The maximally diverse grouping problem (MDGP) seeks to partition the vertices of a complete graph into a fixed number of groups under capacity constraints, maximizing the sum of edge weights within each group. MDGP is an NP-hard combinatorial optimization problem and has wide real-world applications. In this paper, we propose an adaptive configuration-aware simulated annealing (ACSA) algorithm to solve MDGP. First, ACSA adopts a relaxation-based insertion strategy, which temporarily relaxes capacity constraints to expand the neighborhood and allow effective exploration of promising regions. Second, a memory-based swap mechanism is introduced to integrate high-potential suboptimal swap moves into the conventional best-swap operation, thereby achieving a better balance between diversification and intensification of the search. Finally, ACSA employs a vertex-wise sequential coordination strategy to dynamically organize the insertion and swap moves, which enhances the search flexibility. Experiments on 500 benchmark instances demonstrate the strong competitiveness of ACSA, as it improves the best results among the state-of-the-art algorithms on 460 instances and matches them on 39 instances.

Given m elements and n subsets of elements, the maximum intersection of k -subsets (kMIS) problem is to select k subsets of elements to maximize the number of elements simultaneously covered by all of the selected subsets. As a general model, kMIS can be used to formulate some practical problems including data privacy control, community detection, and deoxyribonucleic acid microarray technology. This paper presents an opposition-based learning memetic algorithm that integrates opposition-based learning initialization, adaptive crossover, and solution-based tabu search. Experimental results on 608 instances show that the algorithm competes favorably with the state-of-the-art methods. The importance of the algorithmic components is experimentally validated.

In this paper, we present Alkaid-SDVRP, an open-source C++ package for efficiently solving the Vehicle Routing Problem with Split Deliveries (SDVRP), a classical combinatorial optimization problem which is a variant of the Capacitated Vehicle Routing Problem where the same customer can be served by multiple vehicles. The core algorithm of Alkaid-SDVRP is designed based on the Iterated Local Search and Randomized Variable Neighborhood Descent frameworks, which are highly configurable and extensible. Specifically, we implement a number of predefined neighborhoods, including Swap(p, q), [Formula: see text], SD-[Formula: see text], Cross, Exchange, and Reinsertion, which can be arbitrarily enabled, disabled, and permuted. Moreover, it is easy to develop and integrate new neighborhoods into the current framework. The primary goal of this package is to provide an effective implementation and integration of the state-of-the-art techniques for the SDVRP. Tested on the 12th Implementation Challenge held by the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), Alkaid-SDVRP took first place in the SDVRP track and has been shown beyond any doubt. In addition, we hope that the package can facilitate the research for vehicle routing-related problems by providing high-quality baselines and off-the-shelf implementations. History: Accepted by Ted Ralphs, Area Editor for Software Tools. Funding: Financial support from the National Natural Science Foundation of China [Grant 72101094]; the Special Project for Knowledge Innovation of Hubei Province [Grant 2022013301015175]; and Interdisciplinary Research Program of Huazhong University of Science and Technology [Grant 5003300129] is gratefully acknowledged. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0606 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0606 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

The clique partitioning problem (CPP) aims to find a partition of vertices of a complete graph in order to maximize the sum of edge weights within each partition (clique), which has been proven to be NP-hard and has wide real-world applications. In this paper, we propose an elite-guided weighted simulated annealing algorithm called EWSA to solve the CPP. First, EWSA employs two specific configurations and alternates between them via an oscillation strategy, which balances the exploitation and exploration of the search. Second, a weighting strategy is introduced to improve the scoring function in traditional simulated annealing, which is able to guide the search to explore diverse solutions. Finally, a partition restriction strategy is adopted to reduce search space and increase the search efficiency. Experiments on 255 instances demonstrate the competitiveness of EWSA. For 130 open instances, EWSA discovers new upper bounds in 32 cases and matches the best known results for the others. For the remaining 125 closed instances, EWSA achieves the best known objective values within a short computational time.

The α-neighbor p-center problem (α-pCP) is an extension of the classical p-center problem. It aims to select p centers from a set of candidate centers to minimize the maximum distance between any client and its α service centers. In this paper, we propose a weighting-based fast local search algorithm called WFLS for solving α-pCP. First, WFLS converts the complex α-pCP into a series of decision subproblems by specifying the service radius, effectively mitigating the gradient vanishing issue during the search process, and introduces a new MIP model. Then, it addresses the simpliffed subproblems using a fast local search procedure with a swap-based neighborhood structure. WFLS adopts an efffcient weighting strategy, an incremental evaluation technique, a reffned-grained penaltybased neighborhood evaluation, and two scoring functions of neighborhood evaluation to accelerate and guide the search process. Computational experiments on 154 widely used public benchmark instances demonstrate that WFLS outperforms the state-of-the-art methods in the literature. Speciffcally, WFLS improves 69 previous best known results and matches the best know results for all the remaining ones in less time than other competitors.

Large Language Models (LLMs) have emerged as a promising technology for solving combinatorial optimization problems. However, their direct application to scheduling problems remains limited due to the inherent complexity of these problems. This paper proposes an LLMs-based neighborhood search method that leverages LLMs to tackle the job shop scheduling problem (JSP) and its variants. The main contributions of this work are threefold. First, we introduce a novel LLMs-guided neighborhood evaluation strategy that guides local search by dynamically adjusting operation weights. Second, we develop a verification evolution (VeEvo) framework to mitigate the hallucination effects of LLMs, enabling the generation of high-quality heuristics for weight updates. Third, we integrate this framework with the weighted neighborhood evaluation strategy to effectively guide the search towards promising regions. Extensive experiments are conducted on 349 benchmark instances across three classical scheduling problems. The results demonstrate that our algorithm significantly outperforms existing state-of-the-art methods. For JSP, our algorithm reduces the average optimality gap from 10.46% to 1.35% on Taillard's instances compared to reinforced adaptive staircase curriculum learning. For flexible JSP (FJSP), it reduces the gap from 13.24% to 0.05% on Brandimarte's instances compared to deep reinforcement learning methods. Furthermore, for FJSP with sequence dependent setup time, our algorithm updates 9 upper bounds for benchmark instances.

We present a unified heuristic solver for the PACE 2025 challenge, addressing both the dominating set and hitting set problems by reducing them to the unicost set covering problem. Our solver applies standard reduction rules, a multi-round frequency-based greedy initializer, and a local search guided by adaptive element weights. Additional techniques, such as component-level exact solving and swap restriction, further enhance performance. In the final official evaluation, our proposed solver achieved second place in the heuristic track for the dominating set problem of the PACE 2025 challenge, while securing first place in the heuristic track for the hitting set problem.

The single row facility layout problem aims to position a set of facilities of given lengths on a single line so as to minimize the weighted sum of the distances between all the pairs of facilities, which has wide real-world applications in planning areas. In this paper, we propose an oscillation based simulated annealing algorithm for solving the single row facility layout problem. Our algorithm dynamically oscillates between two simulated annealing algorithms. One employs an exponential descent insertion strategy to capture effective movements, which increases the search efficiency, while the other adopts a radius-constrained neighborhood structure to reduce search space, which significantly enhances the intensification of the search. Besides, a new fast incremental evaluation method based on decomposition and recombination is adopted to speed up the search. Experiments on 110 instances demonstrate the competitiveness of our algorithm. In specific, for all the 110 instances, our algorithm discovers new upper bounds in 32 cases and matches the best known results for other 75 instances, only remaining 3 worse results.

This article studies the two-dimensional (2-D) rectangle packing area minimization problem (RPAMP), a key subproblem in floor planning for very large-scale integration (VLSI) chip design. The goal of RPAMP is to orthogonally pack a set of rectangles into a variable-sized rectangular container without overlap, while minimizing the area of the container. By transforming the original problem into a series of 2-D strip packing problems (2DSPs), we propose a two-stage adaptive search algorithm (TS-ASA) to tackle the RPAMP. TS-ASA incorporates several distinctive features: First, a new candidate width pruning strategy is introduced, which limits the number of rectangles used for width combinations, thus reducing the search space. Second, the packing process is divided into two stages, with distinct scoring rules for each stage to optimize space utilization. Additionally, a multirestart strategy is employed to identify the appropriate switching point for the scoring rules. Tested on 39 public benchmark instances and compared with existing state-of-the-art algorithms, TS-ASA improves the best-known solutions for 27 instances and matches the best results for two instances. The experimental results demonstrate the effectiveness and efficiency of the proposed TS-ASA.

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.

Multi-depot vehicle routing problem (MDVRP) is a variant of the classical VRP, which includes several depots with a fleet of homogeneous vehicles to serve each customer exactly once while satisfying the vehicle capacity and duration constraints. We propose a memetic algorithm called GVTS-DPX which hybridizes the granular variable tabu search (GVTS) with the depot partition crossover (DPX) for solving the MDVRP, where GVTS combines tabu search and the granular neighborhoods with variable neighborhood descent, while DPX treats the solution as the collection of depots and partitions the depots into two groups covering the most customers. The main contributions of this study include proposing the DPX operator, reforming several existing move types used for the VRP and its variants, and designing a granular variable neighborhood consisting of a total of 21 kinds of move types. Experimental results on 33 public MDVRP instances indicate that GVTS-DPX is competitive with the state-of-the-art algorithms in the literature.

The capacitated p-center problem$(\mathrm{C}p\text{CP})$is an extension of the classical p-center problem. It consists of choosing$p$centers from a set of candidate centers and assigning each client to a center such that the total client demand assigned to each center does not exceed its given capacity. The objective of the$\mathrm{C}p\text{CP}$is to minimize the maximum distance between each client and its assigned center. In this paper, we propose a two-stage iterated local search algorithm called TS-ILS to solve the$\mathbf{C}p\mathbf{CP}$. The first stage uses a tabu search procedure to select centers and greedily assign clients to centers, while the second stage adopts a variable neighborhood search procedure to perform the fine-grained assignment of clients. Tested on 39 commonly studied instances in the literature, TS-ILS improves the best known results of the state-of-the-art metaheuristic algorithms on 18 instances and matches the records for the remaining ones within less run time.

Improving productivity at the expense of heavy energy consumption is often no longer possible in modern manufacturing industries. Through efficient scheduling technologies, however, we are able to still maintain high productivity while reducing energy costs. This article addresses a flexible job shop scheduling problem under time-of-use electricity tariffs with the objective of minimizing total energy consumption while considering a predefined makespan constraint. We propose a novel two-individual-based evolutionary (TIE) algorithm, which incorporates several distinguishing features, such as a tabu search procedure, a topological order-based recombination operator, a new neighborhood structure for this specific problem, and an approximate neighborhood evaluation method. Extensive experiments are conducted on widely used benchmark instances, which show that the proposed TIE outperforms traditional trajectory-based and population-based methods. We also analyze the key features of TIE to identify its critical success factors, and discuss the impact of varying key parameters of the problem to derive practical insights.

The p-next center problem (pNCP) is an extension of the classical p-center problem. It consists of locating p centers from a set of candidate centers and allocating both a reference and a backup center to each client, to minimize the maximum cost, which is the length of the path from a client to its reference center and then to its backup center. Among them, the reference center is the closest center to a client and serves it under normal circumstances, while the backup center is the closest center to the reference center and serves the client when the reference center is out of service. In this paper, we propose a weighting-based tabu search algorithm called WTS for solving pNCP. WTS optimizes the pNCP by solving its decision subproblems with given assignment costs with an efficient swap-based neighborhood structure and a hierarchical penalty strategy for neighborhood evaluation. Extensive experimental studies on 413 benchmark instances demonstrate that WTS outperforms the state-of-the-art methods in the literature. Specifically, WTS improves 12 previous best known results and matches the optimal results for all remaining 401 ones in a much shorter time than other algorithms. More importantly, WTS reaches the lower bounds for 10 instances for the first time.

Max-cut is one of the most classic NP-hard combinatorial optimization problems . The symmetry nature of it leads to special difficulty in extracting meaningful configuration information for learning; none of the state-of-the-art algorithms has employed any learning operators. This paper proposes an original learning method for max-cut, namely post-flip edge-state learning (PF-ESL). Different from previous algorithms, PF-ESL regards edge-states (cut or not cut) rather than vertex-positions as the critical information of a configuration, and extracts their statistics over a population for learning. It is based on following observations. 1) Edges are the only factors considered by the objective function. 2) Edge-states keep invariant when rotating a local configuration to its symmetry position, but vertex-positions do not. These suggest that edge-states contain more meaningful information about a configuration than vertex-positions do. It is impossible to set the state of an edge without influencing some other edges’ states due to their dependencies. Therefore, instead of setting edge-states directly, PF-ESL samples the flips on vertices. Flips on vertices are sampled according to their capacities in increasing the similarity on edge-states between the given solution and a population. PF-ESL is employed in an EDA (Estimation of Distribution Algorithm) perturbation operator and a path-relinking operator. Experimental results show that our algorithm is competitive, and show that edge-state learning is value-added for both the two operators. The main contributions of this paper are as follows. Firstly, previous state-of-the-art evolutionary algorithms for max-cut focus on vertex positions in their evolutionary operation, this paper proposes a new and more reasonable perspective suggesting that edge-states are the critical information of divided graphs rather than vertex positions, and introduces a novel method to measure and utilize their similarities based on it. Such a perspective is fundamental to learning based algorithms design for max-cut and other graph partitioning problems, and can shed lights on future researches. Furthermore, since max-cut is one of the most classic and fundamental NP hard problems, many real-world problems involve dividing graph data into different parts to optimize certain functions, this new perspective may inspire related or similar problems. Secondly, besides the original edge-states based perspective, and the post-flip edge-states learning (PFESL) operator based on it, our memetic algorithm also incorporates a novel evolutionary framework which alternates between EDA based Iterated Tabu search (ITS) and path relinking based genetic algorithm . Finally, the proposed algorithm provides competitive results on two mostly used benchmark sets and improves the best-known results of 6 most challenging instances.

The minimum connected dominating set (MCDS) problem consists of selecting a minimum set of vertices from an undirected graph, such that each vertex not in this set is adjacent to at least one of the vertices in it, and the subgraph induced by this vertex set is connected. This paper presents a fast vertex weighting (FVW) algorithm for solving the MCDS problem, which integrates several distinguishing features, such as a vertex weighting-based local search with tabu and perturbation strategies to help the search to jump out of the local optima, as well as a search space reduction strategy to improve the search efficiency. Computational experiments on four sets of 112 commonly used public benchmark instances, as well as 15 newly introduced sparse instances, show that FVW is highly competitive compared with the state-of-the-art algorithms in the literature despite its simplicity. FVW improves the previous best-known results for 20 large public benchmark instances while matching the best-known results for all but 2 of the remaining ones. Several ingredients of FVW are investigated to demonstrate the importance of the proposed ideas and techniques. Summary of Contribution: As a challenging classical NP-hard problem, the minimum connected dominating set (MCDS) problem has been studied for decades in the areas of both operations research and computer science, although there does not exist an exact polynomial algorithm for solving it. Thus, the new breakthrough on this classical NP-hard problem in terms of the computational results on classical benchmark instances is significant. This paper presents a new fast vertex weighting local search for solving the MCDS problem. Computational experiments on four sets of 112 commonly used public benchmark instances show that fast vertex weighting (FVW) is able to improve the previous best-known results for 20 large instances while matching the best-known results for all but 2 of the remaining instances. Several ingredients of FVW are also investigated to demonstrate the importance of the proposed ideas and techniques.

Online ride-hailing platforms have reduced significantly the amounts of the time that taxis are idle and that passengers spend on waiting. As a key component of these platforms, the fleet management problem can be naturally modeled as a Markov Decision Process, which enables us to use the deep reinforcement learning. However, existing studies are proposed based on simplified problem settings that fail to model the complicated supply-dynamics and restrict the performance in the real traffic environment. In this article, we propose a supply-demand-aware deep reinforcement learning algorithm for taxi dispatching, where we use a deep Q-network with action sampling policy, called AS-DQN, to learn an optimal dispatching policy. Furthermore, we utilize a dueling network architecture, called AS-DDQN, to improve the performance of AS-DQN. Extensive experiments on real-world datasets offer insight into the performance of our model and show that it is capable of outperforming the baseline approaches.

The optimal camera placement problem (OCP) aims to accomplish surveillance tasks with the minimum number of cameras, which is one of the topics in the GECCO 2020 Competition and can be modeled as the unicost set covering problem (USCP). This paper presents a weighting-based variable neighborhood search (WVNS) algorithm for solving OCP. First, it simplifies the problem instances with four reduction rules based on dominance and independence. Then, WVNS converts the simplified OCP into a series of decision unicost set covering subproblems and tackles them with a fast local search procedure featured by a swap-based neighborhood structure. WVNS employs an efficient incremental evaluation technique and further boosts the neighborhood evaluation by exploiting the dominance and independence features among neighborhood moves. Computational experiments on the 69 benchmark instances introduced in the GECCO 2020 Competition on OCP and USCP show that WVNS is extremely competitive comparing to the state-of-the-art methods. It outperforms or matches several best performing competitors on all instances in both the OCP and USCP tracks of the competition, and its advantage on 15 large-scale instances are over 10%. In addition, WVNS improves the previous best known results for 12 classical benchmark instances in the literature.

In this paper, we study the quadratic assignment problem with a rank-one cost matrix (QAP-R1). Four integer-programming formulations are introduced of which three are assumed to have partial integer data. Unlike the standard quadratic assignment problem, some of our formulations can solve reasonably large instances of QAP-R1 with impressive running times and are faster than some metaheuristics. Pairwise relative strength of the LP relaxations of these formulations are also analyzed from theoretical and experimental points of view. Finally, we present a new metaheuristic algorithm to solve QAP-R1 along with its computational analysis. Our study offers the first systematic experimental analysis of integer-programming models and heuristics for QAP-R1. The benchmark instances with various characteristics generated for our study are made available to the public for future research work. Some new polynomially solvable special cases are also introduced. Summary of Contribution: This paper aims to advance our knowledge and ability in solving an important special case of the quadratic assignment problem. It shows how to exploit inherent properties of an optimization problem to achieve computational advantages, a strategy that was followed by researchers in model building and algorithm developments for decades. Our computational results attest to this time-tested general philosophy. The paper presents the first systematic computational study of the rank one quadratic assignment problem, along with new mathematical programming models and complexity analysis. We believe the theoretical and computational results of this paper will inspire further research on the topic and will be of significant value to practitioners using rank one quadratic assignment models.

The inventory routing problem (IRP), which is NP-hard, tackles the combination of inventory management and transportation optimization in supply chains. It seeks a minimum-cost schedule which utilizes a single vehicle to perform deliveries in multiple periods, so that no customer runs out of stock. Specifically, the solution of IRP can be represented as how many products should be delivered to which customer during each period, as well as the route in each period. We propose a two-stage matheuristic (TSMH) algorithm to solve the IRP. The first stage optimizes the overall schedule and generates an initial solution by a relax-and-repair method. The second stage employs an iterated tabu search procedure to achieve a fine-grained optimization to the current solution. Tested on 220 most commonly used benchmark instances, TSMH obtains advantages comparing to the state-of-the-art algorithms. The experimental results show that the proposed algorithm can obtain not only the optimal solutions for most small instances, but also better upper bounds for 40 out of 60 large instances. These results demonstrate that the TSMH algorithm is effective and efficient in solving the IRP. In addition, the comparative experiments justify the importance of two optimization stages of TSMH.

The p-center problem consists of choosing p centers from a set of candidates to minimize the maximum cost between any client and its assigned facility. In this paper, we transform the p-center problem into a series of set covering subproblems, and propose a vertex weighting-based tabu search (VWTS) algorithm to solve them. The proposed VWTS algorithm integrates distinguishing features such as a vertex weighting technique and a tabu search strategy to help the search to jump out of the local optima. Computational experiments on 138 most commonly used benchmark instances show that VWTS is highly competitive comparing to the state-of-the-art methods in spite of its simplicity. As a well-known NP-hard problem which has already been studied for over half a century, it is a challenging task to break the records on these classic datasets. Yet VWTS improves the best known results for 14 out of 54 large instances, and matches the optimal results for all remaining 84 ones. In addition, the computational time taken by VWTS is much shorter than other algorithms in the literature.

The routing and wavelength assignment (RWA) problem is a classic and challenging problem in wavelength-division multiplexing (WDM) optical networks and has shown to be NP-hard. This paper studies the min-RWA problem with the objective of minimizing the number of required wavelengths and presents a new powerful neighborhood called Shift-and-Shaking (SAS). The proposed SAS integrates a high-level shift move to change the wavelength of one lightpath and two low-level ejection chain-based shaking (ECS) procedures to find the best routings for the related lightpaths. This new neighborhood is embedded into a simple iterated local search algorithm, called SAS-ILS, for solving min-RWA. The proposed SAS-ILS is tested on three sets of totally 113 widely studied instances in the literature. Comparison with other state-of-the-art algorithms shows that the SAS-ILS is able to improve 22 previous best known results, while matching the best known results for the remaining ones within short computational time.

Population-based evolutionary algorithms usually manage a large number of individuals to maintain the diversity of the search, which is complex and time-consuming. In this paper, we propose an evolutionary algorithm using only two individuals, called master-apprentice evolutionary algorithm (MAE), for solving the flexible job shop scheduling problem (FJSP). To ensure the diversity and the quality of the evolution, MAE integrates a tabu search procedure, a recombination operator based on path relinking using a novel distance definition, and an effective individual updating strategy, taking into account the multiple complex constraints of FJSP. Experiments on 313 widely-used public instances show that MAE improves the previous best known results for 47 instances and matches the best known results on all except 3 of the remaining instances while consuming the same computational time as current state-of-the-art metaheuristics. MAE additionally establishes solution quality records for 10 hard instances whose previous best values were established by a well-known industrial solver and a state-of-the-art exact method.

Learnt clauses in CDCL SAT solvers often contain redundant literals. This may have a negative impact on performance because redundant literals may deteriorate both the effectiveness of Boolean constraint propagation and the quality of subsequent learnt clauses. To overcome this drawback, we define a new inprocessing SAT approach which eliminates redundant literals from learnt clauses by applying Boolean constraint propagation. Learnt clause minimization is activated before the SAT solver triggers some selected restarts, and affects only some learnt clauses during the search process. Moreover, we conducted an empirical evaluation on instances coming from the hard combinatorial and application categories of recent SAT competitions. The results show that a remarkable number of additional instances are solved when the approach is incorporated into five of the best performing CDCL SAT solvers (Glucose, TC_Glucose, COMiniSatPS, MapleCOMSPS and MapleCOMSPS_LRB).

The minimum connected dominating set problem (MCDSP) has become increasingly important in recent years due to its applicability to mobile ad hoc networks and sensor grids. This paper presents a restricted swap-based neighborhood (RSN) tailored for solving MCDSP. This novel neighborhood structure is embedded into tabu Search (TS) and a perturbation mechanism is employed to enhance diversification. The proposed RSN-TS algorithm is tested on four sets of public benchmark instances widely used in the literature. The results demonstrate the efficacy of the proposed algorithm in terms of both solution quality and computational efficiency. In particular, the RSN-TS algorithm was able to improve the best known results on 41 out of the 97 problem instances while matching the best known results on all the remaining 56 instances. Furthermore, the article analyzes some key features of the proposed approach in order to identify its critical success factors. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 69(2), 222–236 2017