We investigate a special case of the maximum quadratic assignment problem where one matrix is a product matrix and the other matrix is the distance matrix of a one-dimensional point set. We show that this special case, which we call the Wiener maximum quadratic assignment problem, is NP-hard in the ordinary sense and solvable in pseudo-polynomial time. Our approach also yields a polynomial time solution for the following problem from chemical graph theory: Find a tree that maximizes the Wiener index among all trees with a prescribed degree sequence. This settles an open problem from the literature.
This paper provides the first meaningful documentation and analysis of an established technique which aims to obtain an approximate solution to linear programming problems prior to applying the primal simplex method. The underlying algorithm is a penalty method with naive approximate minimization in each iteration. During initial iterations an approach similar to augmented Lagrangian is used. Later the technique corresponds closely to a classical quadratic penalty method. There is also a discussion of the extent to which it can be used to obtain fast approximate solutions of LP problems, in pa…
The quadratic assignment problem is a well-known optimization problem with numerous applications. A common strategy to solve it is to use one of its linearizations and then apply the toolbox of mixed integer linear programming methods. One measure of quality of a mixed integer formulation is the quality of its linear relaxation. In this paper, we compare two linearizations of the quadratic assignment problem and prove that the linear relaxation of the linearization of Adams and Johnson is contained in the linear relaxation of the linearization of Xia and Yuan. We furthermore develop a Branch a…
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, numeri…
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 e…
Relative-Interior Solution for the (Incomplete) Linear Assignment Problem with Applications to the Quadratic Assignment Problem
Faculty of Electrical Engineering, Czech Technical University in Prague（捷克技术大学布拉格分校电子工程系）；Computer Vision and Learning Lab, IWR, University of Heidelberg（海德堡大学计算机视觉与学习实验室）
The routing and wavelength assignment with protection is an important problem in telecommunications. Given an optical network and incoming connection requests, a commonly studied variant of the problem aims to grant maximum number of requests by assigning lightpaths at minimum network resource usage level, while ensuring the provided services remain functional in case of a single-link failure through dedicated path protection. We consider a practically relevant version where alternative lightpaths for requests are assumed to be given as a precomputed set, and show that it is NP-hard. We formul…
Variational quantum algorithms for permutation-based combinatorial problems: Optimal ansatz generation with applications to quadratic assignment problems and beyond
Institut Polytechnique de Paris（巴黎理工学院）；ENSTA（国立高等先进技术学院）；UMA（应用数学实验室）；Conservatoire National des Arts et Métiers（国立工艺博物馆）；CEDRIC（计算机科学与数字系统研究中心）；École Nationale Supérieure d’Informatique pour l’Industrie et l’Entreprise（工业与企业高级信息学院）
In this paper it is shown that the compact linearization approach, that has been previously proposed only for binary quadratic problems with assignment constraints, can be generalized to arbitrary linear equations with positive coefficients which considerably enlarges its applicability. We discuss special cases of prominent quadratic combinatorial optimization problems where the obtained compact linearization yields a continuous relaxation that is provably as least as strong as the one obtained with an ordinary linearization.
We study a general class of quadratic capacitated $p$-location problems facility location problems with single assignment where a non-separable, non-convex, quadratic term is introduced in the objective function to account for the interaction cost between facilities and customer assignments. This problem has many applications in the field of transportation and logistics where its most well-known special case is the single-allocation hub location problem and its many variants. The non-convex, binary quadratic program is linearized by applying a reformulation-linearization technique and the resu…
Hardware-Compatible Single-Shot Feasible-Space Heuristics for Solving the Quadratic Assignment Problem
QB Information Technologies (1QBit)（1QB信息技术公司）；Institute for Neuromorphic Compute Nodes (PGI-14), Peter Grünberg Institute, Forschungszentrum Jülich GmbH（于利希研究中心彼得·格林贝格研究所神经形态计算节点研究所）；RWTH Aachen University（亚琛工业大学）；HPE Labs, Hewlett Packard Enterprise（惠普企业实验室）；University of California, Santa Barbara（加州大学圣塔芭芭拉分校）
A quantum Karhunen-Loeve expansion and quadratic-exponential functionals for linear quantum stochastic systems
Research School of Electrical, Energy and Materials Engineering, College of Engineering and Computer Science, Australian National University（澳大利亚国立大学工程与计算机科学学院电气、能源与材料工程研究学院）
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 564…
Learning to Solve the Quadratic Assignment Problem with Warm-Started MCMC Finetuning
School of Mathematical Science, Peking University（北京大学数学科学学院）；State Key Laboratory of Mathematical Sciences, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学国家重点实验室）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）
The rail freight flow assignment (RFFA) is a key point of network design and operations management. Unlike a road network, the freight flow assignment pattern in a rail system has unique characteristics, and the most significant one is the tree-shaped path. More specifically, when two or more shipments arrive at a rail yard, those intended for the same destination will be treated as a single freight flow and be shipped through the same path during the remainder of the trip, regardless of their origins. This paper reveals the forming mechanism of the tree-shaped path from the perspective of rai…
In this paper, the compact linearization approach originally proposed for binary quadratic programs with assignment constraints is generalized to such programs with arbitrary linear equations and inequalities that have positive coefficients and right hand sides. Quadratic constraints may exist in addition, and the technique may as well be applied if these impose the only nonlinearities, i.e., the objective function is linear. We present special cases of linear constraints (along with prominent combinatorial optimization problems where these occur) such that the associated compact linearization…
Eigenvalue, Quadratic Programming, and Semidefinite Programming Bounds for a Cut Minimization Problem
考虑将图的节点集划分为给定大小的$k$集，从而得到使用（去除）第$k$集得到的\emph{minimize the cut}的问题.如果得到的割具有值$0$，那么我们得到了顶点分隔符。这个问题与图的划分问题密切相关。事实上，我们使用的模型与图划分问题的模型相同，除了不同的\emph{quadratic}目标函数。我们期待在已知的和新的界限，从各种放宽这个NP-困难的问题。这包括：标准特征值界、使用邻接矩阵和拉普拉斯算子的投影特征值界、基于最近成功的二次指派问题的二次规划（QP）界的QP界、以及半定规划界。我们包括大型和\emph{huge}的问题，说明在强度和时间方面的边界的效率的数值测试。
Recently, Sinkhorn's algorithm was applied for approximately solving linear programs emerging from optimal transport very efficiently. This was accomplished by formulating a regularized version of the linear program as Bregman projection problem onto the polytope of doubly-stochastic matrices, and then computing the projection using the efficient Sinkhorn algorithm, which is based on alternating closed-form Bregman projections on the larger polytopes of row-stochastic and column-stochastic matrices. In this paper we suggest a generalization of this algorithm for solving a well-known lifted lin…
