Balancing exploration and exploitation remain a fundamental challenge in artificial intelligence-based optimization, particularly when addressing discrete combinatorial problems such as the quadratic assignment problem (QAP). The gorilla troops optimizer (GTO), inspired by the collective social behavior of gorillas, has shown promising results in continuous domains but faces limitations when directly applied to discrete optimization. To address this, the present study introduces a modified gorilla troops optimizer (MGTO), a novel discrete adaptation designed specifically for the QAP. The proposed MGTO strategically integrates a swapping-based diversification mechanism to enhance exploration within discrete solution spaces, while a modified uniform crossover operator promotes effective exploitation of high-quality solutions. Extensive experiments on benchmark instances from the quadratic assignment problem library (QAPLIB) show that MGTO achieves superior convergence behavior and solution quality compared with several state-of-the-art algorithms. These results demonstrate MGTO’s capacity to maintain a balanced equilibrium between exploration and exploitation, effectively navigating complex discrete landscapes to yield high-quality solutions with strong computational efficiency.

Qubit allocation is a central step in adapting abstract quantum circuits to noisy intermediate-scale quantum devices, yet exact approaches for solving it face severe scalability limitations. In this work, we revisit the formulation of qubit allocation as a permutation-based quadratic assignment problem and develop a branch-and-bound algorithm for its exact resolution. We first establish a refined sequential implementation that achieves significantly faster runtimes than previous exact approaches on most problem instances, thereby setting a new state-of-the-art for this formulation. Building on this foundation, we extend the approach to a performance-aware parallel implementation that exploits both intra-node and inter-node parallelism on High-Performance Computing (HPC) infrastructures. Our experimental evaluation demonstrates near-linear strong scaling at the intra-node level and substantial scalability in distributed settings across nodes. Leveraging these capabilities, we provide reference optimal solutions for challenging benchmark circuits of up to 26 qubits—significantly larger than previously reported instances. These results show that large-scale parallelization can effectively extend the reach of exact methods for qubit allocation, thereby advancing the integration of combinatorial optimization and HPC techniques in quantum computing.

Preparing arbitrary quantum states requires exponential resources. Matrix product states (MPS) admit more efficient constructions, particularly when accuracy is traded for circuit complexity. Existing approaches to MPS preparation mostly rely on heuristic circuits that are deterministic but quickly saturate in accuracy, or on variational optimization methods that reach high fidelities but scale poorly. This work introduces an end-to-end MPS preparation framework that combines the strengths of both strategies within a single pipeline. Heuristic staircase-like and brick wall disentangler circuits provide warm-start initializations for variational optimization, enabling high-fidelity state preparation for large systems. Target MPSs are either specified as physical quantum states or constructed from classical datasets via amplitude encoding, using step-by-step singular value decompositions or tensor cross interpolation. The framework incorporates entanglement-based qubit reordering, reformulated as a quadratic assignment problem, and low-level optimizations that reduce depths by up to and CNOT counts by . We evaluate the full pipeline on datasets of varying complexity across systems of 19-50 qubits and identify trade-offs between fidelity, gate count, and circuit depth. Optimized brick wall circuits typically achieve the lowest depths, while the optimized staircase-like circuits minimize gate counts. Overall, our results provide principled and scalable protocols for preparing MPSs as quantum circuits, supporting utility-scale applications on near-term quantum devices.

Solving computationally demanding combinatorial optimization problems using conventional computing architectures is slow and energy intensive. Quantum computing could improve optimization efficiency but remains at an early stage. Probabilistic computing offers a practical near-term approach to faster optimization through stochastic techniques. Here, we experimentally demonstrate a scalable spin-transfer-torque-magnetic-tunnel-junction based probabilistic processor for efficiently solving all-to-all connected quadratic assignment problems. Our system integrates 144 compact spintronics tunable random number generators with a massively parallel architecture, achieving a high Monte-Carlo sampling throughput of 14.4 million flips per second. We co-design a parallel trial annealing scheme, and the integrated system achieves a 123× speedup with 98.3% energy savings over conventional Gibbs sampling, and a 3.2× speedup with 58.3% energy savings relative to the central processing unit implementation based on a compiled language. We further benchmark performance across graphics processing unit, and D-Wave quantum annealers, showing gains in solution quality, speed, and energy efficiency.

SFO-QAP: a novel hybrid sailfish optimizer with 2-opt for efficient quadratic assignment problem optimization

A swarm intelligence-based hybrid metaheuristic with tabu search for the quadratic assignment problem

This paper proposes a Simulated Annealing approach for solving equal-area facility layout problems. The technique employs a swap-based neighbourhood moves and Metropolis acceptance criteria to balance exploration and exploitation of the solution space. The algorithm was implemented in Python and evaluated on standard Quadratic Assignment Problem instances from QAPLIB, and its robustness was evaluated by solving the literature problem. Experimental results demonstrate that the proposed algorithm produced an optimal solution with minimal CPU time. Using a suitable Machine Learning concept, exploit the data of a large solution space and extract the minimum material handling cost.

Abstract The Quadratic Assignment Problem (QAP) is an NP-hard fundamental combinatorial optimization problem introduced by Koopmans and Beckmann in 1957. The problem is to assign n facilities to n different locations with the goal of minimizing the cost of the total distances between facilities weighted by the corresponding flows. We initiate the study of using Rydberg arrays to find optimal solutions to the QAP and provide a complementing circuit theory to facilitate an easy representation of other hard problems. We provide an algorithm for finding valid and optimal solutions to the QAP using Rydberg arrays.

In this paper, the generalized quadratic assignment problem (GQAP) is extended to consider multiple time periods and is called the dynamic GQAP (DGQAP). This problem considers assigning a set of facilities to a set of locations for multiple periods in the planning horizon such that the sum of the transportation, assignment, and reassignment costs is minimized. The facilities may have different space requirements (i.e., unequal areas), and the capacities of the locations may vary during a multi-period planning horizon. Also, multiple facilities may be assigned to each location during each period without violating the capacities of the locations. This research was motivated by the problem of assigning multiple facilities (e.g., equipment) to locations during outages at electric power plants. This paper presents mathematical models, construction algorithms, and two simulated annealing (SA) heuristics for solving the DGQAP problem. The first SA heuristic (SAI) is a direct adaptation of SA to the DGQAP, and the second SA heuristic (SAII) is the same as SAI with a look-ahead/look-back search strategy. In computational experiments, the proposed heuristics are first compared to an exact method on a generated data set of smaller instances (data set 1). Then the proposed heuristics are compared on a generated data set of larger instances (data set 2). For data set 1, the proposed heuristics outperformed a commercial solver (CPLEX) in terms of solution quality and computational time. SAI obtained the best solutions for all the instances, while SAII obtained the best solution for all but one instance. However, for data set 2, SAII obtained the best solution for nineteen of the twenty-four instances, while SAI obtained five of the best solutions. The results highlight the effectiveness and efficiency of the proposed heuristics, particularly SAII, for solving the DGQAP.

One of the classic combinatorial optimization problems belonging to the NP-hard class is the quadratic assignment problem. The interest in solving the problem lies in its high computational complexity, as well as its applications in: logistics, electronic circuits, gate assignment in airports, among others. In this work, the Greedy Random Adaptive Search Procedure metaheuristic was implemented to find its solutions. The main contribution of this work is the adaptation of a neighborhood structure contained in k-exchanges in the post-processing phase. The tests were performed for 29 large-scale instances whose dimensions range from 64 to 254 taken from the QAPLIB. The approximate solutions were found through a metaheuristic that bases its search on neighborhoods and local search algorithms. Java was the programming language used for the implementation of metaheuristics; its execution allowed balancing the parameters to obtain competitive results with respect to the values ​​known in the literature. The results reported and presented in this work achieved the proposed objectives.

This paper focuses on Multi-level Facility Location (MFL) problems, a critical component in supply chain design. Foundational tasks involve selecting plants, warehouses, distribution centres, and retail stores (markets) to maximise profits while considering related constraints. The problem has a single-assignment property. In that, a retail store is served via a single bundle of products transferred from a plant to a warehouse, then to a distribution centre, and finally to the store. Furthermore, each retail store has preferences for products from specific plants. Each selected facility incurs a one-time fixed cost, and there are upper bounds on the number of facilities of each type that can be selected. Transporting a single bundle of products from one upper-level facility to a lower-level facility incurs cost. Adapted from hub-location literature, the problem is formulated as a variation of the quadratic assignment problem. We propose two heuristics as solution approaches: (1) a multi-start greedy heuristic, (2) a multi-start tabu search. Extensive computational experiments with the heuristics are provided for randomly generated large-scale problems, and sensitivity analyses, supported by appropriate statistical methods, are used to validate the effectiveness of the heuristics’ results.

The construction site layout is widely acknowledged as a complex challenge in the field of construction management. However, it is crucial for nearly all construction projects, as the layout of site facilities can significantly impact on the project costs. This issue can be formulated as a Quadratic Assignment Problem, which is widely recognized to be NP-hard problem. While numerous methods have been proposed, most existing approaches either lack sufficient global search capabilities or struggle to escape local optima, leading to suboptimal solutions. To address these limitations, this study proposes a hybrid GA-Tabu algorithm that integrates the exploration strength of Genetic Algorithms with the exploitation efficiency of Tabu Search. This integration aims to achieve a better balance between global and local search processes, thereby enhancing the overall solution quality for the CSLP problem. The proposed algorithm has been subsequently tested with three case studies previously utilized in related research, effectively demonstrating its performance and applicability.

Construction site layout planning (CSLP) plays a pivotal role in determining the overall efficiency and cost-effectiveness of construction projects. Material handling operations, which constitute a significant portion of indirect project costs, heavily depend on the spatial arrangement of temporary facilities such as site offices, storage yards, and equipment zones. Poorly planned layouts can lead to excessive travel distances, increased material handling times, and operational delays, all of which contribute to inflated costs and reduced productivity. Therefore, optimizing the layout of construction sites to minimize transportation distances and enhance workflow is a critical task for project managers, contractors, and other stakeholders. The challenge in CSLP lies in the complexity of simultaneously satisfying multiple, often conflicting, requirements such as space constraints, safety regulations, and functional proximities. This complexity is compounded by the dynamic nature of construction activities and the presence of numerous facilities to be allocated within limited and irregularly shaped site boundaries. Mathematically, this problem can be formulated as a Quadratic Assignment Problem (QAP), a well-known NP-hard combinatorial optimization problem. The QAP seeks to assign a set of facilities to specific locations in a manner that minimizes the total cost, typically modeled as the sum of products of flows (e.g., material movement) and distances between assigned locations. However, due to the computational complexity of QAP, exact solutions become impractical for medium to large-scale site layouts. In recent years, metaheuristic algorithms have gained traction for effectively tackling such complex optimization problems. Among these, the Advanced Jaya Algorithm (A-JA), a recent population-based metaheuristic, stands out for its simplicity, parameter-free nature, and robust search capabilities. This study applies the A-JA to solve the CSLP modeled as a QAP, aiming to minimize the total weighted travel distance of material handling within the site. The algorithm’s performance is validated through two realistic case studies, showcasing its strong search capabilities and competitive results compared to traditional optimization methods. This promising approach offers a valuable decision-support tool for construction managers seeking to enhance site operational efficiency.

Generative diffusion models like Stable Diffusion are at the forefront of the thriving field of generative models today, celebrated for their robust training methodologies and high-quality photorealistic generation capabilities. These models excel in producing rich content, establishing them as essential tools in the industry. Building on this foundation, the field has seen the rise of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">personalized content synthesis</i> as a particularly exciting application. However, the large model sizes and iterative nature of inference make it difficult to deploy personalized diffusion models broadly on local devices with heterogeneous computational power. To address this, we propose a novel framework for efficient multi-user offloading of personalized diffusion models. This framework accommodates a variable number of users, each with different computational capabilities, and adapts to the fluctuating computational resources available on edge servers. To enhance computational efficiency and alleviate the storage burden on edge servers, we propose a tailored multi-user hybrid inference approach. This method splits the inference process for each user into two phases, with an optimizable split point. Initially, a cluster-wide model processes low-level semantic information for each user's prompt using batching techniques. Subsequently, users employ their personalized models to refine these details during the later phase of inference. Given the constraints on edge server computational resources and users' preferences for low latency and high accuracy, we model the joint optimization of each user's offloading request handling and split point as an extension of the Generalized Quadratic Assignment Problem (GQAP). Our objective is to maximize a comprehensive metric that balances both latency and accuracy across all users. To solve this NP-hard problem, we transform the GQAP into an adaptive decision sequence, model it as a Markov decision process, and develop a hybrid solution combining deep reinforcement learning with convex optimization techniques. Simulation results validate the effectiveness of our framework, demonstrating superior optimality and low complexity compared to traditional methods. All related code, datasets, and fine-tuned models are available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/wty2011jl/E-MOPDM</uri>.

We analyze three permutation-based combinatorial optimization problems in Fourier space, namely, the quadratic assignment problem, the linear ordering problem (LOP), and the symmetric and nonsymmetric traveling salesperson problem (STSP). In previous studies, one can find a number of theorems with necessary conditions that the Fourier coefficients of the aforementioned problems must satisfy. In this manuscript, we prove the sufficiency of these conditions, which implies that they constitute the exact characterization of the problems in Fourier space. In addition, the Fourier coefficients of the LOP and the symmetric and non-STSP are completely characterized by showing certain proportionality patterns that they must follow. Taking the characterization in Fourier space of the problems as a basis, we study classes of equivalent instances of the LOP and the symmetric and non-STSP, considering that two instances are equivalent if they have the same objective function. Furthermore, we give canonical representations for each problem in such a way that the input matrices have the minimum number of nonzero parameters.

In this study, we address the problem of assigning correlated Stock Keeping Units (SKUs) to storage locations under uncertain SKUs correlation conditions. The objective is to allocate SKUs within the forward picking area of a warehouse to minimize the total picking distance. To quantify the correlation between SKUs, we employ the joint distribution concept, enabling a more systematic representation of their correlations. The problem is formulated as a Quadratic Assignment Problem (QAP), which becomes computationally intractable at large scales due to its complexity. To mitigate this challenge, the QAP model is linearized, and a robust counterpart is developed to effectively handle uncertainty. The robust model was evaluated through various small-scale scenarios. While it yielded optimal results within an efficient time frame for small-scale problems, the solution time increased significantly as the problem size expanded.

Abstract We study the set of optimal solutions of the dual linear programming formulation of the linear assignment problem (LAP) to propose a method for computing a solution from the relative interior of this set. Assuming that an arbitrary dual-optimal solution and an optimal assignment are available (for which many efficient algorithms already exist), our method computes a relative-interior solution in linear time. Since the LAP occurs as a subproblem in the linear programming (LP) relaxation of the quadratic assignment problem (QAP), we employ our method as a new component in the family of dual-ascent algorithms that provide bounds on the optimal value of the QAP. To make our results applicable to the incomplete QAP, which is of interest in practical use-cases, we also provide a linear-time reduction from the incomplete LAP to the complete LAP along with a mapping that preserves optimality and membership in the relative interior. Our experiments on publicly available benchmarks indicate that our approach with relative-interior solution can frequently provide bounds near the optimum of the LP relaxation and its runtime is much lower when compared to a commercial LP solver.

The independent quadratic assignment problem: complexity and polynomially solvable special cases

Stochastic operators are the backbone of many optimization algorithms. Besides the existing theoretical analysis that studies the asymptotic runtime of such algorithms, characterizing their performance using fitness landscape analysis is far away. The fitness landscape approach proceeds by considering multiple characteristics to understand and explain an optimization algorithm's performance or the difficulty of an optimization problem, and hence would provide a richer explanation. This paper analyzes the fitness landscapes of stochastic operators by focusing on the number of local optima and their relation to the optimization performance. The search spaces of two combinatorial problems are studied: the NK-landscape and the Quadratic Assignment Problem, using binary string-based and permutation-based stochastic operators. The classical bit-flip search operator is considered for binary strings, and a generalization of the deterministic exchange operator for permutation representations is devised. We study small instances, ranging from randomly generated to real-like instances, and large instances from the NK-landscape. For large instances, we propose using an adaptive walk process to estimate the number of locally optimal solutions. Given that stochastic operators are usually used within population and single-solution-based evolutionary optimization algorithms, we contrast the performance of the (μ+λ)-EA, and an Iterated Local Search, versus the landscape properties of large size instances of the NK-landscapes. Our analysis shows that characterizing the fitness landscapes induced by stochastic search operators can effectively explain the optimization performances of the algorithms we considered.

Local search methods (LSM) are powerful metaheuristics known for efficiently exploring complex optimization landscapes. However, a key limitation is their “amnesiac” nature – they fail to utilize valuable data from past search iterations. This untapped data contains crucial information that can enhance the efficiency and outcomes of the search process. In this work, we propose a novel reinforcement learning (RL)-based move scoring mechanism to augment LSM, aiming to leverage historical data for improved adaptiveness and intelligence in the search process. We apply this hybrid method to the quadratic 3-dimensional assignment problem (Q3AP), a benchmark in combinatorial optimization that generalizes the quadratic assignment problem (QAP). Notably, few studies have focused on Q3AP, and to our knowledge, no prior work has integrated RL into its solution. Our experimental results demonstrate substantial improvements in both solution quality and execution time, particularly for larger and more complex problem instances. This performance gain is attributed to our unique iterative learning and scoring approach, which effectively guides the search process, balancing exploration and exploitation to escape local optima and find superior solutions.