCTP problem is fundamentally a linear programming challenge, revolving around the determination of optimal shipment quantities from numerous source locations to multiple destinations with the overarching objective of minimizing overall transportation costs. This necessitates consideration of constraints tied to product availability at the sources and demand dynamics at the destinations. In this study, a pioneering approach to addressing the Fixed Charge Transportation Problem (FCTP) using parallel genetic algorithms (PGA) is unveiled. The research introduces two distinct parallel algorithms: The Master-Slave Approach (MS-GA) and the Coarse-Grained Approach (CG-GA). Additionally, investigation into the hybridization of these approaches has led to the development of the NMS-CG-GA approach. The numerical results reveal that our parallelism-based approaches significantly improve the performance of genetic algorithms. Specifically, the Master-Slave (MS-GA) approach demonstrates its advantages in solving smaller instances of the FCTP problem, while the Coarse-Grained (CG-GA) approach exhibits greater effectiveness for larger problem instances. The conclusion reached is that the novel hybrid parallel genetic algorithm approach (NMS-CG-GA) outperforms its predecessors, yielding outstanding results, particularly across diverse FCTP problem instances.

Depth-bounded fuzzy simulations and bisimulations between fuzzy automata

Simulations and bisimulations are well-established notions in crisp/fuzzy automata theory and are widely used to compare the behaviors of automata. Their main drawback is that they compare the behaviors of fuzzy automata in a crisp manner. Recently, fuzzy simulations and fuzzy bisimulations have been defined for fuzzy automata as a kind of approximate simulations and approximate bisimulations that compare the behaviors of fuzzy automata in a fuzzy manner. However, they still suffer from serious shortcomings. First, they still cannot correlate all fuzzy automata that are intuitively “more or less” (bi)similar. Second, the currently known algorithms for computing the greatest fuzzy simulation or bisimulation between two finite fuzzy automata have an exponential time complexity when the Łukasiewicz or the product structure of fuzzy values is used. This work deals with these problems, providing approximations of fuzzy simulations and fuzzy bisimulations. We define such approximations via a novel notion of decreasing sequences of fuzzy relations whose infima are, under some conditions, fuzzy simulations (respectively, bisimulations). We call such a sequence a depth-bounded fuzzy simulation (respectively, bisimulation), as the nth element from the sequence compares the behaviors of fuzzy automata, but only for words with a length bounded by n. We further provide a logical characterization of the greatest depth-bounded fuzzy simulation or bisimulation between two fuzzy automata by proving that it satisfies the corresponding Hennessy-Milner property. Finally, we provide polynomial-time algorithms for computing the nth component of the greatest depth-bounded fuzzy simulation (respectively, bisimulation) between two finite fuzzy automata.

The algorithms for the Eulerian cycle and Eulerian trail problems for a multiple graph

In this paper, we study undirected multiple graphs of any natural multiplicity $k>1$. There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of $k$ linked edges, which connect 2 or $(k+1)$ vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges and it can be the common end of $k$ linked edges of some multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of another multi-edge. We set the problem of finding the eulerian walk (the cycle or the trail) in a multiple graph, which generalizes the classical problem for an ordinary graph. We formulate the necessary conditions for existence of an eulerian walk in a multiple graph and show that these conditions are not sufficient. Besides that, we show that the necessary conditions of existence of an eulerian cycle and eulerian trail are not mutually exclusive for an arbitrary multiple graph, that is why it is possible to construct a multiple graph where two types of eulerian walks exist simultaneously. Any multiple graph can be juxtaposed to the ordinary graph with quasi-vertices, which represents the structure of the initial graph in a simpler form. In particular, each eulerian walk in the multiple graph corresponds to the eulerian walk in the graph with quasi-vertices. The algorithm for getting such a graph is formulated. Also, the auxiliary problem of finding the covering trails with given endpoints in an ordinary graph is studied. Two algorithms are obtained for this problem. We elaborate the algorithm for finding the eulerian walk in a multiple graph, which has the exponential complexity. We suggest the polynomial algorithm for the special case of a multiple graph and show that the necessary conditions are sufficient for existence of an eulerian walk in this special case.