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**1 - 9**of**9**### Quadratic Assignment Problems typeset Exact Solution of Emerging Quadratic Assignment Problems

"... Abstract -We report on a growing class of assignment problems that are increasingly of interest and very challenging in terms of the difficulty they pose to attempts at exact solution. These problems address economic issues in the location and design of factories, hospitals, depots, transportation ..."

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Abstract -We report on a growing class of assignment problems that are increasingly of interest and very challenging in terms of the difficulty they pose to attempts at exact solution. These problems address economic issues in the location and design of factories, hospitals, depots, transportation hubs and military bases. Others involve improvements in communication network design. In this article we survey the latest and best methods available for solving exactly these difficult problems and suggest a taxonomy that provides a framework for combining existing solution methods and sets of computer tools that can be modified and extended to make inroads in solving this growing class of optimization problems.

### Symmetry in RLT cuts for the quadratic assignment and standard quadratic optimization problems

, 2012

"... The reformulation-linearization technique (RLT), introduced in [W.P. Adams, H.D. Sher-ali, A tight linearization and an algorithm for zero-one quadratic programming problems, Management Science, 32(10):1274{1290, 1986], provides a way to compute linear program-ming bounds on the optimal values of NP ..."

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The reformulation-linearization technique (RLT), introduced in [W.P. Adams, H.D. Sher-ali, A tight linearization and an algorithm for zero-one quadratic programming problems, Management Science, 32(10):1274{1290, 1986], provides a way to compute linear program-ming bounds on the optimal values of NP-hard combinatorial optimization problems. In this paper we show that, in the presence of suitable algebraic symmetry in the original problem data, it is sometimes possible to compute level two RLT bounds with additional linear matrix inequality constraints. As an illustration of our methodology, we compute the best-known bounds for certain graph partitioning problems on strongly regular graphs.

### Cutting Planes for RLT Relaxations of Mixed 0-1 Polynomial Programs

"... Abstract The Reformulation-Linearization Technique (RLT), due to Sherali and Adams, can be used to construct hierarchies of linear programming relaxations of mixed 0-1 polynomial programs. As one moves up the hierarchy, the relaxations grow stronger, but the number of variables increases exponentia ..."

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Abstract The Reformulation-Linearization Technique (RLT), due to Sherali and Adams, can be used to construct hierarchies of linear programming relaxations of mixed 0-1 polynomial programs. As one moves up the hierarchy, the relaxations grow stronger, but the number of variables increases exponentially. We present a procedure that generates cutting planes at any given level of the hierarchy, by optimally weakening linear inequalities that are valid at any given higher level. Computational experiments, conducted on instances of the quadratic knapsack problem, indicate that the cutting planes can close a significant proportion of the integrality gap.

### A hierarchical facility layout planning approach for large and complex hospitals

"... Abstract The transportation processes for patients, personnel, and material in large and complex maximum-care hospitals with many departments can consume significant resources and thus induce substantial logistics costs. These costs are largely determined by the allocation of the different departme ..."

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Abstract The transportation processes for patients, personnel, and material in large and complex maximum-care hospitals with many departments can consume significant resources and thus induce substantial logistics costs. These costs are largely determined by the allocation of the different departments and wards in possibly multiple connected hospital buildings. We develop a hierarchical layout planning approach based on an analysis of organizational and operational data from the Hannover Medical School, a large and complex university hospital in Hannover, Germany. The purpose of this approach is to propose locations for departments and wards for a given system of buildings such that the consumption of resources due to those transportation processes is minimized. We apply the approach to this real-world organizational and operational dataset as well as to a fictitious hospital building and analyze the algorithmic behavior and resulting layout.

### Using Symmetry to Optimize Over the Sherali-Adams Relaxation

, 2013

"... In this paper we examine the impact of using the Sherali-Adams procedure on highly symmetric integer programming problems. Linear relaxations of the extended formulations gen-) many variables for the level-d erated by Sherali-Adams can be very large, containing O ( ( n d closure. When large amounts ..."

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In this paper we examine the impact of using the Sherali-Adams procedure on highly symmetric integer programming problems. Linear relaxations of the extended formulations gen-) many variables for the level-d erated by Sherali-Adams can be very large, containing O ( ( n d closure. When large amounts of symmetry are present in the problem instance however, the symmetry can be used to generate a much smaller linear program that has an identical objective value. We demonstrate this by computing the bound associated with the level 1, 2, and 3 relaxations of several highly symmetric binary integer programming problems. We also present a class of constraints, called counting constraints, that further improves the bound, and in some cases provides a tight formulation. A major advantage of the Sherali-Adams formulation over the traditional formulation is that symmetry-breaking constraints can be more efficiently implemented. We show that using the Sherali-Adams formulation in conjunction with the symmetry breaking tool isomorphism pruning can lead to the pruning of more nodes early on in the branch-and-bound process.

### Improving Lower Bounds for the Quadratic Assignment Problem by applying a Distributed Dual Ascent Algorithm

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### Semi-definite Programming Relaxation of Quadratic Assignment Problems based on Nonredundant Matrix Splitting

, 2013

"... Quadratic Assignment Problems (QAPs) are known to be among the most challenging discrete optimization problems. Recently, a new class of semi-definite relaxation (SDR) models for QAPs based on matrix splitting has been proposed [25, 28]. In this paper, we consider the issue of how to choose an appro ..."

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Quadratic Assignment Problems (QAPs) are known to be among the most challenging discrete optimization problems. Recently, a new class of semi-definite relaxation (SDR) models for QAPs based on matrix splitting has been proposed [25, 28]. In this paper, we consider the issue of how to choose an appropriate matrix splitting scheme so that the result-ing relaxation model is easy to solve and able to provide a strong bound. For this, we first introduce a new notion of the so-called redundant and non-redundant matrix splitting and show that the relaxation based on a non-redundant matrix splitting can provide a stronger bound than a redundant one. Then we propose to follow the minimal trace principle to find a non-redundant matrix splitting via solving an auxiliary semi-definite programming problem (SDP). We show that applying the minimal trace principle directly leads to the so-called orthogonal matrix splitting introduced in [28]. To find other non-redundant matrix splitting schemes whose resulting relaxation models are relatively easy to solve, we elaborate on two splitting schemes based on the so-called one-matrix and the sum-matrix. We analyze the solutions from the auxiliary problems for these two cases and characterize when they can pro-

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"... Amino acid residues within EHEC O157:H7 Tir involved in phosphorylation, α-actinin 1 ..."

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Amino acid residues within EHEC O157:H7 Tir involved in phosphorylation, α-actinin 1

### A Data-guided Lexisearch Algorithm for the Quadratic Assignment Problem

, 2014

"... This paper considers the well-known Quadratic Assignment Problem (QAP) for the study. It is NP-hard combinatorial optimizations that can be defined as follows. There is n facilities and n locations. A distance is specified for each pair of locations, and a flow is specified for each pair of facilit ..."

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This paper considers the well-known Quadratic Assignment Problem (QAP) for the study. It is NP-hard combinatorial optimizations that can be defined as follows. There is n facilities and n locations. A distance is specified for each pair of locations, and a flow is specified for each pair of facilities. The objective of problem is to allocate all facilities to different locations such that the sum of the flows multiplied by the corresponding distances is minimized. We develop a data-guided lexisearch algorithm based on an existing reformulation to find exact solution to the problem. For this we first modify alphabet table according to the number of zeros in the rows of the surplus matrix, thus, renaming rows (facilities), and then we apply lexisearch algorithm. It is shown that before applying lexisearch algorithm, this minor preprocessing of the data improves computational time significantly. Finally, we present a comparative study between data-guided lexisearch algorithm and two existing algorithms on some QAPLIB instances of various sizes. The computational study shows the effectiveness of our proposed data-guided lexisearch algorithm.