A Data-guided Lexisearch Algorithm for the Quadratic Assignment Problem

In this paper, we consider the quadratic assignment problem (QAP), one of the hardest NP-hard combinatorial optimization problems. We first present a new reformulation of the problem. Then a Lexisearch Algorithm (LSA) is developed for obtaining exact optimal solution to the problem. Finally, a comparative study has been carried out to show the efficiency of the algorithm against an existing algorithm for some medium sized instances from the quadratic assignment problem library, QAPLIB. The comparative study shows the efficiency of the proposed LSA based on the reformulation.

Currently several thousands of objects are being tracked in the MEO and GEO regions through optical means. The problem faced in this framework is that of Multiple Target Tracking (MTT). In this context both, the correct associations among the observations and the orbits of the objects have to be determined. The complexity of the MTT problem is defined by its dimension S. The number S corresponds to the number of fences involved in the problem. Each fence consists of a set of observations where each observation belongs to a different object. The S ≥ 3 MTT problem is an NP-hard combinatorial optimization problem. There are two general ways to solve this. One way is to seek the optimum solution, this can be achieved by applying a branch-and- bound algorithm. When using these algorithms the problem has to be greatly simplified to keep the computational cost at a reasonable level. Another option is to approximate the solution by using meta-heuristic methods. These methods aim to efficiently explore the different possible combinations so that a reasonable result can be obtained with a reasonable computational effort. To this end several population-based meta-heuristic methods are implemented and tested on simulated optical measurements. With the advent of improved sensors and a heightened interest in the problem of space debris, it is expected that the number of tracked objects will grow by an order of magnitude in the near future. This research aims to provide a method that can treat the correlation and orbit determination problems simultaneously, and is able to efficiently process large data sets with minimal manual intervention.

-Quadratic assignment problems (QAPs) are NP-hard combinatorial optimization problems often used to compare the performance of meta-heuristics. In this paper, we propose a QAP problem instance generator whose instances can be used as benchmark for meta-heuristics. Our generator aggregates various QAP instances into a larger size QAP instance in order to obtain an large size, additively decomposable QAP. We assume that a QAP instance that is difficult for local search has many local optima from which local search needs to escape from. We call the resulting QAPs the composite QAP instances (cQAPs). We use numerical and empirical techniques for the landscape analysis of generated composite QAPs. The comparison between our QAP instances with the other QAPs from the literature classify cQAPs as difficult. We show that heuristic algorithms that exploit the particularities of the cQAP search space, like iterated local search, can outperform other heuristics that do not consider the structure of this search space, like the multi-restart local search.

Introduction. The quadratic assignment problem is a well-established NP-hard problem in combinatorial optimization with applications in diverse fields like economics, archaeology, and chemistry. Due to its complexity, research on efficient solution methods remains active, including efforts for parallelization on multiprocessor computing systems. However, effective parallel algorithms are crucial to fully leverage these computational resources. In this context, algorithm unions (portfolios and teams) play a significant role in achieving parallel execution for solving such problems. Research objectives. This work investigates the application of portfolios constructed from modifications of the repeated iterated tabu search algorithm to the quadratic assignment problem. The effectiveness of these portfolios was evaluated through experimental computations. Results. The portfolios, derived from modifications of the repeated iterated tabu search algorithm, were applied to the quadratic assignment problem. For the most demanding test instances, the proposed algorithms were evaluated on the SCIT-4 supercomputer, alongside previously published results from the authors, confirming their competitive performance. Additionally, we assessed the parallel efficiency of these portfolios in solving instances of the quadratic assignment problem. The results demonstrate their ability to accelerate the optimization process (with speedup dependent on problem size and utilized processors), enabling the solution of large-scale problems. Conclusions. The conducted studies demonstrate that employing algorithm portfolios significantly accelerates the solution process for the quadratic assignment problem. Analysis of the results reveals a near-linear speedup factor achieved by the portfolio. For the challenging test instance tai100a, a new best solution value of 21040996 was obtained using a portfolio of 16 algorithms. Keywords: quadratic assignment problem, algorithm portfolios, experimental research, algorithm portfolios efficiency.

Dear SCPE Reader, We present a selection of papers which are extensions of papers presented at the 7-th International Symposium on Parallel and Distributed Computing, 1–5 July 2008, in Krakow, Poland. The motivation for publishing the selection in the SCPE Journal was, on the one hand, to present the flavour of the research reported at the conference and on the other hand to present some of the most relevant topics currently focused on the research on parallel and distributed computing in general. The selection contains only 6 papers out of about 60 presented at the conference, and thus, is far from covering all relevant topics represented at the ISPDC 2008. This is because not all of the invited authors were patient enough to accept a fairly long paper publishing process. Nevertheless, we hope that the presented papers will bring you closer to the research covered by the ISPDC conferences and will encourage you to participate in future ISPDC editions. The first paper ``The Impact of Workload Variability on Load Balancing Algorithms'' is by Marta Beltran and Antonio Guzman from King Juan Carlos University in Spain. It concerns an important topic of load balancing in cluster systems, namely adaptativity of the load balancing algorithms to changes of the workload in the system. Adequate accounting for additional load in the hosting system is of great relevance for correct optimization effects. The paper presents a thorough formal analysis of the workload variability metrics and their influence on the quality of load balancing algorithms. Four basic activities appearing in load balancing algorithms are identified, and based on them some algorithmic solutions are proposed to correctly deal with workload variability in system load balancing. The problem of dynamic load balancing algorithms robustness has been discussed. Two different robustness metrics sensitive to the applied type of opimization: local task-oriented or a global one enable selecting task remote execution or migration as load balancing operations. The proposed approach is illustrated with experiments. The second paper ``Model-Driven Engineering and Formal Validation of High-Performance Embedded Systems'' is by Abdoulaye Gamatie, Eric Rutten, Huafeng Yu, Pierre Boulet, Jean-Luc Dekeyser, from University of Lille and INRIA in France. The paper is concerned with a very advanced methodology of designing correct parallel embedded systems for intensive data-parallel computing. In their previous research, the authors of the paper designed the GASPARD embedded system design framework. It is based on the hardware/software co-design approach through model-driven engineering. The framework is based on an UML-like model specification language in which hardware and software elements are modelled using a component approach with special mechanisms for repetitive structures. This paper tries to combine the modelling framework of GASPARD with the mechanisms of synchronous languages to achieve design verifiability provided for such languages. The paper shows how GASPARD models can be translated into synchronous models based on data flow equations in order to formally check their correctness. The proposed approach is illustrated with an example of a video processing system. The third paper ``Relations Between Several Parallel Computational Models'' is by Stephan Bruda and Yuanqiao Zhang from Bishop’s University in Canada. The paper is concerned with theoretical aspects of shared memory systems described by the parallel random access machine PRAM model and aims in studying performance properties of different types of PRAM systems. The attention is focussed on analysing the computational power of two more sophisticated PRAM models (Combining CRCW and Broadcast Selective Reduction), which include data reduction in case of concurrent writes. The paper shows that these two models have equivalent computational power, which is a new result comparing the existing literature. The performance of both models applied to reconfigurable multiple bus machines was studied as a possible architectural solution for current VLSI processor implementations. It was shown that in such systems under reasonable assumptions concurrent-write does not enhance performance comparing the exclusive-write model. Another result important for the VLSI technology is that the Combining CRCW PRAM model (in which data of concurrent writes are arthmetically or logically combined before write) and the exclusive-write on directed reconfigurable busses perform in equivalent way under strong real-time requirements. The fourth paper ``Experiences with Mesh-Like Computations Using Prediction Binary Trees'' is by Gennaro Cordasco, Biagio Cosenza, Rosario de Chiara, Ugo Erra and Vittorio Scarano from the University ``degli Studi'' of Salerno and the University ``degli Studi della Biasilicata'' of Potenza in Italy. The paper concerns optimization methods for mesh-like computations in clusters of processors. The computations are perfomed assuming a phase-like program execution control using a tiling approach which reduces inter-processor communication. A temporal coherence is also assumed, which means that task sizes provide similar execution times in consecutive phases. Temporary coherent computations are structured in a Prediction Binary Tree, in which leaves represent computing tiles to be mapped to processors. A phase-by-phase semi-static load balancing is introduced to the scheduling algorithm. The scheduling algorithm is equipped with a predictor, which estimates the computation time of next phase tiles based on previous execution times and modifies the tiles to achieve balanced execution in phases. For this, two heuristics are used to leverage on data locality in processors. The proposed approach is illustrated by the example of interactive rendering with Parallel Ray Tracing algorithm. The fifth paper ``The Influence of the IBM pSeries Servers Virtualization Mechanism on Dynamic Resource Allocation in AIX 5L'' is by Maciej Mlynski from ASpartner Limited in Poland. The paper concerns a very up-to-date problem of system virtualization and presents the results of research carried on IBM pSeries servers. IBM is strongly developing the virtualization technique especially on IBM pSeries servers enabling an improved and flexible sharing of system resources between applications. The paper investigates novel facilities for dynamic resource management such as micro-partitioning and partition load manager. They enable dynamic creation of workload logical partitions of system resources and their dynamic mangement. It includes run-time resource re-alocation between logical partitions including setting of sharing specifications as well as run-time adding/removing/setting parameters of resources in the system. It remains an open question how to properly tune parameters of the operating system using the provided virtualization facilities to obtain the best efficiency for a given application program. The paper presents the results of experiments which study the effects of tuning the disk subsystem parameters under the IBM AIX 5L operating system with the use of the provided virtualization facilities on the resulting application execution performance. The results show that even small deterioration in the resource pool status requires an immediate adaptation of the operating system parameters to maintain the required performance. The sixth paper ``HeteroPBLAS: A Set of Parallel Basic Linear Algebra Subprograms Optimized for Heterogeneous Computational Clusters'' is by Ravi Reddy, Alexey Lastovetsky and Pedro Alonso from University College Dublin in Ireland and Polytechnic University of Valencia in Spain. The paper concerns the methodology for parallelization of linear algebra computations for execution in heterogeneous cluster environments. The design of the HeteroPBLAS library (Parallel Basic Linear Algebra Subprograms) for heterogeneous computational clusters is presented. The main contribution of the paper is the automation of the parallelization and optimization of the PBLAS, which is done by means of a special user interface and the underlying set of functions. An important element is here a performance model that is based on program code instrumentation, which determines parameters of the application and the executive heterogeneous platform relevant for execution performance of parallel code. The parameter values specified for or returned by execution of the performance model functions are next used for generation and optimal mapping of the parallel code of the library subroutines. The proposed approach is illustrated by experimental results of execution of optimized HeteroPBLAS programs on homogeneous and heterogeneous computing clusters. Marek Tudruj

Quadratic assignment problems (QAPs) is a NP-hard combinatorial optimization problem. QAPs are often used to compare the performance of meta-heuristics. In this paper, we propose a QAP problem instance generator that can be used for benchmarking for heuristic algorithms. Our QAP generator combines small size QAPs with known optimum solution into a larger size QAP instance. We call these instances composite QAPs (cQAPs), and we show that the cost function of cQAPs is additively decomposable. We give mild conditions for which a cQAP instance has known optimum solution. We generate cQAP instances using uniform distributions with different bounds for the component QAPs and for the rest of the cQAP elements. Numerical and analytical techniques that measure the difficulty of the cQAP instances in comparison with other QAPs from the literature are introduced. These methods point out that some cQAP instances are difficult for local search with many local optimum of various values, low epistasis and non-trivial asymptotic behaviour.

Proteins are cellular macromolecules made up of linear chains of amino acids that adopt a unique three-dimensional structure to carry out specific biological functions. Amongst various computational approaches developed for Protein Structure Prediction (PSP), ab initio methods perform the prediction without prior knowledge of known structures. However, due to huge computational complexity incurred by these methods, simplified hydrophobic-polar (HP) models are often used for the investigations. Despite this simplicity, PSP continues to be a hard combinatorial optimization problem requiring development of efficient heuristic search techniques. Memetic algorithm (MA), with its flexible architecture, encompasses the strength of both local and global search and has shown promise in solving problems with complex search landscape, including PSP. This thesis is focused on developing an effective ab initio technique under a suitable MA framework for solving the PSP problem using low resolution HP models. To enhance the effectiveness of the MA, a mechanism is devised to include knowledge from the problem domain by exploiting the concept of hydrophobic-core (H-core) formation. The size of the maximal possible H-core is estimated and the approximation is validated by our proposed deterministic search technique. Further, a new knowledge-based initial population generation technique ensures generating conformations which are not only valid, but also maintain diversity in the population. This helps to commence the search with good seeds leading towards faster convergence. While solving the PSP problem using MA, the greatest intricacy arises from the complex nature of its multi-modal landscape. Like all the evolutionary algorithms (EAs), MAs have the tendency to get trapped into local minima due to selection pressure and genetic drift induced by the genetic operators, and techniques are necessary to surmount this difficulty. A novel parental selection technique called “Adaptive Strategy for Assortative Mating” is developed that dynamically distributes the re production opportunity among the individuals of the population to boost the effectiveness of cross-over. Further, a new survival selection strategy, namely “Sib-based Survival Selection”, prevents the rapid spread of genetic material from a particular individual to other members in the population, ensuring the concurrent preservation of several potential solutions, and restricting premature convergence. A new diversification mechanism, namely “Memory-based Diversification”, systematically diversifies the population by introducing new genetic material whenever it gets stuck at local peaks and allows it to proceed for further progress. An extensive experimental analysis is also reported on the influence of all three techniques proposed to maintain the diversity of population. Furthermore, in order to carry out the conformational search efficiently, especially with the increase in sequence length, the optimization technique is designed such that it provides an effective balance between exploration and exploitation. A robust Multimodal Memetic Framework has been developed where optimization is accomplished in hierarchical stages. The population of each stage is partitioned into three different states: an Exploratory state to locate promising peaks, an Exploitative state to fine-tune the solutions found in the Exploratory state of the previous stage, and finally, a Central state to carry out the formation of potential solutions to explore the undiscovered regions. Non-identical genetic operators have been implemented in each state to balance the exploration and exploitation by employing necessary selection pressure and diversity. A non-isomorphic encoding based technique has been developed to capture the sub-structures that are representing a particular region. Furthermore, a novel local search technique exploits already explored fit individuals by employing the knowledge of sub-structures. The efficacy of the proposed framework is demonstrated with an adequate number of distinctive data sets and comparison with state-of-the-art methods.

A noticeable fraction of Algorithms papers in the last few decades improve the running time of well-known algorithms for fundamental problems by logarithmic factors. For example, the {O}(n^2) dynamic programming solution to the Longest Common Subsequence problem (LCS) was improved to O(n^2/log^{2}n) in several ways and using a variety of ingenious tricks. This line of research, also known as the art of shaving log factors, lacks a tool for proving negative results. Specifically, how can we show that it is unlikely that LCS can be solved in time O(n^2/log^3n)?\nPerhaps the only approach for such results was suggested in a recent paper of Abboud, Hansen, Vassilevska W. and Williams (STOC'16). The authors blame the hardness of shaving logs on the hardness of solving satisfiability on boolean formulas (Formula-SAT) faster than exhaustive search. They show that an O(n^2/log^{1000} n) algorithm for LCS would imply a major advance in circuit lower bounds. Whether this approach can lead to tighter barriers was unclear.\nIn this paper, we push this approach to its limit and, in particular, prove that a well-known barrier from complexity theory stands in the way for shaving five additional log factors for fundamental combinatorial problems. For LCS, regular expression pattern matching, as well as the Fréchet distance problem from Computational Geometry, we show that an O(n^2/log^{7+epsilon}{n}) runtime would imply new Formula-SAT algorithms.\nOur main result is a reduction from SAT on formulas of size s over n variables to LCS on sequences of length N=2^{n/2} * s^{1+o(1)}. Our reduction is essentially as efficient as possible, and it greatly improves the previously known reduction for LCS with N=2^{n/2} * s^c, for some c >= 100.

Characterizing linearizable QAPs by the level-1 reformulation-linearization technique

This paper proposes a new heuristic approach for solving facility layout problem (FLP) which is traditionally formulated as quadratic assignment problem (QAP). In this paper, FLP is formulated as linear assignment problem (LAP) consisting of facility pair and location pair. Being linear in nature LAP can be solved efficiently. Solution of LAP provides lower bound on corresponding QAP formulation of FLP. Heuristic procedure is applied to solve FLP from sets of LAP solution. Proposed heuristic is tested on benchmark instances taken from literature and compared with other heuristics available in literature. Computational results show that proposed heuristic provides a good quality approximate solution. This solution can be taken as initial solution for any improvement heuristic to get optimal/near optimal solution for FLP.

Quadratic assignment problems (QAPs) and quadratic assignment matchings (QAMs) recently gained a lot of interest in computer graphics and vision, e.g. for shape and graph matching. Literature describes several convex relaxations to approximate solutions of the NP-hard QAPs in polynomial time. We compare the convex relaxations recently introduced in computer graphics and vision to established approaches in discrete optimization. Building upon a unified constraint formulation we theoretically analyze their solution spaces and their approximation quality. Experiments on a standard benchmark as well as on instances of the shape matching problems support our analysis. It turns out that often the bounds of a tight linear relaxation are competitive with the bounds of semidefinite programming (SDP) relaxations, while the linear relaxation is often much faster to calculate. Indeed, for many instances the bounds of the linear relaxation are only slightly worse than the SDP relaxation of Zhao [ZKRW98,PR09], which itself is at least as accurate as the relaxations currently used in computer graphics and vision. Solving the SDP relaxations can often be accelerated considerably from hours to minutes using the recently introduced approximation method for trace bound SDPs [WSvdHT16], but nonetheless calculating linear relaxations is faster in most cases. For the shape matching problem all relaxations generate the optimal solution, only that the linear relaxation does so faster. Our results generalize as well to QAMs for which we deliver new relaxations. Furthermore by interpreting the Product Manifold Filter [VLR∗17] in the context of QAPs we show how to automatically calculate correspondences between shapes of several hundred points.

This thesis proposes a new parallel computing network--a binary relation inference network. The network topology is set up to match conventional optimization procedures. The network transforms some optimization problems (shortest path, transitive closure, assignment, etc.) into network convergence problems and divides the work load evenly among all processors. Link weights are known. It can operate in synchronous discrete-time, asynchronous discrete-time, and continuous-time domains. Inference network algorithms are derived directly from problem formulation, and are effective for a large range of problem sizes. The inference network consists of interconnected components; each component is composed of a unit, a number of sites attached to the unit, and links from the unit to sites on other units. Each unit is a simple computational engine. The topology of the inference network matches naturally to many optimization problems, since its deduction sites produce candidates to compete for the optimum and its decision-making units select the optimum from the site values. Either directly or through a transformation mapping to a systolic structure, discrete inference network algorithms can be implemented on a commercially available parallel processing facility, such as the Connection Machine. Analog inference network can be built using integrated circuits which solve problems efficiently in the continuous-time domain. In this thesis, mathematical analysis and numerical simulation were conducted for the synchronous, asynchronous, and analog inference network. Various applications has been discussed. The inference network has shown to solve the all-pair shortest path problem efficiently in various updating schemes. The inference network algorithm for the transitive closure problem parts derived straightforwardly from the formulation of the problem and the topology of the inference network. Efficient continuous-time solution was obtained from a non-synchronized logical circuit. The convergence of the network for the shortest path or transitive closure problems was shown to be independent of the problem size. It was demonstrated that the inference network can solve the assignment problem in a way similar to the Hopfield net's solution to the traveling salesman problem. However, the former was shown to converge for all test cases with problem size up to 128. (Abstract shortened by UMI.)

Signed graphs are graphs with an assignment of a positive or a negative sign to each edge. These graphs are helpful to represent different types of networks. For instance, they have been used in social networks, where a positive sign in an edge represents friendship between the two endpoints of that edge, while a negative sign represents enmity. Given a signed graph, an important question is how to embed such a graph in a metric space so that in the embedding every vertex is closer to its positive neighbors than to its negative ones. This problem is known as Sitting Arrangement (SA) problem and it was introduced by Kermarrec et al. (Proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS), pp. 388---399, 2011). Cygan et al. (Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS), 2012) proved that the decision version of SA problem is NP-Complete when the signed graph has to be embedded into the Euclidean line. In this work, we study the minimization version of SA (MinSA) problem in the Euclidean line. We relate MinSA problem to the well known quadratic assignment (QA) problem. We establish such a relation by proving that local minimums in MinSA problem are equivalent to local minimums in a particular case of QA problem. In this document, we design two heuristics based on the combinatorial structure of MinSA problem. We experimentally compare their performances against heuristics designed for QA problem. This comparison favors the proposed heuristics.

In this paper, we describe an implementation of the iterated tabu search (ITS) algorithm for the quadratic assignment problem (QAP), which is one of the well-known problems in combinatorial optimization. The medium- and large-scale QAPs are not, to this date, practically solvable to optimality, therefore heuristic algorithms are widely used. In the proposed ITS approach, intensification and diversification mechanisms are combined in a proper way. The goal of intensification is to search for good solutions in the neighbourhood of a given solution, while diversification is responsible for escaping from local optima and moving towards new regions of the search space. In particular, the following enhancements were implemented: new formula for fast evaluation of the objective function and efficient data structure; extended intensification mechanisms (including randomized tabu criterion, combination of tabu search and local search, dynamic tabu list maintaining); enhanced diversification strategy using periodic tabu tenure and special mutation procedure. The ITS algorithm is tested on the different instances taken from the QAP library QAPLIB. The results from the experiments demonstrate promising efficiency of the proposed algorithm, especially for the random QAP instances.

The power of genetic algorithms (GAs) has been demonstrated for various domains of the computer science, including combinatorial optimization. In this paper, we propose a new conceptual modification of the genetic algorithm entitled a “restart-based genetic algorithm” (RGA). An effective implementation of RGA for a well-known combinatorial optimization problem, the quadratic assignment problem (QAP), is discussed. The results obtained from the computational experiments on the QAP instances from the publicly available library QAPLIB show excellent performance of RGA. This is especially true for the real-life like QAPs.