Showing papers on "Covering problems published in 2014"
TL;DR: A new branch and bound method is proposed for solving a subclass of MOMIP problems, where only two objectives are allowed, the integer variables are binary, and one of the two objectives has only integer variables.
Abstract: Most real-world optimization problems are multiobjective by nature, involving noncomparable objectives Many of these problems can be formulated in terms of a set of linear objective functions that should be simultaneously optimized over a class of linear constraints Often there is the complicating factor that some of the variables are required to be integral The resulting class of problems is named multiobjective mixed integer programming (MOMIP) problems Solving these kinds of optimization problems exactly requires a method that can generate the whole set of nondominated points (the Pareto-optimal front) In this paper, we first give a survey of the newly developed branch and bound methods for solving MOMIP problems After that, we propose a new branch and bound method for solving a subclass of MOMIP problems, where only two objectives are allowed, the integer variables are binary, and one of the two objectives has only integer variables The proposed method is able to find the full set of nondominat
109 citations
TL;DR: This work considers a class of packing problems with uncertain data, which it refers to as the chance-constrained binary packing problem, and proposes a problem formulation in its original space based on the so-called probabilistic covers.
Abstract: We consider a class of packing problems with uncertain data, which we refer to as the chance-constrained binary packing problem. In this problem, a subset of items is selected that maximizes the total profit so that a generic packing constraint is satisfied with high probability. Interesting special cases of our problem include chance-constrained knapsack and set packing problems with random coefficients. We propose a problem formulation in its original space based on the so-called probabilistic covers. We focus our solution approaches on the special case in which the uncertainty is represented by a finite number of scenarios. In this case, the problem can be formulated as an integer program by introducing a binary decision variable to represent feasibility of each scenario. We derive a computationally efficient coefficient strengthening procedure for this formulation, and demonstrate how the scenario variables can be efficiently projected out of the linear programming relaxation. We also study how method...
92 citations
TL;DR: It is APX-hard to compute a minimum cover of a set of points in the plane by a family of axis-aligned fat rectangles, even when each rectangle is an @e-perturbed copy of a single unit square.
Abstract: We study several geometric set cover and set packing problems involving configurations of points and geometric objects in Euclidean space. We show that it is APX-hard to compute a minimum cover of a set of points in the plane by a family of axis-aligned fat rectangles, even when each rectangle is an @e-perturbed copy of a single unit square. We extend this result to several other classes of objects including almost-circular ellipses, axis-aligned slabs, downward shadows of line segments, downward shadows of graphs of cubic functions, fat semi-infinite wedges, 3-dimensional unit balls, and axis-aligned cubes, as well as some related hitting set problems. We also prove the APX-hardness of a related family of discrete set packing problems. Our hardness results are all proven by encoding a highly structured minimum vertex cover problem which we believe may be of independent interest. In contrast, we give a polynomial-time dynamic programming algorithm for geometric set cover where the objects are pseudodisks containing the origin or are downward shadows of pairwise 2-intersecting x-monotone curves. Our algorithm extends to the weighted case where a minimum-cost cover is required. We give similar algorithms for several related hitting set and discrete packing problems.
77 citations
TL;DR: A hyper-heuristic methodology is described that can generate a fast, deterministic algorithm capable of producing results comparable to that of using the best problem-specific heuristic, and sometimes even better, but without the cost of trying all the heuristics.
Abstract: One- and two-dimensional packing and cutting problems occur in many commercial contexts, and it is often important to be able to get good-quality solutions quickly. Fairly simple deterministic heuristics are often used for this purpose, but such heuristics typically find excellent solutions for some problems and only mediocre ones for others. Trying several different heuristics on a problem adds to the cost. This paper describes a hyper-heuristic methodology that can generate a fast, deterministic algorithm capable of producing results comparable to that of using the best problem-specific heuristic, and sometimes even better, but without the cost of trying all the heuristics. The generated algorithm handles both one- and two-dimensional problems, including two-dimensional problems that involve irregular concave polygons. The approach is validated using a large set of 1417 such problems, including a new benchmark set of 480 problems that include concave polygons.
67 citations
18 Jun 2014
TL;DR: The set cover problem, belongs to the branch of combinatorial optimization problems, whose complexity is exponential theoretically established as NP-complex problems, and is solved by binary Firefly algorithm, based on the flashing behavior of fireflies, using binary representation.
Abstract: The non-unicost Set Covering Problem is a well-known NP-hard problem with many practical applications. In this work, a new approach based on Binary Firefly Algorithm is proposed to solve this problem. The Firefly Algorithm has attracted much attention and has been applied to many optimization problems. Here, we demonstrate that is also able to produce very competitive results solving the portfolio of set covering problems from the OR-Library.
50 citations
TL;DR: AUGMECON2 a Multi-Objective Mathematical Programming (MOMP) method is used which is capable of generating the exact Pareto set in Multi- Objective Integer Programming (MOIP) problems for producing all the Pare to optimal solutions in two popular MOCO problems.
49 citations
TL;DR: In this paper, a non-linear multi-objective formula-tions for single and multiple allocation hub maximal covering problems as well as the linearized versions are proposed, which substantially mitigate the complexity of the existing models due to the fewer number of constraints and variables.
Abstract: Hubs are centers for collection, rearrangement, and redistribution of commodities in transportation net- works. In this paper, non-linear multi-objective formula- tions for single and multiple allocation hub maximal covering problems as well as the linearized versions are proposed. The formulations substantially mitigate com- plexity of the existing models due to the fewer number of constraints and variables. Also, uncertain shipments are studied in the context of hub maximal covering problems. In many real-world applications, any link on the path from origin to destination may fail to work due to disruption. Therefore, in the proposed bi-objective model, maximizing safety of the weakest path in the network is considered as the second objective together with the traditional maximum coverage goal. Furthermore, to solve the bi-objective model, a modified version of NSGA-II with a new dynamic immigration operator is developed in which the accurate number of immigrants depends on the results of the other two common NSGA-II operators, i.e. mutation and cross- over. Besides validating proposed models, computational results confirm a better performance of modified NSGA-II versus traditional one.
45 citations
TL;DR: The use of dual information significantly improves the efficacy of the heuristic in terms of both solution time and accuracy and concludes that set covering problems of various characteristics and sizes may be solved to near optimality without resorts to custom solution methods.
Abstract: This paper investigates the role of dual information on the
performances of heuristics designed for solving the set covering
problem. After solving the linear programming relaxation of the
problem, the dual information is used to obtain the two main
approaches proposed here: (i) The size of the original problem is
reduced and then the resulting model is solved with exact
methods. We demonstrate the effectiveness of this approach on a rich
set of benchmark instances compiled from the literature. We conclude
that set covering problems of various characteristics and sizes may
reliably be solved to near optimality without resorting to custom
solution methods. (ii) The dual information is embedded into an
existing heuristic. This approach is demonstrated on a well-known
local search based heuristic that was reported to obtain successful
results on the set covering problem. Our results demonstrate that
the use of dual information significantly improves the efficacy of
the heuristic in terms of both solution time and accuracy.
43 citations
TL;DR: The experiments show that the proposed solution method competes well against commercial software and that can be competitive with specialized procedures in some problem classes.
Abstract: The goal of this work is the development of a black-box solver based on the scatter search methodology. In particular, we seek a solver capable of obtaining high quality outcomes to optimization problems for which solutions are represented as a vector of integer values. We refer to these problems as integer optimization problems. We assume that the decision variables are bounded and that there may be constraints that require that the black-box evaluator is called in order to know whether they are satisfied. Problems of this type are common in operational research areas of applications such as telecommunications, project management, engineering design and the like.Our experimental testing includes 171 instances within four classes of problems taken from the literature. The experiments compare the performance of the proposed method with both the best context-specific procedures designed for each class of problem as well as context-independent commercial software. The experiments show that the proposed solution method competes well against commercial software and that can be competitive with specialized procedures in some problem classes.
29 citations
TL;DR: This work compares three types of random-key genetic algorithms: the unbiased algorithm of Bean (1994); the biased algorithm of Goncalves and Resende (2010); and a greedy version of Bean's algorithm on 12 instances from four types of covering problems: general-cost set covering, Steiner triple covering, general- cost set k -covering, and unit-cost covering by pairs.
Abstract: Random key genetic algorithms are heuristic methods for solving combinatorial optimization problems. They represent solutions as vectors of randomly generated real numbers, the so-called random keys. A deterministic algorithm, called a decoder, takes as input a vector of random keys and associates with it a feasible solution of the combinatorial optimization problem for which an objective value or fitness can be computed. We compare three types of random-key genetic algorithms: the unbiased algorithm of Bean (1994); the biased algorithm of Goncalves and Resende (2010); and a greedy version of Bean's algorithm on 12 instances from four types of covering problems: general-cost set covering, Steiner triple covering, general-cost set k -covering, and unit-cost covering by pairs. Experiments are run to construct runtime distributions for 36 heuristic/instance pairs. For all pairs of heuristics, we compute probabilities that one heuristic is faster than the other on all 12 instances. The experiments show that, in 11 of the 12 instances, the greedy version of Bean's algorithm is faster than Bean's original method and that the biased variant is faster than both variants of Bean's algorithm.
27 citations
TL;DR: This paper gives the following simple and intuitive template for robust covering problems: having built some anticipating solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat.
Abstract: In a two-stage robust covering problem, one of several possible scenarios will appear tomorrow and require to be covered, but costs are higher tomorrow than today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? We consider the $$k$$ k -robust model where the possible scenarios tomorrow are given by all demand-subsets of size $$k$$ k . In this paper, we give the following simple and intuitive template for $$k$$ k -robust covering problems: having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat. We show that this template gives good approximation algorithms for $$k$$ k -robust versions of many standard covering problems: set cover, Steiner tree, Steiner forest, minimum-cut and multicut. Our $$k$$ k -robust approximation ratios nearly match the best bounds known for their deterministic counterparts. The main technical contribution lies in proving certain net-type properties for these covering problems, which are based on dual-rounding and primal---dual ideas; these properties might be of some independent interest. As a by-product of our techniques, we also get algorithms for max---min problems of the form: "given a covering problem instance, which $$k$$ k of the elements are costliest to cover?" For the problems mentioned above, we show that their $$k$$ k -max---min versions have performance guarantees similar to those for the $$k$$ k -robust problems.
Posted Content•
TL;DR: This work considers optimization problems that are formulated and solved in the framework of tropical mathematics, and presents direct solutions to the problems in a closed compact vector form suitable for further analysis and applications.
Abstract: We consider optimization problems that are formulated and solved in the framework of tropical mathematics. The problems consist in minimizing or maximizing functionals defined on vectors of finite-dimensional semimodules over idempotent semifields, and may involve constraints in the form of linear equations and inequalities. The objective function can be either a linear function or nonlinear function calculated by means of multiplicative conjugate transposition of vectors. We start with an overview of known tropical optimization problems and solution methods. Then, we formulate certain new problems and present direct solutions to the problems in a closed compact vector form suitable for further analysis and applications. For many problems, the results obtained are complete solutions.
TL;DR: This article considers the cooperative maximum covering location problem on a network, and develops a finite dominant set, a mixed-integer programming formulation, and two heuristics that can be used for large instances of the p-facility problem.
Abstract: In this article, we consider the cooperative maximum covering location problem on a network. In this model, it is assumed that each facility emits a certain "signal" whose strength decays over distance according to some "signal strength function." A demand point is covered if the total signal transmitted from all the facilities exceeds a predefined threshold. The problem is to locate facilities so as to maximize the total demand covered. For the 2-facility problem, we present efficient polynomial algorithms for the cases of linear and piecewise linear signal strength functions. For the p-facility problem, we develop a finite dominant set, a mixed-integer programming formulation that can be used for small instances, and two heuristics that can be used for large instances. The heuristics use the exact algorithm for the 2-facility case. We report results of computational experiments. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 634, 334-349 2014
TL;DR: In this paper, the trade-off between the quality of the polygonal representation and its derived number of circles is discussed, showing the impact on the resolution of the problem, in terms of computational efficiency.
TL;DR: In this article, an iterative algorithm for equilibrium problems and split feasibility problems in Hilbert spaces is introduced. But the main purpose of this paper is not to solve the split feasibility problem; rather, it is to prove that the sequence converges strongly to a common element of the set of solutions of equilibrium problems.
Abstract: The main purpose of this paper is to introduce an iterative algorithm for equilibrium problems and split feasibility problems in Hilbert spaces. Under suitable conditions we prove that the sequence converges strongly to a common element of the set of solutions of equilibrium problems and the set of solutions of split feasibility problems. Our result extends and improves the corresponding results of some others.
TL;DR: This paper addresses the problem of classifying positive and negative data with the technique known as box clustering, and model both problems as set covering problems with column generation.
Dissertation•
12 Mar 2014
04 Sep 2014
TL;DR: A generic online allocation problem that generalizes the classical online set cover framework by considering requests comprising a set of elements rather than a single element, which has multiple applications in cloud computing, crowd sourcing, facility planning, etc.
Abstract: We consider a generic online allocation problem that generalizes the classical online set cover framework by considering requests comprising a set of elements rather than a single element. This problem has multiple applications in cloud computing, crowd sourcing, facility planning, etc. Formally, it is an online covering problem where each online step comprises an offline covering problem. In addition, the covering sets are capacitated, leading to packing constraints. We give a randomized algorithm for this problem that has a nearly tight competitive ratio in both objectives: overall cost and maximum capacity violation. Our main technical tool is an online algorithm for packing/covering LPs with nested constraints, which may be of interest in other applications as well.
23 Jun 2014
TL;DR: This work uses techniques based on permutations of firing sequences and on canonical firing sequences to approach computational problems involving classes of Petri nets with arbitrary arc multiplicities to obtain PSPACE-completeness for the reachability and the covering problems of conservative Petrinets.
Abstract: We present a framework based on permutations of firing sequences and on canonical firing sequences to approach computational problems involving classes of Petri nets with arbitrary arc multiplicities. As an example of application, we use these techniques to obtain PSPACE-completeness for the reachability and the covering problems of conservative Petri nets, generalizing known results for ordinary 1-conservative Petri nets. We also prove PSPACE-completeness for the RecLFS and the liveness problems of conservative Petri nets, for which, in case of ordinary 1-conservative Petri nets, PSPACE-membership but no matching lower bound has been known. Last, we show PSPACE-completeness for the containment and equivalence problems of conservative Petri nets. PSPACE-hardness of the problems mentioned above still holds if they are restricted to ordinary 1-conservative Petri nets.
TL;DR: The semi-continuous inflow set with variable upper bounds as a relaxation of general semi-Continuous network flow problems is introduced and the computational efficacy of the developed polyhedral results in solving randomly generated instances of semi- Continuous transportation problems is studied.
Abstract: We consider semi-continuous network flow problems, that is, a class of network flow problems where some of the variables are restricted to be semi-continuous. We introduce the semi-continuous inflow set with variable upper bounds as a relaxation of general semi-continuous network flow problems. Two particular cases of this set are considered, for which we present complete descriptions of the convex hull in terms of linear inequalities and extended formulations. We consider a class of semi-continuous transportation problems where inflow systems arise as substructures, for which we investigate complexity questions. Finally, we study the computational efficacy of the developed polyhedral results in solving randomly generated instances of semi-continuous transportation problems.
TL;DR: A new packing problem is formulated, named the Generalized Bin Packing Problem (GBPP), characterized by both compulsory and non-compulsory items, and multiple item and bin attributes, and two stochastic variants of the GBPP are studied.
Abstract: Packing problems make up a fundamental topic of combinatorial optimization. Their importance is confirmed both by their wide range of scientific and technological applications they are able to address and by their theoretical implications. In fact, they are exploited in many fields such as computer science and technologies, industrial applications, transportation and logistics, and telecommunications. From a theoretical perspective, packing problems often appear as sub-problems in order to iteratively solve bigger problems. Although packing problems play a fundamental role in all these settings, there is a gap in terms of comprehensive study in the literature. In fact, the joint presence of both compulsory and non-compulsory items has not been considered yet. This particular setting arises in many real-life applications, not yet addressed or only partially addressed by the current state-of-the-art packing problems. Furthermore, little has been done in terms of unified methodologies, and different techniques have been used in order to solve packing problems with different objective functions. In particular, none of these techniques is able to address the presence of compulsory and non-compulsory items at the same time. In order to overcome a noteworthy portion of this gap, we formulated a new packing problem, named the Generalized Bin Packing Problem (GBPP), characterized by both compulsory and non-compulsory items, and multiple item and bin attributes. Packing problems have also been studied within stochastic settings where the items are affected by uncertainty. In these settings, there are fundamentally two kinds of stochasticity concerning the items: 1) stochasticity of the item attributes, where one attribute is affected by uncertainty and modeled as a random variable or 2) stochasticity of the item availability, i.e., the items are not known a priori but they arrive on-line in an unpredictable way to a decision maker. Although packing problems have been studied according to these stochastic variants, the GBPP with uncertainty on the items is still an open problem. Therefore, we have also studied two stochastic variants of the GBPP, named the Stochastic Generalized Bin Packing Problem (S-GBPP) and the On-line Generalized Bin Packing Problem (OGBPP). Our main results concern the development of models and unified methodologies of these new packing problems, making up, as done for the Vehicle Routing Problem (VRP) with the definition of the so called Rich Vehicle Routing Problems, a new family of advanced packing problems named Generalized Bin Packing Problems
Proceedings Article•
01 Jan 2014
TL;DR: In this paper, a new approach based on Binary Firefly Algorithm is proposed to solve the non-unicost set covering problem, which is a well-known NP-hard problem with many practical applications.
Abstract: The non-unicost Set Covering Problem is a well-known NP-hard problem with many practical applications. In this work, a new approach based on Binary Firefly Algorithm is proposed to solve this problem. The Firefly Algorithm has attracted much attention and has been applied to many optimization problems. Here, we demonstrate that is also able to produce very competitive results solving the portfolio of set covering problems from the OR-Library.
30 Sep 2014
TL;DR: In this paper, an implicit net with perturbations for solving the mixed equilibrium problems and fixed point problems has been constructed and it is shown that the proposed net converges strongly to a common solution of the mixed equilibria and fixed points problems.
Abstract: In this paper, an implicit net with perturbations for solving the mixed equilibrium problems and fixed point problems has been constructed and it is shown that the proposed net converges strongly to a common solution of the mixed equilibrium problems and fixed point problems. Also, as applications, some corollaries for solving the minimum-norm problems are also included. MSC: 47J05; 47J25; 47H09
Dissertation•
24 Jan 2014
TL;DR: This thesis systematically investigates multidimensional covering mechanism- design problems, and gives a black-box method to transform any Lagrangian-multiplier-preserving ρ- approximation algorithm for UFL into a truthful-in-expectation, π-approximation mecha- nism.
Abstract: Algorithmic mechanism design deals with efficiently-computable algorithmic constructions in the presence of strategic players who hold the inputs to the problem and may misreport their input if doing so benefits them. Algorithmic mechanism design finds applications in a variety of internet settings such as resource allocation, facility location and e-commerce, such as sponsored search auctions. There is an extensive amount of work in algorithmic mechanism design on packing problems such as single-item auctions, multi-unit auctions and combinatorial auctions. But, surprisingly, covering problems, also called procurement auctions, have almost been completely unexplored, especially in the multidimensional setting. In this thesis, we systematically investigate multidimensional covering mechanism- design problems, wherein there are m items that need to be covered and n players who provide covering objects, with each player i having a private cost for the covering objects he provides. A feasible solution to the covering problem is a collection of covering objects (obtained from the various players) that together cover all items. Two widely considered objectives in mechanism design are: (i) cost-minimization (CM) which aims to minimize the total cost incurred by the players and the mechanism designer; and (ii) payment minimization (PayM), which aims to minimize the payment to players. Covering mechanism design problems turn out to behave quite differently from pack- ing mechanism design problems. In particular, various techniques utilized successfully for packing problems do not perform well for covering mechanism design problems, and this necessitates new approaches and solution concepts. In this thesis we devise various tech- niques for handling covering mechanism design problems, which yield a variety of results for both the CM and PayM objectives. In our investigation of the CM objective, we focus on two representative covering problems: uncapacitated facility location (UFL) and vertex cover. For multi-dimensional UFL, we give a black-box method to transform any Lagrangian-multiplier-preserving ρ- approximation algorithm for UFL into a truthful-in-expectation, ρ-approximation mecha- nism. This yields the first result for multi-dimensional UFL, namely a truthful-in-expectation
Posted Content•
TL;DR: In this paper, a primal-dual LP relaxation was proposed for the graph covering problem, where a set of edges in an edge-and node-weighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights.
Abstract: This paper discusses the graph covering problem in which a set of edges in an edge- and node-weighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights. In this problem, because of the large integrality gap of a natural linear programming (LP) relaxation, LP rounding algorithms based on the relaxation yield poor performance. Here we propose a stronger LP relaxation for the graph covering problem. The proposed relaxation is applied to designing primal-dual algorithms for two fundamental graph covering problems: the prize-collecting edge dominating set problem and the multicut problem in trees. Our algorithms are an exact polynomial-time algorithm for the former problem, and a 2-approximation algorithm for the latter problem, respectively. These results match the currently known best results for purely edge-weighted graphs.
Journal Article•
TL;DR: In this paper, an adaptive division and an intelligent master are proposed to solve the unate covering problem in the context of graph-covering problems in the process of circuit synthesis.
Abstract: This paper aims at better possibilities to solve problems of exponential complexity. Our special focus is the combination of the computational power of four cores of a standard PC with better approaches in the application domain. As the main example we selected the unate covering problem which must be solved, among others, in the process of circuit synthesis and for graph-covering (domination) problems. We introduce into the wide field of problems that can be solved using Boolean models. We explain the models and the classic solutions, and discuss the results of a selected model by using a benchmark set. Subsequently we study sources of parallelism in the application domain and explore improvements given by the parallel utilization of the available four cores of a PC. Starting with a uniform splitting of the problem, we suggest improvements by means of an adaptive division and an intelligent master. Our experimental results confirm that the combination of improvements of the application models and of the algorithmic domain leads to a remarkable speedup and an overall improvement factor of more than 35 millions in comparison with the improved basic approach.
TL;DR: This paper proposes a new approach for finding solutions to LP problems by using a part of the constraints with the help of intercept and projection values of the each constraint, which is more efficient when compared with the existing method.
Abstract: Linear programs (LP) play an important role in the theory and practice of optimization problems. Linear programming deals with problems such as maximising profits, minimising costs or ensuring you make the best use of available resources. While formulating a linear programming model, system analyst and researchers often tend to include all the possible constraints, although some of them may not be binding at the optimal solution. It is well known that, for most of the large scale LP problems, only a relatively small percentage of constraints are binding at the optimal solutions. Researchers have proposed methods which identify those constraints most likely to be tight at optimality. This paper proposes a new approach for finding solutions to LP problems by using a part of the constraints with the help of intercept and projection values of the each constraint. This method is more efficient when compared with the existing method. The developed algorithm is implemented by programming language Java and the computational results are presented. It shows that the proposed method reflects a significant decrease in the computational effort and is one of the best alternatives to select the necessary constraints prior to solving an LP problem. 1312 P. Sumathi and A. Gangadharan
TL;DR: The procedure for the sequential recalculation of the coefficients of the objective function in this algorithm can be applied to other problems with unimodular matrices, in particular, to various transportation problems.
Abstract: Network optimization problems are considered Their statements include numerous variables and equipments Decomposition methods are used for their solution In specific situations, intermediate problems solved by the algorithms have the form of the knapsack problem In the case when the constraints have a staircase structure, an efficient algorithm can be constructed The procedure for the sequential recalculation of the coefficients of the objective function in this algorithm can be applied to other problems with unimodular matrices, in particular, to various transportation problems
TL;DR: A general, unified framework to resolve geometric covering problems, reduced to a set cover search in parametric space, allowing for flexible trade-offs between solution quality and computation time is presented.