Peter M. Hahn

A survey for the quadratic assignment problem

The quadratic assignment problem (QAP), one of the most difficult problems in the NP-hard class, models many real-life problems in several areas such as facilities location, parallel and distributed computing, and combinatorial data analysis. Combinatorial optimization problems, such as the traveling salesman problem, maximal clique and graph partitioning can be formulated as a QAP. In this paper, we present some of the most important QAP formulations and classify them according to their mathematical sources. We also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms. We then give a detailed discussion of the progress made in both exact and heuristic solution methods, including those formulated according to metaheuristic strategies. Finally, we analyze the contributions brought about by the study of different approaches.

Theoretical Diversity Improvement in Multiple Frequency Shift Keying

Multiple frequency shift keying (MFSK) is a modulation suitable for transmitting digital data under fading conditions. A quantitative analysis of MFSK-with-diversity is presented. The MFSK signals on the several diversity channels are presumed to be perturbed independently by Rayleigh fading and additive white Gaussian noise. Also, it is assumed that fading is slow and that envelope, cross-correlation (matched filter) detection is used. The diversity combining method is chosen so that the receiver performs a likelihood-ratio test in deciding which one of K frequencies was transmitted. This optimum comibining method is to square and add the detected outputs of corresponding filters from each diversity channel. Theoretical error probability as a function of signal-energy-per-bit received is derived, and curves are plotted for two-, four-, and eight-frequency MFSK-with-diversity. Bandwidth requirements, as a function of type and order of diversity, are determined. Eight-frequency MFSK with triple diversity has a 21.8-db advantage over simple FSK for transmitting 6-bit characters with a 0.001 error probability.

Recent Advances for the Quadratic Assignment Problem with Special Emphasis on Instances that are Difficult for Meta-Heuristic Methods

This paper reports heuristic and exact solution advances for the Quadratic Assignment Problem (QAP).QAPinstances most often discussed in the literature are relatively well solved by heuristic approaches. Indeed, solutions at a fraction of one percent from the best known solution values are rapidly found by most heuristic methods. Exact methods are not able to prove optimality for these instances as soon as the problem size approaches 30 to 40. This article presents new QAP instances that are ill conditioned for many metaheuristic-based methods. However, these new instances are shown to be solved relatively well by some exact methods, since problem instances up to a size of 75 have been exactly solved.

A branch-and-bound algorithm for the quadratic assignment problem based on the Hungarian method

This paper presents a new branch-and-bound algorithm for solving the quadratic assignment problem (QAP). The algorithm is based on a dual procedure (DP) similar to the Hungarian method for solving the linear assignment problem. Our DP solves the QAP in certain cases, i.e., for some small problems (N< 7) and for numerous larger problems (7≤ N ≤16) that arise as sub-problems of a larger QAP such as the Nugent 20. The DP, however, does not guarantee a solution. It is used in our algorithm to calculate lower bounds on solutions to the QAP. As a result of a number of recently developed improvements, the DP produces lower bounds that are as tight as any which might be useful in a branch-and-bound algorithm. These are produced relatively cheaply, especially on larger problems. Experimental results show that the computational complexity of our algorithm is lower than known methods, and that its actual runtime is significantly shorter than the best known algorithms for QAPLIB test instances of size 16 through 22. Our method has the potential for being improved and therefore can be expected to aid in solving even larger problems.

A level-2 reformulation-linearization technique bound for the quadratic assignment problem

Abstract This paper studies polyhedral methods for the quadratic assignment problem. Bounds on the objective value are obtained using mixed 0–1 linear representations that result from a reformulation–linearization technique (rlt). The rlt provides different “levels” of representations that give increasing strength. Prior studies have shown that even the weakest level-1 form yields very tight bounds, which in turn lead to improved solution methodologies. This paper focuses on implementing level-2. We compare level-2 with level-1 and other bounding mechanisms, in terms of both overall strength and ease of computation. In so doing, we extend earlier work on level-1 by implementing a Lagrangian relaxation that exploits block-diagonal structure present in the constraints. The bounds are embedded within an enumerative algorithm to devise an exact solution strategy. Our computer results are notable, exhibiting a dramatic reduction in nodes examined in the enumerative phase, and allowing for the exact solution of large instances.

A hospital facility layout problem finally solved

This paper presents a history of a difficult facility layout problem that falls into the category of the Koopmans–Beckmann variant of the quadratic assignment problem (QAP), wherein 30 facilities are to be assigned to 30 locations. The problem arose in 1972 as part of the design of a German University Hospital, Klinikum Regensburg. This problem, known as the Krarup 30a upon its inclusion in the QAP library of QAP instances, has remained an important example of one of the most difficult to solve. In 1999, two approaches provided multiple optimum solutions. The first was Thomas Stützles analysis of fitness–distance correlation that resulted in the discovery of 256 global optima. The second was a new branch-and-bound enumeration that confirmed 133 of the 256 global optima found and proved that Stützles 256 solutions were indeed optimum solutions. We report here on the steps taken to provide in-time heuristic solutions and the methods used to finally prove the optimum.

Symbol mapping diversity design for multiple packet transmissions

In this paper, we present a simple, but effective method of enhancing and exploiting diversity from multiple packet transmissions in systems that employ nonbinary linear modulations such as phase-shift keying (PSK) and quadrature amplitude modulation (QAM). This diversity improvement results from redesigning the symbol mapping for each packet transmission. By developing a general framework for evaluating the upper bound of the bit error rate (BER) with multiple transmissions, a criterion to obtain optimal symbol mappings is attained. The optimal adaptation scheme reduces to solutions of the well known quadratic assignment problem (QAP). Symbol mapping adaptation only requires a small increase in receiver complexity but provides very substantial BER gains when applied to additive white Gaussian noise (AWGN) and flat-fading channels.

An algorithm for the generalized quadratic assignment problem

Abstract This paper reports on a new algorithm for the Generalized Quadratic Assignment problem (GQAP). The GQAP describes a broad class of quadratic integer programming problems, wherein M pair-wise related entities are assigned to N destinations constrained by the destinations’ ability to accommodate them. This new algorithm is based on a Reformulation Linearization Technique (RLT) dual ascent procedure. Experimental results show that the runtime of this algorithm is as good or better than other known exact solution methods for problems as large as M=20 and N=15.

Optimal symbol mapping diversity for multiple packet transmissions

We present a simple, but effective, method of creating and exploiting diversity from packet retransmissions in systems that employ nonbinary modulations such as PSK and QAM. This diversity results from differing the symbol mapping for each packet retransmission. By developing a general framework for evaluating the bit error rate (BER) upper bound with multiple transmissions, a criterion to obtain optimal symbol (re)mappings is attained for memoryless AWGN channels. The optimal adaptation scheme reduces to solutions of the quadratic assignment problem (QAP). Symbol mapping adaptation only requires a small increase in receiver complexity but provides very substantial BER gains.

The quadratic three-dimensional assignment problem: Exact and approximate solution methods

Abstract This paper reports on algorithm development for solving the quadratic three-dimensional assignment problem (Q3AP). The Q3AP arises, for example, in the implementation of a hybrid ARQ (automatic repeat request) scheme for enriching diversity among multiple packet re-transmissions, by optimizing the mapping of data bits to modulation symbols. Typical practical problem sizes would be 8, 16, 32 and 64. We present an exact solution method based upon a reformulation linearization technique that is one of the best available for solving the quadratic assignment problem (QAP). Our current exact algorithm is useful for Q3AP instances of size 13 or smaller. We also investigate four stochastic local search algorithms that provide optimum or near optimum solutions for large and difficult QAP instances and adapt them for solving the Q3AP. The results of our experiments make it possible to get good solutions to signal mapping problems of size 8 and 16.