NP-hard Problem Research Articles

SMITS: Social and Mobility aware Intelligent Task Scheduling in Vehicular Fog Computing — A Federated DRL Approach

Vehicular fog computing (VFC) has become an enticing research hot spot to provide resources for the Internet of Vehicles (IoV) application requests. Moreover, with the massive expansion of services in vehicular fog networks such as augmented reality, 3-D gaming, and autonomous driving, the allocation of fog computing resources to IoV applications constraining the deadline and energy consumption of the vehicles is becoming challenging. In this paper, we investigate a task scheduling problem in vehicular fog networks to jointly optimize the success rate of vehicular tasks and energy consumption of vehicles by considering the moving paths of the vehicles and the social relationships among the vehicles. In this problem, we model the social relationship among the vehicles based on their communication patterns and social characteristics to improve the success rate of the tasks. We also incorporate the mobility of vehicles using the Markov renewal process (MRP) technique. Initially, we formulate a mixed integer nonlinear programming (MINLP) problem for the proposed problem and further prove it to be an NP-hard problem. To solve the proposed problem, we design a federated deep reinforcement learning (FDRL) mechanism by converting the problem into a Markov decision process (MDP) problem. We compare our proposed algorithm with existing scheduling approaches, and simulation results show that the proposed algorithm performs efficiently for task scheduling problems. The proposed algorithm achieved a substantial improvement of 12% in the maximization of the success rate and 36% in the minimization of the energy consumption compared to the existing algorithms.

Binary rat swarm optimizer algorithm for computing independent domination metric dimension problem

In this article, we look at the NP-hard problem of determining the minimum independent domination metric dimension of graphs. A vertex set B of a connected graph G(V,E) resolves G if every vertex of G is uniquely recognized by its vector of distances to the vertices in B. If there are no neighboring vertices in a resolving set B of G, then B is independent. Every vertex of G that does not belong to B must be a neighbor of at least one vertex in B for a resolving set to be dominant. The metric dimension of G, independent metric dimension of G, and independent dominant metric dimension of G are, respectively, the cardinality of the smallest resolving set of G, the minimal independent resolving set, and the minimal independent domination resolving set. We propose the first attempt to use a binary version of the Rat Swarm Optimizer Algorithm (BRSOA) to heuristically calculate the smallest independent dominant resolving set of graphs. The search agent of BRSOA are binary-encoded and used to identify which one of the vertices of the graph belongs to the independent domination resolving set. The feasibility is enforced by repairing search agent such that an additional vertex created from vertices of G is added to B, and this repairing process is repeated until B becomes the independent domination resolving set. Using theoretically computed graph findings and comparisons to competing methods, the proposed BRSOA is put to the test. BRSOA surpasses the binary Grey Wolf Optimizer (BGWO), the binary Particle Swarm Optimizer (BPSO), the binary Whale Optimizer (BWOA), the binary Gravitational Search Algorithm (BGSA), and the binary Moth-Flame Optimization (BMFO), according to computational results and their analysis.

Dynamic resource allocation for energy-efficient downlink NOMA systems in 5G networks

Non-Orthogonal Multiple Access (NOMA) is a promising energy-efficient technology designed to satisfy the demands of future networks by efficiently sharing radio resources. In NOMA, the same radio resource is simultaneously assigned to two users at different power levels based on the NOMA-power domain. Resource allocation in NOMA presents a non-convex challenge, characterized as a non-deterministic polynomial (NP-hard) problem. This involves user and channel assignment and power allocation, making it an extraordinarily complex task to achieve an optimal solution. In this work, Simulated Annealing (SA) is proposed as an optimization technique for resource allocation in an energy-efficient downlink NOMA system. This resource allocation scheme addresses user and channel assignment, as well as power allocation, using SA as an efficient standalone approach to maximize energy efficiency in NOMA. SA is utilized to execute the assignment of users to channels, distribute the necessary power for each channel, and determine the power ratio among users sharing the same channel. The results obtained demonstrate a significant improvement in energy efficiency, outperforming the existing numerical methods by 22 %. The proposed SA scheme not only achieves a close optimal solution but also in less computational time, offering sufficient reliability in terms of energy efficiency enhancement when compared to the existing numerical method.

Big Data Scheduling with Order Acceptance Consideration in Supply Chain

The efficient job scheduling schemes can make full use of resources, and then achieve different goals, such as maximizing efficiency, minimizing cost and saving energy. In supply chain, there are a lot of members and enormous data. Therefore, a suitable scheduling scheme has become the most common and effective method to optimize the execution of big data in supply chain. A scheduling problem with production and delivery consideration, which is usually in supply chain has been considered. For the reason of highly time emergency and random coming orders in quick production and delivery system, the effective algorithms for order acceptance scheduling problem are required. This paper addressed the production and delivery problem which has one manufacturer and multiple customers. There is single machine for order production at the manufacturer. For the consideration of order acceptance or not, the manufacturer need to choose order set to be accepted for processing. The paper aims at finding a balance between orders profit, delivery cost and time-based cost to maximize the total revenue. We consider three main objective functions in scheduling theory, analyze the problem complexity. The complexity of two of the problems are weakly NP hard, and the other one is strongly NP hard. For three weakly NP hard problems, we give pseudo-polynomial time optimal algorithms.

A new early warning system based on metaheuristics in CPS architecture

ABSTRACT Cyber security is the most studied in recent years by the association of metaheuristic techniques thanks to their ability to solve optimization problems in a limited time and NP-hard problems. The Early Warning System (EWS) can identify malicious exercises and anomalies in the CPS and provide robust network systems protection. In addition, grouping attacks in EWS is important for defining defense policies. Information security in the CPS architecture is about securing information during data aggregation, large-scale processing and sharing in the network environment, especially low-linkage open networks. Motivated by these considerations, we propose a new security approach based on the Artificial Bee Colony (ABC) metaheuristic algorithm to strengthen the security framework of the CPS architecture against the system and within CPS attacks. In this work, we propose a new Early Warning System approach based on metaheuristics. Based on two fundamental concepts of clustering, coherence and separation, an integrated index was proposed based on Davies-Bouldin (DB) and Silhouette Index (SI). An improved ABC algorithm was proposed based on these indices. After having identified two different types of an objective function, based on the above concepts, and after having tested the algorithm on DBSI, the results of the experiments found are auspicious.

Research on Genetic Algorithm Optimization with Fusion Tabu Search Strategy and Its Application in Solving Three-Dimensional Packing Problems

Symmetry is an important principle and characteristic that is prevalent in nature and artificial environments. In the three-dimensional packing problem, leveraging the inherent symmetry of goods and the symmetry of the packing space can enhance packing efficiency and utilization.The three-dimensional packing problem is an NP-hard combinatorial optimization problem in the field of modern logistics, with high computational complexity. This paper proposes an improved genetic algorithm by incorporating a fusion tabu search strategy to address this problem. The algorithm employs a three-dimensional loading mathematical model and utilizes a wall-building method under residual space constraints for stacking goods. Furthermore, adaptation of fitness variation strategy, chromosome adjustment, and tabu search algorithm are introduced to balance the algorithm’s global and local search capabilities, as well as to enhance population diversity and convergence speed. Through testing on benchmark cases such as Bischoff and Ratcliff, the improved algorithm demonstrates an average increase of over 3% in packing space utilization compared to traditional genetic algorithms and other heuristic algorithms, validating its feasibility and effectiveness. The proposed improved genetic algorithm provides new insights for solving three-dimensional packing problems and optimizing logistics loading schedules, offering promising prospects for various applications.

Deep Meta Q-Learning Based Multi-Task Offloading in Edge-Cloud Systems

Resource-constrained edge devices can not efficiently handle the explosive growth of mobile data and the increasing computational demand of modern-day user applications. Task offloading allows the migration of complex tasks from user devices to the remote edge-cloud servers thereby reducing their computational burden and energy consumption while also improving the efficiency of task processing. However, obtaining the optimal offloading strategy in a multi-task offloading decision-making process is an NP-hard problem. Existing Deep learning techniques with slow learning rates and weak adaptability are not suitable for dynamic multi-user scenarios. In this article, we propose a novel deep meta-reinforcement learning-based approach to the multi-task offloading problem using a combination of first-order meta-learning and deep Q-learning methods. We establish the meta-generalization bounds for the proposed algorithm and demonstrate that it can reduce the time and energy consumption of IoT applications by up to 15%. Through rigorous simulations, we show that our method achieves near-optimal offloading solutions while also being able to adapt to dynamic edge-cloud environments.

A genetic scheduling strategy with spatial reuse for dense wireless networks

Novel networking technologies such as massive Internet-of-Things and 6G-and-beyond cellular networks are based on ultra-dense wireless communications. A wireless communication channel is a shared medium that demands access control, such as proper transmission scheduling. The SINR model can improve the performance of ultra-dense wireless networks by taking into consideration the effects of interference to allow multiple simultaneous transmissions in the same coverage area and using the same frequency band. However, scheduling in wireless networks under the SINR model is an NP-hard problem. This work presents a bioinspired solution based on a genetic heuristic to solve that problem. The proposed solution, called Genetic-based Transmission Scheduler (GeTS) produces a complete transmission schedule optimizing size, increasing the number of simultaneous transmissions (i.e., spatial reuse) thus allowing devices to communicate as soon as possible. Simulation results are presented for GeTS, including a convergence test and comparisons with other alternatives. Results confirm the ability of the solution to produce near-optimal schedules.

Detecting k-Vertex Cuts in Sparse Networks via a Fast Local Search Approach

The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -vertex cut ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -VC) problem belongs to the family of the critical node detection problems, which aims to find a minimum subset of vertices whose removal decomposes a graph into at least <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> connected components. It is an important NP-hard problem with various real-world applications, e.g., vulnerability assessment, carbon emissions tracking, epidemic control, drug design, emergency response, network security, and social network analysis. In this article, we propose a fast local search (FLS) approach to solve it. It integrates a two-stage vertex exchange strategy based on neighborhood decomposition and cut vertex, and iteratively executes operations of addition and removal during the search. Extensive experiments on both intersection graphs of linear systems and coloring/DIMACS graphs are conducted to evaluate its performance. Empirical results show that it significantly outperforms the state-of-the-art (SOTA) algorithms in terms of both solution quality and computation time in most of the instances. To evaluate its generalization ability, we simply extend it to solve the weighted version of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -VC problem. FLS also demonstrates its excellent performance.

Decomposition-based multi-objective approach for a green hybrid flowshop rescheduling problem with consistent sublots

Rescheduling in a hybrid flowshop holds significant importance in modern industries that face uncertain events. Moreover, real-world manufacturing scenarios often utilise lot streaming to enhance market competitiveness. In light of escalating energy demands and their consequential environmental impacts, contemporary manufacturing companies are placing a heightened emphasis on energy efficiency. This study addressed a green hybrid flowshop rescheduling problem with consistent sublots (GHFRP_CS) in the context of urgent lot insertion. Initially, we establish an optimisation model aimed at minimising the makespan, total energy consumption, and system stability. To tackle this NP-hard multi-objective optimization problem, we develop a constructive heuristic generating promising solutions based on lot split, sequence, and local search rules. Further improvement is achieved through a multi-objective discrete artificial bee colony algorithm (MDABC). MDABC decomposes the problem into sub-problems, initiating solutions with the constructive heuristic and refining them through employed bee, onlooker bee, and scout bee phases. Computational experiments compare MDABC with other multi-objective evolutionary algorithms (MOEAs) on small- and large-scale problems. Results demonstrate MDABC's superiority, achieving fourfold accuracy and efficiency enhancement for small-scale instances and sixfold improvement for large-scale problems at low cost compared to other MOEAs.

Scheduling of twin automated stacking cranes based on Deep Reinforcement Learning

Effective scheduling of twin automated stacking cranes (ASCs) in automated storage yard is critical to maximize operational efficiency. While Deep Reinforcement Learning (DRL) is promising in solving NP-hard scheduling problems, twin ASCs scheduling is challenging due to its unique properties including sequence-dependent setup and potential ASC interferences. In this paper, we propose a novel DRL method to learn high-quality policy for scheduling twin ASCs. We propose a Markov Decision Process model that enables the DRL agent to learn to minimize makespan and possible interferences. Based on the problem characteristics, we design a self-attention based neural architecture to effectively capture the relationships between containers under certain block state. Experiments show that the agent with the proposed feature extraction network can learn high-quality policies from training instances. These learned policies can be employed to produce effective scheduling solutions within seconds. Compared to traditional scheduling methods, the learned policy performs best in most problem sizes, and the performance improvement amplifies as the scales increase. Moreover, the policies show remarkable generalization ability on unseen instances with different distributions or scales.

Preprocessing to reduce the search space: Antler structures for feedback vertex set

The goal of this paper is to open up a new research direction aimed at understanding the power of preprocessing in speeding up algorithms that solve NP-hard problems exactly. We explore this direction for the classic Feedback Vertex Set problem on undirected graphs, leading to a new type of graph structure called antler decomposition, which identifies vertices that belong to an optimal solution. It is an analogue of the celebrated crown decomposition which has been used for Vertex Cover. We develop the graph structure theory around such decompositions and develop fixed-parameter tractable algorithms to find them, parameterized by the number of vertices for which they witness presence in an optimal solution. This reduces the search space of fixed-parameter tractable algorithms parameterized by the solution size that solve Feedback Vertex Set.

EnMatch: Matchmaking for Better Player Engagement via Neural Combinatorial Optimization

Matchmaking is a core task in e-sports and online games, as it contributes to player engagement and further influences the game's lifecycle. Previous methods focus on creating fair games at all times. They divide players into different tiers based on skill levels and only select players from the same tier for each game. Though this strategy can ensure fair matchmaking, it is not always good for player engagement. In this paper, we propose a novel Engagement-oriented Matchmaking (EnMatch) framework to ensure fair games and simultaneously enhance player engagement. Two main issues need to be addressed. First, it is unclear how to measure the impact of different team compositions and confrontations on player engagement during the game considering the variety of player characteristics. Second, such a detailed consideration on every single player during matchmaking will result in an NP-hard combinatorial optimization problem with non-linear objectives. In light of these challenges, we turn to real-world data analysis to reveal engagement-related factors. The resulting insights guide the development of engagement modeling, enabling the estimation of quantified engagement before a match is completed. To handle the combinatorial optimization problem, we formulate the problem into a reinforcement learning framework, in which a neural combinatorial optimization problem is built and solved. The performance of EnMatch is finally demonstrated through the comparison with other state-of-the-art methods based on several real-world datasets and online deployments on two games.

Equity-Transformer: Solving NP-Hard Min-Max Routing Problems as Sequential Generation with Equity Context

Min-max routing problems aim to minimize the maximum tour length among multiple agents as they collaboratively visit all cities, i.e., the completion time. These problems include impactful real-world applications but are known as NP-hard. Existing methods are facing challenges, particularly in large-scale problems that require the coordination of numerous agents to cover thousands of cities. This paper proposes Equity-Transformer to solve large-scale min-max routing problems. First, we model min-max routing problems into sequential planning, reducing the complexity and enabling the use of a powerful Transformer architecture. Second, we propose key inductive biases that ensure equitable workload distribution among agents. The effectiveness of Equity-Transformer is demonstrated through its superior performance in two representative min-max routing tasks: the min-max multi-agent traveling salesman problem (min-max mTSP) and the min-max multi-agent pick-up and delivery problem (min-max mPDP). Notably, our method achieves significant reductions of runtime, approximately 335 times, and cost values of about 53% compared to a competitive heuristic (LKH3) in the case of 100 vehicles with 1,000 cities of mTSP. We provide reproducible source code: https://github.com/kaist-silab/equity-transformer.

Threshold-Based Responsive Simulated Annealing for Directed Feedback Vertex Set Problem

As a classical NP-hard problem and the topic of the PACE 2022 competition, the directed feedback vertex set problem (DFVSP) aims to find a minimum subset of vertices such that, when vertices in the subset and all their adjacent edges are removed from the directed graph, the remainder graph is acyclic. In this paper, we propose a threshold-based responsive simulated annealing algorithm called TRSA for solving DFVSP. First, we simplify the problem instances with two new reduction rules proposed in this paper and eight reduction rules from the literature. Then, based on a new solution representation, TRSA solves DFVSP with a fast local search procedure featured by a swap-based neighborhood structure and three neighborhood acceleration strategies. Finally, all these strategies are incorporated into a threshold-based responsive simulated annealing framework. Computational experiments on 140 benchmark instances show that TRSA is highly competitive compared to the state-of-the-art methods. Specifically, TRSA can improve the best known results for 53 instances, while matching the best known results for 79 ones. Furthermore, some important features of TRSA are analyzed to identify its success factors.

Learning Small Decision Trees for Data of Low Rank-Width

We consider the NP-hard problem of finding a smallest decision tree representing a classification instance in terms of a partially defined Boolean function. Small decision trees are desirable to provide an interpretable model for the given data. We show that the problem is fixed-parameter tractable when parameterized by the rank-width of the incidence graph of the given classification instance. Our algorithm proceeds by dynamic programming using an NLC decomposition obtained from a rank-width decomposition. The key to the algorithm is a succinct representation of partial solutions. This allows us to limit the space and time requirements for each dynamic programming step in terms of the parameter.

Scheduling on parallel dedicated machines with job rejection

We study scheduling problems on parallel dedicated machines. Thus, each job can be processed on one specific machine only. The option of job-rejection is considered, and the total permitted rejection cost of all the jobs is bounded. Six scheduling problems are solved: ( i ) minimising makespan, ( ii ) minimising makespan with release-dates, ( iii ) minimising total completion time, ( iv ) minimising total weighted completion time, ( v ) minimising total load, and ( vi ) minimising maximum tardiness. Pseudo-polynomial dynamic programming algorithms are introduced for all these NP-hard problems.

Optimization for cost-effective design of water distribution networks: a comprehensive learning approach

The Comprehensive Learning Gravitational Search Algorithm (CLGSA) has demonstrated its effectiveness in solving continuous optimization problems. In this research, we extended the CLGSA to tackle NP-hard combinatorial problems and introduced the Discrete Comprehensive Learning Gravitational Search Algorithm (D-CLGSA). The D-CLGSA framework incorporated a refined position and velocity update scheme tailored for discrete problems. To evaluate the algorithm's efficiency, we conducted two sets of experiments. Firstly, we assessed its performance on a diverse range of 24 benchmarks encompassing unimodal, multimodal, composite, and special discrete functions. Secondly, we applied the D-CLGSA to a practical optimization problem involving water distribution network planning and management. The D-CLGSA model was coupled with the hydraulic simulation solver EPANET to identify the optimal design for the water distribution network, aiming for cost-effectiveness. We evaluated the model's performance on six distribution networks, namely Two-loop network, Hanoi network, New-York City network, GoYang network, BakRyun network, and Balerma network. The results of our study were promising, surpassing previous studies in the field. Consequently, the D-CLGSA model holds great potential as an optimizer for economically and reliably planning and managing water networks.

Agile earth observation satellite constellations scheduling for large area target imaging using heuristic search

Demand for imaging of large area targets by space-based assets such as constellations of Earth Observation Satellites (EOS) is on the rise for different applications. The area targets coverage cannot be achieved by a single satellite in a single observation opportunity, as the region for imaging spreads wider than the coverage swaths of imaging sensors. Hence, the imaging satellites are scheduled in a cooperative way to achieve maximum coverage of the area in a given planning time horizon. The scheduling algorithms also aim to maximize or minimize some specific objectives of interest while simultaneously complying with all resource constraints for an efficient use of precious space resources. As the current generation satellites are highly agile, there are many ways of choosing an imaging location in the target area for each single observation opportunity of a satellite. Even though the possible imaging region in continuous space is discretized into a finite number of strips for each observation, the huge combinatorial search space formed with all satellites’ imaging opportunities makes the problem intractable. Thus, the area target imaging scheduling is a complex combinatorial NP-hard optimization problem. Many exact and heuristic methods were evolved to provide a solution to this scheduling problem. Even though the exact methods solve the problem to optimality, they are limited to smaller problem instances because of the high computational complexities involved in those methods. More focus is seen in the literature on heuristic approaches as they guarantee scheduling performance with feasible solutions within an acceptable computational complexity. This paper presents our efficient heuristic approach proposed for the scheduling problem for imaging a large area target using a constellation of multiple satellites. Our approach is an extension of a widely used greedy approach in combination with insights from dynamic programming. In this paper, first we addressed the scheduling problem by describing it in detail and formulating it into a non-linear integer programming problem, considering the multi-objectives of achieving maximum coverage and minimum imaging resolution. The solution to the problem comprises two phases, namely the area decomposition phase and the scheduling phase. In the area decomposition phase, the area target is divided into strips dynamically for each observation opportunity of a satellite. Then exact observation period of each strip is computed using a simplified semi-analytical computational method. In the scheduling phase, we applied our heuristic search strategy, namely Forward and Backward Heuristic Search (FBHS) to obtain a near optimal solution within an acceptable computational time. Through the extensive simulations conducted under various scenarios, the effectiveness of the proposed method is verified in comparison with the baseline greedy approach.

Air cargo load and route planning in pickup and delivery operations

In the aerial pickup and delivery of goods in a distribution network, transport aviation faces risks of load imbalance due to the urgency required for loading, immediate take-off, and mission accomplishment. Transport planners deal with trip itineraries, prioritization of items, building up pallets, and balanced loading, but there are no commercially available systems that can integrally assist in all these requirements. This enables other risks, such as improper delivery, excessive fuel burn, and possible safety issues due to cargo imbalance, as well as a longer than necessary turn-around time. This NP-hard problem, named Air Cargo Load Planning with Routing, Pickup, and Delivery Problem (ACLP+RPDP), is mathematically modelled using standardized pallets in fixed positions. We developed a strategy to solve this problem, considering historical transport data from some Brazilian hub networks, and performed several experiments with a commercial solver, five known meta-heuristics, and a new heuristic designed specifically for this problem. By using a portable computer, our strategy quickly found practical solutions to a wide range of real problems in much less than operationally acceptable time.