Combinatorial Optimization

A main purpose of this work is to give a good algorithm for a certain welldescribed class of integer linear programming problems, called matching problems (or the matching problem). Methods developed for simple matching [2,3], a special case to which these problems can be reduced , are applied directly to the larger class. In the process, we derive a description of a system of linear inequalities whose polyhedron is the convex hull of the admissible solution vectors to the given matching problem. At the same time, various combinatorial results about matchings are derived and discussed in terms of graphs.

Combinatorial optimization model for railway engine assignment problem. In: Proceedings of the 5th Workshop on Algorithmic Methods and Models for Optimization of Railways. Dagstuhl Research Online. Strathprints is designed to allow users to access the research output of the University of Strathclyde.

The Quadratic Assignment Problem (QAP) is one of the classical combinatorial optimization problems and is known for its diverse applications. In this paper, we suggest a genetic algorithm for the QAP and report its computational behavior. The genetic algorithm incorporates many greedy principles in its design and, hence, we refer to it as a greedy genetic algorithm. The ideas we incorporate in the greedy genetic algorithm include (i) generating the initial population using a randomized construction heuristic; (ii) new crossover schemes; (iii) a special purpose immigration scheme that promotes diversity; (iv) periodic local optimization of a subset of the population; (v) tournamenting among different populations; and (vi) an overall design that attempts to strike a balance between diversity and a bias towards fitter individuals. We test our algorithm on all the benchmark instances of QAPLIB, a well-known library of QAP instances. Out of the 132 total instances in QAPLIB of varied sizes, the greedy genetic algorithm obtained the best known solution for 103 instances, and for the remaining instances (except one) found solutions within 1 % of the best known solutions.

This paper introduces Phased Local Search (PLS), a new stochastic reactive dynamic local search algorithm for the maximum clique problem. PLS interleaves sub-algorithms which alternate between sequences of iterative improvement, during which suitable vertices are added to the current clique, and plateau search, where vertices of the current clique are swapped with vertices not contained in the current clique. The sub-algorithms differ in their vertex selection techniques in that selection can be solely based on randomly selecting a vertex, randomly selecting within highest vertex degree or randomly selecting within vertex penalties that are dynamically adjusted during the search. In addition, the perturbation mechanism used to overcome search stagnation differs between the sub-algorithms. PLS has no problem instance dependent parameters and achieves state-of-the-art performance for the maximum clique problem over a large range of the commonly used DIMACS benchmark instances.

We propose a technique that we call HodgeRank for ranking data that may be incomplete and imbalanced, characteristics common in modern datasets coming from e-commerce and internet applications. We are primarily interested in cardinal data based on scores or ratings though our methods also give specific insights on ordinal data. From raw ranking data, we construct pairwise rankings, represented as edge flows on an appropriate graph. Our statistical ranking method exploits the graph X. Jiang acknowledges support from ARO Grant W911NF-04-R-0005 BAA and the School of Engineering fellowship at Stanford. L.-H. Lim acknowledges support from the Gerald J. Liebermann fellowship at Stanford and the Charles B. Morrey assistant professorship at Berkeley.

School timetabling is a classical combinatorial optimization problem, which consists in assigning lessons to time slots, satisfying a set of constraints of various kinds. Due mostly to the constraints this problem falls in the category of NP-Complete problems. In this paper we try to show an implementation of a decision support system that solves real timetabling problems from various schools in Portugal. This implementation is based on the Simulated Annealing meta-heuristic. The constraints we use were obtained after inquiries made to several schools in Portugal. We show the results on three schools from different levels of teaching.

W e propose a tabu search algorithm for the generalized assignment problem, which is one of the representative combinatorial optimization problems known to be NP-hard. The algorithm features an ejection chain approach, which is embedded in a neighborhood construction to create more complex and powerful moves. We also incorporate an adaptive mechanism for adjusting search parameters, to maintain a balance between visits to feasible and infeasible regions. Computational results on benchmark instances of small sizes show that the method obtains solutions that are optimal or that deviate by at most 0.16% from the best known solutions. Comparisons with other approaches from the literature show that, for instances of larger sizes, our method obtains the best solutions among all heuristics tested.

This work compares the effectiveness of eight evolutionary heuristic algorithms applied to the Quadratic Assignment Problem (QAP), reputedly one of the most difficult combinatorial optimization problems. QAP is merely used as a carrier for the comparison: we do not attempt to compare any heuristics with solving algorithms specific for it. Results are given, both with respect to the best result achieved by each algorithm in a limited time span and to its speed of convergence to that result.

We discuss the connection between facility location prob- lems and covering problems, and present approaches to solving facility location problems by reducing them to cov- ering problems. In particular, we discuss the following NP-hard optimization problems: the set cover and the re- lated set cover facility location problem, the k-center and k-suppliers problem, the dominating set problem, the max- imum

A pebbling move consists of removing two pebbles from one vertex and then placing one pebble at an adjacent vertex. If a distribution δ of pebbles lets us move at least one pebble to each vertex by applying pebbling moves repeatedly(if necessary), then δ is called a pebbling of G. The optimal pebbling number f (G) of G is the minimum number of pebbles used in a pebbling of G. In this paper, we improve the known upper bound for the optimal pebbling number of the hypercubes Q n . Mainly, we prove for large n, f (Q n ) = O(n 3/2 ( 4 3 ) n ) by a probabilistic argument.

In this paper, we apply the tabu-search technique to the job-shop scheduling problem, a notoriously difficult problem in combinatorial optimization. We show that our implementation of this method dominates both a previous approach with tabu search and the other heuristics based on iterative improvements.

We study the polyhedron P(G) defined by the convex hull of 2-edge-connected subgraphs of G where multiple copies of edges may be chosen. We show that each vertex of P(G) is also a vertex of the LP relaxation. Given the close relationship with the Graphical Traveling Salesman problem (GTSP), we examine how polyhedral results for GTSP can be modified and applied to P(G). We characterize graphs for which P(G) is integral and study how this relates to a similar result for GTSP. In addition, we show how one can modify some classes of valid inequalities for GTSP and produce new valid inequalities and facets for P(G).

Semidefinite programming(SDP) relaxations for the quadratic assignment problem (QAP)are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalentrepresentations of QAP. These relaxations result in the interesting, special, case where onlythe dual problem of the SDP relaxation has strict interior, i.e. the Slater constraint qualificationalways fails for the primal problem. Although there is no duality gap in theory,

The subject of this paper is the formulation and solution of a variation of the classical binary knapsack problem. The variation that is addressed is termed the “fixed-charge knapsack problem”, in which sub-sets of variables (activities) are associated with fixed costs. These costs may represent certain set-ups and/or preparations required for the associated sub-set of activities to be scheduled. Several

This paper presents the SR-GCWS-CS probabilistic algorithm that combines Monte Carlo simulation with splitting techniques and the Clarke and Wright savings heuristic to find competitive quasi-optimal solutions to the Capacitated Vehicle Routing Problem (CVRP) in reasonable response times. The algorithm, which does not require complex fine-tuning processes, can be used as an alternative to other metaheuristics—such as Simulated Annealing, Tabu Search, Genetic Algorithms, Ant Colony Optimization or GRASP, which ...

In this paper, we present an improved hybrid optimization algorithm, which was applied to the hard combinatorial optimization problem, the quadratic assignment problem (QAP). This is an extended version of the earlier hybrid heuristic approach proposed by the author. The new algorithm is distinguished for the further exploitation of the idea of hybridization of the well‐known efficient heuristic algorithms, namely, simulated annealing (SA) and tabu search (TS). The important feature of our algorithm is the so‐called “cold restart mechanism”, which is used in order to avoid a possible “stagnation” of the search. This strategy resulted in very good solutions obtained during simulations with a number of the QAP instances (test data). These solutions show that the proposed algorithm outperforms both the “pure” SA/TS algorithms and the earlier author's combined SA and TS algorithm. Key words: hybrid optimization, simulated annealing, tabu search, quadratic assignment problem, simulat...

The Graph is a powerful mathematical tool applied in many fields as transportation, communication, informatics, economy, … In an ordinary graph, the weights of edges and vertexes are considered independently where the length of a path is the sum of weights of the edges and the vertexes on this path. However, in many practical problems, weights at a vertex are not the same for all paths passing this vertex but depend on coming and leaving edges. The presented paper develops a model of the extended linear multi-commodity multi-cost network that can be more exactly and effectively applied to model many practical problems. Then, maximal limit cost flow problems are modeled as implicit linear programming problems. On the base of dual theory in linear rogramming, an effective approximate algorithm is developed.

Answer Set Programming is a very convenient framework to represent various problems issued from Artificial Intelligence (nonmonotonic reasoning, planning, diagnosis...). Furthermore, it can be used to neatly encode combinatorial problems. In all cases, the solutions are obtained as sets of literals: the Answer Sets. Ant Colony Optimization is a general metaheuristics that has been already successfully used to solve hard combinatorial problems (traveling salesman problem, graph coloring, quadratic assignment...). It is based on the collective behavior of artificial ants exploring a graph and exchanging pieces of information by means of pheromone traces. The purpose of this work is to show how Ant Colony Optimization can be used to compute an answer set of a logic program.

In manufacturing facilities, shipping and receiving traditionally have been concentrated in one or two areas of the building. The widespread introduction of just-in-time shipping policies in US manufacturing companies during the past 15 years, as well as the common use of decentralized receiving in the Japanese automobile industry, has motivated the consideration (and occasionally the adoption) of dock configurations to support decentralized receiving (also known as perimeter receiving). By providing multiple access points along the perimeter of the building, decentralized receiving generally reduces the time and cost associated with moving purchased parts to the locations within the facility where they are used, but also results in increased cost of construction, maintenance, and operation for docks and associated truck access/egress, as well as additional interior floorspace requirements. We develop an optimization-based procedure to determine which of several possible dock areas to construct, how many doors or gates each should have, how many unloading and internal material handling staff should be assigned to each group, and which inbound materials each group should handle. We consider the amortized cost of constructing and equipping the docks, the amortized cost of material handling equipment to support the unloaders and material handlers, the cost of operating the material handling equipment, and labor costs for both unloading and internal material handling.

We call a matrix A nearly totally unimodular if it can be obtained from a totally unimodular matrixà by adding to each row ofà an integer multiple of some fixed row a T ofÃ. For an integer vector b and a nearly totally unimodular matrix A, we denote by P A,b the integer hull of the set {x ∈ R n | Ax ≤ b}. We show that P A,b has the integer decomposition property and that we can find a decomposition of a given integer vector x ∈ kP A,b in polynomial time.

The most well-known theorem in combinatorial optimization is the classical max-flow min-cut theorem of Ford and Fulkerson. This theorem serves as the basis for deriving efficient algorithms for finding max-flows and min-cuts. Starting with the work of Leighton and Rao, significant effort was directed towards finding approximate analogs for the undirected multicommodity flow problem. In this paper we consider an approximate max-flow min-cut theorem for directed graphs. We prove a polylogarithmic bound on the worst case ratio between the minimum multicut and the value of the maximum multicommodity flow in the special case when the demands are symmetric. The method presented in this paper can be used to give polynomial time polylogarithmic approximation algorithms for the corresponding minimum directed multicut problems. The problem with symmetric demands extends the only previously known special case concerning directed graphs due to Leighton and Rao, who proved an O(log n) bound for the Research supported by NSF PYI award CCR-9157620, together with PYI matching funds from Thinking Machines Corporation and Xerox Corporation. Additional support provided by ARPA contract N00014-91-J-4052 ARPA Order No. 8225.

In this contribution, we will study the influence of the three components of the Best-Worst Ant System (BWAS) algorithm. As the importance of each of them as the fact whether all of them are necessary will be analyzed. Besides, we will introduce a new algorithm called Best-Worst Ant Colony System by combining the basis of the Ant Colony System with the special components of the BWAS. The performance of different variants of these algorithms will be tested when solving different instances of the QAP.

A lot of minimization covering problems on graphs consist in covering vertices or edges by subgraphs verifying a certain property. These problems can be seen as particular cases of set-covering. If the number of subgraphs is polynomial in the order n of the input-graph, then these problems can be approximated within logarithmic ratio by the classical greedy set-covering algorithm. We extend the class of problems approximable by this approach to covering problems where the number of candidate subgraphs is exponential in n, by revisiting an old technique called "master-slave" and extending it to weighted master or/and slave problems. Finally, we use the master-slave tool to prove some positive approximation results for two network-design and a VLSI-design graph-problems.

The chain rule -fundamental to any kind of analytical differentiation -can be applied in various ways to computational graphs representing vector functions. These variants result in different operations counts for the calculation of the corresponding Jacobian matrices. The minimization of the number of arithmetic operations required for the calculation of the complete Jacobian leads to a hard combinatorial optimization problem.

In this paper we present a population declining ant colony optimization (PDACO) multiuser detector for asynchronous CDMA communications. Ant colony optimization (ACO) algorithms have already been used in multiuser detection in CDMA systems; however, as the pheromone accumulates, we may not get a global optimum because it stops searching early. PDACO can enlarge searching range through increasing the initial population of the ant colony, and the population declines in successive iterations. So, the performance of PDACO is superior with the same computational complexity. PDACO is applied to multiuser detection in asynchronous CDMA systems over slowly multipath Rayleigh-fading channels in this paper. Via computer simulations it is shown that the performance of PDACO detector is much better in bit-error rate and near-far effect resistance than conventional detector, ACO detector and genetic algorithm detector, and is close the optimal multiuser detector.

Cooperative game theory is concerned primarily with groups of players who coordinate their actions and pool their winnings. One of the main concerns is how to divide the extra earnings (or cost savings) among the members of the coalitions. Thus a number of solution concepts for cooperative games have been proposed. In this chapter, a selection of basic notions and solution concepts for cooperative games are presented and analyzed in detail. The paper is particularly concerned with cost allocation methods in problems that arise from the field of combinatorial (discrete) optimization.

Scope and Purpose-In several physical distribution problems, goods must be delivered from several depots to a set of geographically dispersed customers, under capacity or route length constraints. Documented examples include the delivery of meals, of chemical products, of machines, of industrial gases, of petroleum products, of packaged food, etc. This article describes a powerful heuristic for this difficult problem.

In order to obtain a competitive level of productivity in a manufacturing system, efficient machine or department arrangements and appropriate transportation path structures are of considerable importance. By defining a production system's basic structure and material flows, the layout determines its operational performance over the long term. However, most approaches proposed in the literature provide only a block layout, which neglects important operational details. By contrast, in this paper, we introduce approaches to planning layouts at a more detailed level. Hence, this present paper introduces an integrated approach which allows a more detailed layout planning by simultaneously determining machine arrangement and transportation paths. Facilities to be arranged as well as the entire layout may have irregular shapes and sizes. By assigning specific attributes to certain layout subareas, application-dependent barriers within the layout, like existing walls or columns, can be incorporated. We introduce a new mathematical layout model and develop several improvement procedures. An analysis of the computational experiments shows that more elaborate heuristics using variable neighborhoods can generate promising layout configurations.

The linear ordering problem with cumulative costs (LOPCC) is a variant of the well-known linear ordering problem, in which a cumulative propagation makes the objective function highly non-linear. The LOPCC has been recently introduced in the context of mobile-phone telecommunications. In this paper we propose two metaheuristic methods for this NP-hard problem. The first one is based on the GRASP methodology, while the second one implements an Iterated Greedy-Strategic Oscillation procedure. We also propose a post-processing based on Path Relinking to obtain improved outcomes. We compare our methods with the state-ofthe-art procedures on a set of 218 previously reported instances. The comparison favors the Iterated Greedy-Strategic Oscillation with the Path Relinking post-processing, which is able to identify 87 new best objective function values.

We consider the Variable Cost and Size Bin Packing Problem, a generalization of the well-known bin packing problem, where a set of items must be packed into a set of heterogeneous bins characterized by possibly different volumes and fixed selection costs. The objective of the problem is to select bins to pack all items at minimum total binselection cost. The paper introduces lower bounds and heuristics for the problem, the latter integrating lower and upper bound techniques. Extensive numerical tests conducted on instances with up to 1000 items show the effectiveness of these methods in terms of computational effort and solution quality. We also provide a systematic numerical analysis of the impact on solution quality of the bin selection costs and the correlations between these and the bin volumes. The results show that these correlations matter and that solution methods that are unbiased toward particular correlation values perform better.

The location routing problem (LRP) appears as a combination of two difficult problems: the facility location problem (FLP) and the vehicle routing problem (VRP). In this work, we consider a discrete LRP with two levels: a set of potential capacitated distribution centres (DC) and a set of ordered customers. In our problem we intend to determine the set of installed DCs as well as the distribution routes (starting and ending at the DC). The problem is also constrained with capacities on the vehicles. Moreover, there is a homogeneous fleet of vehicles, carrying a single product and each customer is visited just once. As an objective we intend to minimize the routing and location costs.

Harmony search (HS) algorithm was applied to solving Sudoku puzzle. The HS is an evolutionary algorithm which mimics musicians’ behaviors such as random play, memory-based play, and pitch-adjusted play when they perform improvisation. Sudoku puzzles in this study were formulated as an optimization problem with number-uniqueness penalties. HS could successfully solve the optimization problem after 285 function evaluations, taking 9 seconds. Also, sensitivity analysis of HS parameters was performed to obtain a better idea of algorithm parameter values.

In this paper we present a general genetic algorithm to address a wide variety o f sequencing and optimization problems including multiple machine scheduling, reso urce allocation and the quadratic assignment problem. When addressing such probl ems, genetic algorithms typically have difficulty maintaining feasibility from parent to offs pring. This is overcome with a robust representation technique called random keys. Computati onal results are shown for multiple machine scheduling, resource allocation and quadr atic assignment problems.

The school choice problem concerns the design and implementation of matching mechanisms that produce school assignments for students within a given public school district. In this note we define a simple student-optimal criterion that is not met by any previously employed mechanism in the school choice literature. We then use this criterion to adapt a well-known combinatorial optimization technique (Hungarian algorithm) to the school choice problem.

This paper aims at describing the state of the art on quadratic assignment problems (QAPs). It discusses the most important developments in all aspects of the QAP such as linearizations, QAP polyhedra, algorithms to solve the problem to optimality, heuristics, polynomially solvable special cases, and asymptotic behavior. Moreover, it also considers problems related to the QAP, e.g. the biquadratic assignment problem, and discusses the relationship between the QAP and other well known combinatorial optimization problems, e.g. the traveling salesman problem, the graph partitioning problem, etc. The paper will appear in the Handbook of Combinatorial Optimization to be published by Kluwer Academic Publishers, P. Pardalos and D.-Z. Du, eds.

Fast algorithms can be created for many graph problems when instances are confined to classes of graphs that are recursively constructed. This paper first describes some basic conceptual notions regarding the design of such fast algorithms, and then the coverage proceeds through several recursive · 3 example, G.x = maximum-cardinality independent set, then G.x carries with it two pieces of information: primarily, the size of a maximum cardinality independent set, and secondarily, a pointer to a particular instance of such a set. This information is carried forward in computations and assignments involving G.x. Moreover, our pseudo-code will generally entail only the primary information, such as existence, size, or weight, and suppress the secondary information, that is, the pointer to a particular instance.

The Benders decomposition algorithm has been successfully applied to a wide range of difficult optimization problems.This paper presents a state-of-the-art survey of this algorithm, with an emphasis on its use in combinatorial optimization. We discuss the classical algorithm, the impact of the problem formulation on its convergence, and the relationship to other decomposition methods. We introduce a taxonomy of algorithmic enhancements and acceleration strategies based on the main components of the algorithm. The taxonomy provides the framework to synthesize the literature, and to identify shortcomings, trends and potential research directions. We also discuss the use of the BD to develop efficient (meta-)heuristics, describe the limitations of the classical algorithm, and present extensions enabling its application to a broader range of problems.

Prioritizing and selecting a few critical transportation projects from several competing projects is a multiobjective combinatorial optimization problem (MOCO). Transportation planners and managers are always interested in analyzing and visualizing the tradeoffs involved, but equity issues in distribution of resources are given much less attention. This paper develops a methodology for integrating equity metrics with traditional metrics for planning and prioritizing a large and diverse portfolio of transportation investment projects. The methodology serves to help planners, managers, and engineers to visualize and compare measures of the distributed equity of the allocation along with cost-benefit tradeoffs. It is based on incorporating network-level equity metrics along with traditional metrics in formulating a generic multiobjective combinatorial optimization (MOCO) problem and visualizing multiobjective tradeoffs on the spatial network. A case study of a region demonstrates the use of the methodology in tradeoff analysis for prioritizing and selecting transportation projects. The approach is adaptable to other manufacturing and service industries where consideration of the distributed equity of allocation is an important issue.

The problem of voltage collapse in power systems due to increased loads can be solved by adding renewable energy sources like wind and photovoltaic (PV) to some bus-bars. This option can reduce the cost of the generated energy and increase the system efficiency and reliability. In this paper, a modified smart technique using particle swarm optimization (PSO) has been introduced to select the hourly optimal load flow with renewable distributed generation (DG) integration under different operating conditions in the 30-bus IEEE system. Solar PV and wind power plants have been introduced to selected buses to evaluate theirs benefits as DG. Different solar radiation and wind speeds for the Dammam site in Saudi Arabia have been used as an example to study the feasibility of renewable energy integration and its effect on power system operation. Sensitivity analysis to the load and the other input data has been carried out to predict the sensitivity of the results to any deviation in the input data of the system. The obtained results from the proposed system prove that using of renewable energy sources as a DG reduces the generation and operation cost of the overall power system.

Computing the chromatic number of a graph is an NP-hard problem. For random graphs and some other classes of graphs, estimators of the expected chromatic number have been well studied. In this paper, a new 0–1 integer programming formulation for the graph coloring problem is presented. The proposed new formulation is used to develop a method that generates graphs of known chromatic number by using the KKT optimality conditions of a related continuous nonlinear program.

Ant-Q is an algorithm belonging to the class of ant colony based methods, that is, of combinatorial optimization methods in which a set of simple agents, called ants, cooperate to find good solutions to combinatorial optimiza tion problems. The main focus of this article is on the experimental study of the sensitivity of the Ant-Q algorithm to its parameters and on the investigation of synergistic effects when using more than a single ant. We conclude comparing Ant-Q with its ancestor Ant System, and with other heuristic algorithms.

A non-convex optimization problem involving minimization of the sum of max and min concave functions over a transportation polytope is studied in this paper. Based upon solving at most (g+1)(p) cost minimizing transportation problems with m sources and n destinations, a polynomial time algorithm is proposed which minimizes the concave objective function where, p is the number of pairwise disjoint entries in the m× n time matrix {t ij } sorted decreasingly and T g is the minimum value of the max concave function. An exact global minimizer is obtained in a finite number of iterations. A numerical illustration and computational experience on the proposed algorithm is also included.

We consider a container loading problem that occurs at a typical furniture manufacturer. Each furniture item has an associated profit. Given container dimensions and a set of furniture items, the problem is to determine a subset of items with maximal profit sum that is loadable in the container. In the studied company, the problem arises hundreds of times daily during