Search

We demonstrate that the search space of the quadratic assignment problem (QAP), known as an NP-hard combinatorial optimization problem, can be reduced using Grover adaptive search (GAS) with Dicke state operators. To that end, we first revise the traditional quadratic formulation of the QAP into a higher-order formulation, introducing a binary encoding method ordered by a descending Hamming weight, such that the number of terms in the objective function is reduced. We also show that the phase gate in the GAS can be replaced by a rotation gate about the Z axis, simplifying the quantum circuit without any penalty. Algebraic analyses in terms of the number of qubits, quantum gates, and query complexity are performed, which indicate that our proposed approach significantly reduces the search space size, improving convergence performance to the optimal solution compared to the conventional one. Interestingly, it is suggested that the higher-order formulation is effective for problems whose size are powers of two, while the quadratic formulation is more effective for other sizes, indicating that switching between the two formulations can enhance the feasibility of the GAS-solved QAP., Comment: 12 pages, 6 figures

The Quadratic Assignment Problem (QAP) is an important discrete optimization instance that encompasses many well-known combinatorial optimization problems, and has applications in a wide range of areas such as logistics and computer vision. The QAP, unfortunately, is NP-hard to solve. To address this difficulty, a number of semidefinite relaxations of the QAP have been developed. These techniques are known to be powerful in that they compute globally optimal solutions in many instances, and are often deployed as sub-routines within enumerative procedures for solving QAPs. In this paper, we investigate the strength of these semidefinite relaxations. Our main result is a deterministic set of conditions on the input matrices -- specified via linear inequalities -- under which these semidefinite relaxations are exact. Our result is simple to state, in the hope that it serves as a foundation for subsequent studies on semidefinite relaxations of the QAP as well as other related combinatorial optimization problems. As a corollary, we show that the semidefinite relaxation is exact under a perturbation model whereby the input matrices differ by a suitably small perturbation term. One technical difficulty we encounter is that the set of dual feasible solutions is not closed. To circumvent these difficulties, our analysis constructs a sequence of dual feasible solutions whose objective value approaches the primal objective, but never attains it.

Quadratic assignment problem (QAP) is a fundamental problem in combinatorial optimization and finds numerous applications in operation research, computer vision, and pattern recognition. However, it is a very well-known NP-hard problem to find the global minimizer to the QAP. In this work, we study the semidefinite relaxation (SDR) of the QAP and investigate when the SDR recovers the global minimizer. In particular, we consider the two input matrices satisfy a simple signal-plus-noise model, and show that when the noise is sufficiently smaller than the signal, then the SDR is exact, i.e., it recovers the global minimizer to the QAP. It is worth noting that this sufficient condition is purely algebraic and does not depend on any statistical assumption of the input data. We apply our bound to several statistical models such as correlated Gaussian Wigner model. Despite the sub-optimality in theory under those models, empirical studies show the remarkable performance of the SDR. Our work could be the first step towards a deeper understanding of the SDR exactness for the QAP.

Recently various optimization problems, such as Mixed Integer Linear Programming Problems (MILPs), have undergone comprehensive investigation, leveraging the capabilities of machine learning. This work focuses on learning-based solutions for efficiently solving the Quadratic Assignment Problem (QAPs), which stands as a formidable challenge in combinatorial optimization. While many instances of simpler problems admit fully polynomial-time approximate solution (FPTAS), QAP is shown to be strongly NP-hard. Even finding a FPTAS for QAP is difficult, in the sense that the existence of a FPTAS implies $P = NP$. Current research on QAPs suffer from limited scale and computational inefficiency. To attack the aforementioned issues, we here propose the first solution of its kind for QAP in the learn-to-improve category. This work encodes facility and location nodes separately, instead of forming computationally intensive association graphs prevalent in current approaches. This design choice enables scalability to larger problem sizes. Furthermore, a \textbf{S}olution \textbf{AW}are \textbf{T}ransformer (SAWT) architecture integrates the incumbent solution matrix with the attention score to effectively capture higher-order information of the QAPs. Our model's effectiveness is validated through extensive experiments on self-generated QAP instances of varying sizes and the QAPLIB benchmark., Comment: Accepted by ICML 2024

The Quadratic Assignment Problem (QAP) is an NP-hard problem which has proven particularly challenging to solve: unlike other combinatorial problems like the traveling salesman problem (TSP), which can be solved to optimality for instances with hundreds or even thousands of locations using advanced integer programming techniques, no methods are known to exactly solve QAP instances of size greater than 30. Solving the QAP is nevertheless important because of its many critical applications, such as electronic wiring design and facility layout selection. We propose a method to solve the original Koopmans-Beckman formulation of the QAP using deep reinforcement learning. Our approach relies on a novel double pointer network, which alternates between selecting a location in which to place the next facility and a facility to place in the previous location. We train our model using A2C on a large dataset of synthetic instances, producing solutions with no instance-specific retraining necessary. Out of sample, our solutions are on average within 7.5% of a high-quality local search baseline, and even outperform it on 1.2% of instances.

The generalized quadratic assignment problem (GQAP) is one of the hardest problems to solve in the operations research area. The GQAP addressed in this work is defined as the task of minimizing the assignment and transportation costs of assigning a set of facilities to a set of locations. The facilities have different space requirements, and the locations have different space capacities. Multiple facilities can be assigned to each location if the space capacity is not violated. In this work, three instances of GQAP in different situations are presented. Then, a genetic algorithm is developed to solve the GQAP instances. Finally, the local neighborhood search with the steepest descend strategy is constructed and applied to the final solution obtained by the GA, and the final solution is compared with the best solution found by MPL/CPLEX software and reference papers. The results show that the developed GA heuristic is effective for solving the GQAP.

The Quadratic Assignment Problem (QAP) is an important combinatorial optimization problem with applications in many areas including logistics and manufacturing. QAP is known to be NP-hard, a computationally challenging problem, which requires the use of sophisticated heuristics in finding acceptable solutions for most real-world data sets. In this paper, we present GPU-accelerated implementations of a 2opt and a tabu search algorithm for solving the QAP. For both algorithms, we extract parallelism at multiple levels and implement novel code optimization techniques that fully utilize the GPU hardware. On a series of experiments on the well-known QAPLIB data sets, our solutions, on average run an order-of-magnitude faster than previous implementations and deliver up to a factor of 63 speedup on specific instances. The quality of the solutions produced by our implementations of 2opt and tabu is within 1.03% and 0.15% of the best known values. The experimental results also provide key insight into the performance characteristics of accelerated QAP solvers. In particular, the results reveal that both algorithmic choice and the shape of the input data sets are key factors in finding efficient implementations., Comment: 25 pages, 9 figures; parts of this work appeared as short papers in XSEDE14 and XSEDE15 conferences. This version of the paper is a substantial extension of previous work with optimizations for newer GPU platforms and extended experimental results

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 quantum guesswork quantifies the minimum number of queries needed to guess the state of a quantum ensemble if one is allowed to query only one state at a time. Previous approaches to the computation of the guesswork were based on standard semi-definite programming techniques and therefore lead to approximated results. In contrast, we show that computing the quantum guesswork of qubit ensembles with uniform probability distribution corresponds to solving a quadratic assignment problem and we provide an algorithm that, upon the input of any qubit ensemble over a discrete ring, after finitely many steps outputs the exact closed-form expression of its guesswork. While in general the complexity of our guesswork-computing algorithm is factorial in the number of states, our main result consists of showing a more-than-quadratic speedup for symmetric ensembles, a scenario corresponding to the three-dimensional analog of the maximization version of the turbine-balancing problem. To find such symmetries, we provide an algorithm that, upon the input of any point set over a discrete ring, after finitely many steps outputs its exact symmetries. The complexity of our symmetries-finding algorithm is polynomial in the number of points. As examples, we compute the guesswork of regular and quasi-regular sets of qubit states., Comment: 12 pages, 3 tables, published version

Quantum annealing and D-Wave quantum annealer attracted considerable attention for their ability to solve combinatorial optimization problems. In order to solve other type of optimization problems, it is necessary to apply certain kinds of mathematical transformations. However, linear constraints reduce the size of problems that can be represented in quantum annealers, owing to the sparseness of connections between qubits. For example, the quadratic assignment problem (QAP) with linear equality constraints can be solved only up to size 12 in the quantum annealer D-Wave Advantage, which has 5640 qubits. To overcome this obstacle, Ohzeki developed a method for relaxing the linear equality constraints and numerically verified the effectiveness of this method with some target problems, but others remain unsolvable. In particular, it is difficult to obtain feasible solutions to problems with hard constraints, such as the QAP. We therefore propose a method for solving the QAP with quantum annealing by applying a post-processing bit-flip heuristic algorithm to solutions obtained by the Ohzeki method. In a numerical experiment, we solved a sparse QAP by the proposed method. This sparse QAP has been used in areas such as item listing on an E-commerce website. We successfully solved a QAP of size 19 with high accuracy for the first time using D-Wave Advantage. We also confirmed that the bit-flip heuristic algorithm moves infeasible solutions to nearby feasible solutions in terms of Hamming distance with good computational efficiency., Comment: 11 pages. The first page includes the initial screen displaying IEEE-copyrighted material

A quadratic assignment problem (QAP) is a combinatorial optimization problem that belongs to the class of NP-hard ones. So, it is difficult to solve in the polynomial time even for small instances. Research on the QAP has thus focused on obtaining a method to overcome this problem. Heuristics and meta-heuristics algorithm are prevalent solution methods for this problem. This paper is one of comparative studies to apply different metaheuristic algorithms for solving the QAP. One of the most popular approaches for categorizing meta-heuristic algorithms is based on a search strategy, including (1) local search improvement meta-heuristics and (2) global search-based meta-heuristics. The matter that distinguishes this paper from the other is the comparative performance of local and global search (both EA and SI), in which meta-heuristics that consist of genetic algorithm (GA), particle swarm optimization (PSO), hybrid GA-PSO, grey wolf optimization (GWO), harmony search algorithm (HAS) and simulated annealing (SA). Also, one improvement heuristic algorithm (ie, 2-Opt) is used to compare with others. The PSO, GWO and 2-Opt algorithms are improved to achieve the better comparison toward the other algorithms for evaluation. In order to analysis the comparative advantage of these algorithms, eight different factors are presented. By taking into account all these factors, the test is implemented in six test problems of the QAP Library (QAPLIB) from different sizes. Another contribution of this paper is to measure a strong convergence condition for each algorithm in a new way.

Graph matching involves combinatorial optimization based on edge-to-edge affinity matrix, which can be generally formulated as Lawler's Quadratic Assignment Problem (QAP). This paper presents a QAP network directly learning with the affinity matrix (equivalently the association graph) whereby the matching problem is translated into a constrained vertex classification task. The association graph is learned by an embedding network for vertex classification, followed by Sinkhorn normalization and a cross-entropy loss for end-to-end learning. We further improve the embedding model on association graph by introducing Sinkhorn based matching-aware constraint, as well as dummy nodes to deal with unequal sizes of graphs. To our best knowledge, this is one of the first network to directly learn with the general Lawler's QAP. In contrast, recent deep matching methods focus on the learning of node/edge features in two graphs respectively. We also show how to extend our network to hypergraph matching, and matching of multiple graphs. Experimental results on both synthetic graphs and real-world images show its effectiveness. For pure QAP tasks on synthetic data and QAPLIB benchmark, our method can perform competitively and even surpass state-of-the-art graph matching and QAP solvers with notable less time cost. We provide a project homepage at http://thinklab.sjtu.edu.cn/project/NGM/index.html., Comment: Accepted by TPAMI

The Quadratic Assignment Problem (QAP) is a well-known permutation-based combinatorial optimization problem with real applications in industrial and logistics environments. Motivated by the challenge that this NP-hard problem represents, it has captured the attention of the optimization community for decades. As a result, a large number of algorithms have been proposed to tackle this problem. Among these, exact methods are only able to solve instances of size $n<40$. To overcome this limitation, many metaheuristic methods have been applied to the QAP. In this work, we follow this direction by approaching the QAP through Estimation of Distribution Algorithms (EDAs). Particularly, a non-parametric distance-based exponential probabilistic model is used. Based on the analysis of the characteristics of the QAP, and previous work in the area, we introduce Kernels of Mallows Model under the Hamming distance to the context of EDAs. Conducted experiments point out that the performance of the proposed algorithm in the QAP is superior to (i) the classical EDAs adapted to deal with the QAP, and also (ii) to the specific EDAs proposed in the literature to deal with permutation problems., Comment: 23 pages

In this paper, we show that the quadratic assignment problem (QAP) can be reformulated to an equivalent rank constrained doubly nonnegative (DNN) problem. Under the framework of the difference of convex functions (DC) approach, a semi-proximal DC algorithm (DCA) is proposed for solving the relaxation of the rank constrained DNN problem whose subproblems can be solved by the semi-proximal augmented Lagrangian method (sPALM). We show that the generated sequence converges to a stationary point of the corresponding DC problem, which is feasible to the rank constrained DNN problem. Moreover, numerical experiments demonstrate that for most QAP instances, the proposed approach can find the global optimal solutions efficiently, and for others, the proposed algorithm is able to provide good feasible solutions in a reasonable time., Comment: 24 pages, 1 figure

We consider three known bounds for the quadratic assignment problem (QAP): an eigenvalue, a convex quadratic programming (CQP), and a semidefinite programming (SDP) bound. Since the last two bounds were not compared directly before, we prove that the SDP bound is stronger than the CQP bound. We then apply these to improve known bounds on a discrete energy minimization problem, reformulated as a QAP, which aims to minimize the potential energy between repulsive particles on a toric grid. Thus we are able to prove optimality for several configurations of particles and grid sizes, complementing earlier results by Bouman, Draisma and Van Leeuwaarden [ SIAM Journal on Discrete Mathematics, 27(3):1295--1312, 2013]. The semidefinite programs in question are too large to solve without pre-processing, and we use a symmetry reduction method by Parrilo and Permenter [Mathematical Programming, 181:51--84, 2020] to make computation of the SDP bounds possible.