DOCSLIB.ORG

Mathematical Models and Decomposition Algorithms for Cutting and Packing Problems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematical Models and Decomposition Algorithms for Cutting and Packing Problems Alma Mater Studiorum - Universit`adi Bologna Dottorato di Ricerca in Automatica e Ricerca Operativa Ciclo XXIX Settore concorsuale di afferenza: 01/A6 - RICERCA OPERATIVA Settore scientifico disciplinare: MAT/09 - RICERCA OPERATIVA Mathematical Models and Decomposition Algorithms for Cutting and Packing Problems Presentata da Maxence Delorme Coordinatore Dottorato Relatore Prof. Daniele Vigo Prof. Silvano Martello Co-relatore Prof. Manuel Iori Esame finale anno 2017 Contents Acknowledgments v 1 Introduction 1 2 BPP and CSP: Mathematical Models and Exact Algorithms 7 2.1 Introduction.................................... 7 2.2 Formalstatement................................. 10 2.3 Upperandlowerbounds............................. 11 2.3.1 Approximation algorithms . 12 2.3.2 Lowerbounds............................... 13 2.3.3 Heuristics and metaheuristics . 15 2.4 Pseudo-polynomial formulations . ...... 17 2.4.1 ConsiderationsonthebasicILPmodel. 17 2.4.2 One-cutformulation ........................... 19 2.4.3 DP-flowformulation ........................... 21 2.4.4 Arc-flowformulations . .. .. .. .. .. .. .. 23 2.5 Enumerationalgorithms . 24 2.5.1 Branch-and-bound ............................ 24 2.5.2 Constraint programming approaches . 26 2.6 Branch-and-price ................................ 26 2.6.1 Set covering formulation and column generation . ...... 26 2.6.2 Integerround-upproperty. 29 2.6.3 Branch(-and-cut)-and-price algorithms . ....... 30 2.7 Experimentalevaluation . 33 2.7.1 Benchmarks................................ 33 2.7.2 Computercodes ............................. 35 2.7.3 Experiments ............................... 37 2.8 Conclusions .................................... 46 i ii CONTENTS 3 BPPLIB: A Library for Bin Packing and Cutting Stock Problems 47 3.1 Introduction.................................... 47 3.2 Computercodes ................................. 49 3.3 Benchmarks.................................... 53 3.4 Computationalexperiments . 54 3.4.1 GIinstances................................ 56 4 Enhanced PP Formulations for Bin Packing and Cutting Stock Problems 59 4.1 Introduction.................................... 60 4.2 The BPP, the CSP, and their well-known formulations . ........ 62 4.2.1 Problem description and notation . 63 4.2.2 Pattern-based formulations . 63 4.2.3 Pseudo-polynomial formulations . 65 4.3 Relationsamongmodels . .. .. .. .. .. .. .. .. 67 4.4 Reflect, an improved arc-flow formulation . ...... 72 4.4.1 Adapting reflect to solve large size instances: Reflect+........ 75 4.5 Generalizations ................................. 78 4.5.1 VariablesizedBPP............................ 78 4.5.2 BPP with item fragmentation . 80 4.6 Computationalresults . 82 4.6.1 ResultsonBPPandCSP .. .. .. .. .. .. .. 82 4.6.2 ResultsontheVSBPP. .. .. .. .. .. .. .. 86 4.6.3 ResultsontheBPPIF .......................... 88 4.7 Conclusion .................................... 89 Supplementary material 4.A Details for Lemma 1 . ...... 91 Supplementarymaterial4.B ProofofLemma2 . 91 Supplementary material 4.C Proof of Theorem 1 . ...... 93 Supplementary material 4.D Proof of Theorem 2 . 101 Supplementary material 4.E Proof of Theorem 3 . 102 Supplementary material 4.F Proof of Theorem 4 . 104 Supplementary material 4.G Algorithms for reflect . ......... 104 Supplementary material 4.H Proof of Theorem 5 . 108 CONTENTS iii 5 Logic Based Benders’ Decomposition for Orthogonal Stock Cutting Prob- lems 109 5.1 Introduction.................................... 109 5.2 Literaturereview ................................ 110 5.3 Mathematicalmodel ............................... 112 5.4 Preprocessing................................... 114 5.5 Decomposition algorithm . 115 5.6 Masterproblem.................................. 117 5.7 Slave problem and cut generation . 120 5.8 The case of identical item copies . 122 5.9 Computationalexperiments . 123 5.9.1 SCPinstances .............................. 124 5.9.2 Rectanglepackings. .. .. .. .. .. .. .. .. 126 5.9.3 Palletloading............................... 127 5.10Conclusion .................................... 129 6 A Training Software for Orthogonal Packing Problems 131 6.1 Introduction.................................... 131 6.2 Orthogonalpackingproblems . 134 6.3 Software...................................... 137 6.4 Experiments.................................... 140 6.4.1 Setup ................................... 141 6.4.2 Results .................................. 142 6.5 Conclusions .................................... 146 7 Mathematical Models and Decomposition Algorithms for the PRPP 147 7.1 Introduction.................................... 148 7.2 ProblemDescription .............................. 151 7.3 A Compact Mathematical Formulation . 153 7.4 Lower Bounds based on a Decomposition Method . 155 7.4.1 Cuttingcomponent(CC) . 156 7.4.2 Schedulingcomponent(SC) . 159 7.5 UpperBoundingProcedures. 161 7.5.1 A Heuristic Based on a Generalized Assignment Problem . 162 iv CONTENTS 7.5.2 An Iterated Local Search Algorithm . 163 7.6 ComputationalExperiments. 165 7.6.1 Instances ................................. 165 7.6.2 Algorithmperformance . 167 7.7 ConcludingRemarks ............................... 169 8 ILP and CP for Project Scheduling Problems 171 8.1 Introduction.................................... 171 8.2 Literaturereview ................................ 172 8.3 Mathematicalmodels.. .. .. .. .. .. .. .. .. 173 8.3.1 RCPSPformulations. 174 8.3.2 DTCTPformulations . 175 8.3.3 MMRCPSPformulations . 176 8.4 Proposedapproaches.............................. 176 8.4.1 RCPSPimprovedalgorithm. 177 8.4.2 DTCTPimprovedalgorithm . 179 8.4.3 MMRCPSP improved algorithm . 180 8.5 Computationalexperiments . 180 8.5.1 Computational experiments for the RCPSP . 181 8.5.2 Computational experiments for the DTCTP . 182 8.5.3 Computational experiments for the MMRCPSP . 183 8.6 Conclusion .................................... 184 Bibliography 185 Acknowledgments First of all, I would like to thank my tutors Prof. Silvano Martello and Prof. Manuel Iori for the help they provided during three years of Ph.D. studies, for giving me the opportunity to work on many different and interesting research projects, and for teaching me a lot about Operations Research and Combinatorial Optimization. I would also like to thank all the other components of the group of Operations Research of the Department of Electrical, Electronic and Information Engineering ”Guglielmo Mar- coni” (DEI) of the University of Bologna, namely Prof. Paolo Toth, Prof. Daniele Vigo, Prof. Andrea Lodi, Prof. Enrico Malaguti, Prof. Michele Monaci, Dr. Valentina Cacchiani, Dr. Paolo Tubertini, Ph.D. students Carlos Emilio Contreras Bolton, Alberto Santini, and the former members Dr. Claudo Gambella, Dr. Tiziano Parriani, Dr. Dimitri Thomopulos, and Dr. Sven Wiese. I am grateful to my co-authors, who helped me in writing the reports that form part of this thesis: Prof. Jean Fran¸cois Cˆot´efrom the Universit´eLaval, and Prof. Anand Subra- manian from the Federal University of Para´ıba. Thanks are also due to the undergraduate and master students I collaborated with during these years, in particular Gianluca Costa and Vitor Nesello. I would also like to thank the Ministero dell’Istruzione, dell’Universit`ae della Ricerca (MIUR) for the financial support given to my Ph.D. course Special thanks are addressed to Prof. Fran¸cois Clautiaux, Prof. Joshua D. Habiger, and Prof. Alexandru-Adrian Tantar for encouraging me to pursue a Ph.D. and to Prof. Clarisse Dhaenens, who made me discover and enjoy Combinatorial Optimization. Finally, I would like to thank my friends Damien and Julien, my mother, my father, and all my family for always supporting me in my decisions. Bologna, March 27, 2017 Maxence Delorme v List of Figures 1.1 Number of papers dealing with bin packing and cutting stock problems, 1991-2016 ..................................... 2 2.1 DP-flow graph construction for Example 1 . ..... 22 2.2 Arc-flow representation of the graph of Figure 2.1 . ......... 23 3.1 The interactive visual solver . ..... 52 4.1 L(FAF ) solution and path decomposition for Example 2 . 68 4.2 An L(FOC ) solution of Example 2 represented as a set of trees (¯zt = value of tree t). ..................................... 69 4.3 An L(FDP ) solution of Example 2 (selected arcs in bold, values taken by the selected variables on the arcs) . 69 4.4 Graphical representation of relations among CSP formulations. 70 4.5 Set of arcs required by the standard arc-flow (above) and by reflect (below) for Example 2 (item arcs are depicted in straight lines, loss arcs in dotted lines)........................................ 73 4.6 Solution of L(FRE ) for Example 2 (selected item arcs are depicted in straight lines, selected loss arcs in dotted lines, variable values onthearcs) . 74 4.7 Set of arcs required by reflect for Example 3 (item arcs are depicted in straightlines)................................... 79 4.8 Set of arcs built by arc-flow for the BPPIF Example 4 (item arcs are depicted instraightlines) ................................. 80 4.9 Construction of an L(FAF ) solution for Example 2 through Algorithm 5. Each iteration, from top to bottom, processes the cut associated with they ¯ variablegivenontheleft. 100 4.10 An invalid L(FAF ) solution of Example 2 with a negative flow (cycle), ob- tained by executing Algorithm 5 on an input L(FOC ) solution that does not satisfy(4.51). ................................... 101 4.11 Example 2 shows that FAF is not included in FDP ............... 102 vii
Load more

Recommended publications
  • A Branch-And-Price Approach with Milp Formulation to Modularity Density Maximization on Graphs
    A BRANCH-AND-PRICE APPROACH WITH MILP FORMULATION TO MODULARITY DENSITY MAXIMIZATION ON GRAPHS KEISUKE SATO Signalling and Transport Information Technology Division, Railway Technical Research Institute. 2-8-38 Hikari-cho, Kokubunji-shi, Tokyo 185-8540, Japan YOICHI IZUNAGA Information Systems Research Division, The Institute of Behavioral Sciences. 2-9 Ichigayahonmura-cho, Shinjyuku-ku, Tokyo 162-0845, Japan Abstract. For clustering of an undirected graph, this paper presents an exact algorithm for the maximization of modularity density, a more complicated criterion to overcome drawbacks of the well-known modularity. The problem can be interpreted as the set-partitioning problem, which reminds us of its integer linear programming (ILP) formulation. We provide a branch-and-price framework for solving this ILP, or column generation combined with branch-and-bound. Above all, we formulate the column gen- eration subproblem to be solved repeatedly as a simpler mixed integer linear programming (MILP) problem. Acceleration tech- niques called the set-packing relaxation and the multiple-cutting- planes-at-a-time combined with the MILP formulation enable us to optimize the modularity density for famous test instances in- cluding ones with over 100 vertices in around four minutes by a PC. Our solution method is deterministic and the computation time is not affected by any stochastic behavior. For one of them, column generation at the root node of the branch-and-bound tree arXiv:1705.02961v3 [cs.SI] 27 Jun 2017 provides a fractional upper bound solution and our algorithm finds an integral optimal solution after branching. E-mail addresses: (Keisuke Sato) [email protected], (Yoichi Izunaga) [email protected].
    [Show full text]
  • Generating Single and Multiple Cooperative Heuristics for the One Dimensional Bin Packing Problem Using a Single Node Genetic Programming Island Model
    Generating Single and Multiple Cooperative Heuristics for the One Dimensional Bin Packing Problem Using a Single Node Genetic Programming Island Model Kevin Sim Emma Hart Institute for Informatics and Digital Innovation Institute for Informatics and Digital Innovation Edinburgh Napier University Edinburgh Napier University Edinburgh, Scotland, UK Edinburgh, Scotland, UK [email protected] [email protected] ABSTRACT exhibit superior performance than similar human designed Novel deterministic heuristics are generated using Single Node heuristics and can be used to produce collections of heuris- Genetic Programming for application to the One Dimen- tics that collectively maximise the potential of selective HH. sional Bin Packing Problem. First a single deterministic The papers contribution is two fold. Single heuristics ca- heuristic was evolved that minimised the total number of pable of outperforming a range of well researched deter- bins used when applied to a set of 685 training instances. ministic heuristics are generated by combining features ex- Following this, a set of heuristics were evolved using a form tracted from those heuristics. Secondly an island model[19] of cooperative co-evolution that collectively minimise the is adapted to use multiple SNGP implementations to gener- number of bins used across the same set of problems. Re- ate diverse sets of heuristics which collectively outperform sults on an unseen test set comprising a further 685 prob- any of the single heuristics when used in isolation. Both ap- lem instances show that the single evolved heuristic out- proaches are trained and evaluated using equal divisions of a performs existing deterministic heuristics described in the large set of 1370 benchmark problem instances sourced from literature.
    [Show full text]
  • Optimal Placement by Branch-And-Price
    Optimal Placement by Branch-and-Price Pradeep Ramachandaran1 Ameya R. Agnihotri2 Satoshi Ono2;3;4 Purushothaman Damodaran1 Krishnaswami Srihari1 Patrick H. Madden2;4 SUNY Binghamton SSIE1 and CSD2 FAIS3 University of Kitakyushu4 Abstract— Circuit placement has a large impact on all aspects groups of up to 36 elements. The B&P approach is based of performance; speed, power consumption, reliability, and cost on column generation techniques and B&B. B&P has been are all affected by the physical locations of interconnected applied to solve large instances of well known NP-Complete transistors. The placement problem is NP-Complete for even simple metrics. problems such as the Vehicle Routing Problem [7]. In this paper, we apply techniques developed by the Operations We have tested our approach on benchmarks with known Research (OR) community to the placement problem. Using optimal configurations, and also on problems extracted from an Integer Programming (IP) formulation and by applying a the “final” placements of a number of recent tools (Feng Shui “branch-and-price” approach, we are able to optimally solve 2.0, Dragon 3.01, and mPL 3.0). We find that suboptimality is placement problems that are an order of magnitude larger than those addressed by traditional methods. Our results show that rampant: for optimization windows of nine elements, roughly suboptimality is rampant on the small scale, and that there is half of the test cases are suboptimal. As we scale towards merit in increasing the size of optimization windows used in detail windows with thirtysix elements, we find that roughly 85% of placement.
    [Show full text]
  • 13. Mathematics University of Central Oklahoma
    Southwestern Oklahoma State University SWOSU Digital Commons Oklahoma Research Day Abstracts 2013 Oklahoma Research Day Jan 10th, 12:00 AM 13. Mathematics University of Central Oklahoma Follow this and additional works at: https://dc.swosu.edu/ordabstracts Part of the Animal Sciences Commons, Biology Commons, Chemistry Commons, Computer Sciences Commons, Environmental Sciences Commons, Mathematics Commons, and the Physics Commons University of Central Oklahoma, "13. Mathematics" (2013). Oklahoma Research Day Abstracts. 12. https://dc.swosu.edu/ordabstracts/2013oklahomaresearchday/mathematicsandscience/12 This Event is brought to you for free and open access by the Oklahoma Research Day at SWOSU Digital Commons. It has been accepted for inclusion in Oklahoma Research Day Abstracts by an authorized administrator of SWOSU Digital Commons. An ADA compliant document is available upon request. For more information, please contact [email protected]. Abstracts from the 2013 Oklahoma Research Day Held at the University of Central Oklahoma 05. Mathematics and Science 13. Mathematics 05.13.01 A simplified proof of the Kantorovich theorem for solving equations using scalar telescopic series Ioannis Argyros, Cameron University The Kantorovich theorem is an important tool in Mathematical Analysis for solving nonlinear equations in abstract spaces by approximating a locally unique solution using the popular Newton-Kantorovich method.Many proofs have been given for this theorem using techniques such as the contraction mapping principle,majorizing sequences, recurrent functions and other techniques.These methods are rather long,complicated and not very easy to understand in general by undergraduate students.In the present paper we present a proof using simple telescopic series studied first in a Calculus II class.
    [Show full text]
  • Branch-And-Bound Experiments in Convex Nonlinear Integer Programming
    Noname manuscript No. (will be inserted by the editor) More Branch-and-Bound Experiments in Convex Nonlinear Integer Programming Pierre Bonami · Jon Lee · Sven Leyffer · Andreas W¨achter September 29, 2011 Abstract Branch-and-Bound (B&B) is perhaps the most fundamental algorithm for the global solution of convex Mixed-Integer Nonlinear Programming (MINLP) prob- lems. It is well-known that carrying out branching in a non-simplistic manner can greatly enhance the practicality of B&B in the context of Mixed-Integer Linear Pro- gramming (MILP). No detailed study of branching has heretofore been carried out for MINLP, In this paper, we study and identify useful sophisticated branching methods for MINLP. 1 Introduction Branch-and-Bound (B&B) was proposed by Land and Doig [26] as a solution method for MILP (Mixed-Integer Linear Programming) problems, though the term was actually coined by Little et al. [32], shortly thereafter. Early work was summarized in [27]. Dakin [14] modified the branching to how we commonly know it now and proposed its extension to convex MINLPs (Mixed-Integer Nonlinear Programming problems); that is, MINLP problems for which the continuous relaxation is a convex program. Though a very useful backbone for ever-more-sophisticated algorithms (e.g., Branch- and-Cut, Branch-and-Price, etc.), the basic B&B algorithm is very elementary. How- Pierre Bonami LIF, Universit´ede Marseille, 163 Av de Luminy, 13288 Marseille, France E-mail: [email protected] Jon Lee Department of Industrial and Operations Engineering, University
    [Show full text]
  • Three-Dimensional Bin-Packing Approaches, Issues, and Solutions
    Three Dimensional Bin-Packing Issues and Solutions Seth Sweep University of Minnesota, Morris [email protected] Abstract This paper reviews the three-dimensional version of the classic NP-hard bin-packing, optimization problem along with its theoretical relevance and practical importance. Also, new approximation solutions are introduced based on prior research. The three-dimensional bin-packing optimization problem concerns placing box-shaped objects of arbitrary size and number into a box-shaped, three-dimensional space efficiently. Approximation algorithms become a guide that attempts to place objects in the least amount of space and time. As a NP-hard problem, three-dimensional bin- packing holds much academic interest to computer scientists. However, this problem also has relevance in industrial settings such as shipping cargo and efficiently designing machinery with replaceable parts, such as automobiles. Industry relies on algorithms to provide good solutions to versions of this problem. This research project adapted common approaches to one-dimensional bin-packing, such as next fit and best fit, to three dimensions. Adaptation of these algorithms is possible by dividing the space of a bin. Then, by paying attention strictly to the width of each object, the one-dimensional approaches attempt to efficiently pack into the divided compartments of the original bin. Through focusing entirely on width, these divisions effectively become multiple one-dimensional bins. However, entirely disregarding the volume of the bin by ignoring height and depth may generate inefficient packings. As this paper details in more depth, generally cubical objects may pack well but exceptionally high or long objects leave large gaps of empty space unpacked.
    [Show full text]
  • Algorithmic Combinatorial Game Theory∗
    Playing Games with Algorithms: Algorithmic Combinatorial Game Theory∗ Erik D. Demaine† Robert A. Hearn‡ Abstract Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer. 1 Introduction Many classic games are known to be computationally intractable (assuming P 6= NP): one-player puzzles are often NP-complete (as in Minesweeper) or PSPACE-complete (as in Rush Hour), and two-player games are often PSPACE-complete (as in Othello) or EXPTIME-complete (as in Check- ers, Chess, and Go). Surprisingly, many seemingly simple puzzles and games are also hard. Other results are positive, proving that some games can be played optimally in polynomial time. In some cases, particularly with one-player puzzles, the computationally tractable games are still interesting for humans to play. We begin by reviewing some basics of Combinatorial Game Theory in Section 2, which gives tools for designing algorithms, followed by reviewing the relatively new theory of Constraint Logic in Section 3, which gives tools for proving hardness.
    [Show full text]
  • Exactly Solving Packing Problems with Fragmentation
    Exactly solving packing problems with fragmentation M. Casazza, A. Ceselli, Università degli Studi di Milano, Dipartimento di Informatica, OptLab – Via Bramante 65, 26013 – Crema (CR) – Italy {marco.casazza, alberto.ceselli}@unimi.it April 13, 2015 In packing problems with fragmentation a set of items of known weight is given, together with a set of bins of limited capacity; the task is to find an assignment of items to bins such that the sum of items assigned to the same bin does not exceed its capacity. As a distinctive feature, items can be split at a price, and fractionally assigned to different bins. Arising in diverse application fields, packing with fragmentation has been investigated in the literature from both theoretical, modeling, approximation and exact optimization points of view. We improve the theoretical understanding of the problem, and we introduce new models by exploiting only its combinatorial nature. We design new exact solution algorithms and heuristics based on these models. We consider also variants from the literature arising while including different objective functions and the option of handling weight overhead after splitting. We present experimental results on both datasets from the literature and new, more challenging, ones. These show that our algorithms are both flexible and effective, outperforming by orders of magnitude previous approaches from the literature for all the variants considered. By using our algorithms we could also assess the impact of explicitly handling split overhead, in terms of both solutions quality and computing effort. Keywords: Bin Packing, Item Fragmentation, Mathematical Programming, Branch&Price 1 1 Introduction Logistics has always been a benchmark for combinatorial optimization methodologies, as practitioners are traditionally familiar with the competitive advantage granted by optimized systems.
    [Show full text]
  • Solving Packing Problems with Few Small Items Using Rainbow Matchings
    Solving Packing Problems with Few Small Items Using Rainbow Matchings Max Bannach Institute for Theoretical Computer Science, Universität zu Lübeck, Lübeck, Germany [email protected] Sebastian Berndt Institute for IT Security, Universität zu Lübeck, Lübeck, Germany [email protected] Marten Maack Department of Computer Science, Universität Kiel, Kiel, Germany [email protected] Matthias Mnich Institut für Algorithmen und Komplexität, TU Hamburg, Hamburg, Germany [email protected] Alexandra Lassota Department of Computer Science, Universität Kiel, Kiel, Germany [email protected] Malin Rau Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, LIG, 38000 Grenoble, France [email protected] Malte Skambath Department of Computer Science, Universität Kiel, Kiel, Germany [email protected] Abstract An important area of combinatorial optimization is the study of packing and covering problems, such as Bin Packing, Multiple Knapsack, and Bin Covering. Those problems have been studied extensively from the viewpoint of approximation algorithms, but their parameterized complexity has only been investigated barely. For problem instances containing no “small” items, classical matching algorithms yield optimal solutions in polynomial time. In this paper we approach them by their distance from triviality, measuring the problem complexity by the number k of small items. Our main results are fixed-parameter algorithms for vector versions of Bin Packing, Multiple Knapsack, and Bin Covering parameterized by k. The algorithms are randomized with one-sided error and run in time 4k · k! · nO(1). To achieve this, we introduce a colored matching problem to which we reduce all these packing problems. The colored matching problem is natural in itself and we expect it to be useful for other applications.
    [Show full text]
  • Generic Branch-Cut-And-Price
    Generic Branch-Cut-and-Price Diplomarbeit bei PD Dr. M. L¨ubbecke vorgelegt von Gerald Gamrath 1 Fachbereich Mathematik der Technischen Universit¨atBerlin Berlin, 16. M¨arz2010 1Konrad-Zuse-Zentrum f¨urInformationstechnik Berlin, [email protected] 2 Contents Acknowledgments iii 1 Introduction 1 1.1 Definitions . .3 1.2 A Brief History of Branch-and-Price . .6 2 Dantzig-Wolfe Decomposition for MIPs 9 2.1 The Convexification Approach . 11 2.2 The Discretization Approach . 13 2.3 Quality of the Relaxation . 21 3 Extending SCIP to a Generic Branch-Cut-and-Price Solver 25 3.1 SCIP|a MIP Solver . 25 3.2 GCG|a Generic Branch-Cut-and-Price Solver . 27 3.3 Computational Environment . 35 4 Solving the Master Problem 39 4.1 Basics in Column Generation . 39 4.1.1 Reduced Cost Pricing . 42 4.1.2 Farkas Pricing . 43 4.1.3 Finiteness and Correctness . 44 4.2 Solving the Dantzig-Wolfe Master Problem . 45 4.3 Implementation Details . 48 4.3.1 Farkas Pricing . 49 4.3.2 Reduced Cost Pricing . 52 4.3.3 Making Use of Bounds . 54 4.4 Computational Results . 58 4.4.1 Farkas Pricing . 59 4.4.2 Reduced Cost Pricing . 65 5 Branching 71 5.1 Branching on Original Variables . 73 5.2 Branching on Variables of the Extended Problem . 77 5.3 Branching on Aggregated Variables . 78 5.4 Ryan and Foster Branching . 79 i ii Contents 5.5 Other Branching Rules . 82 5.6 Implementation Details . 85 5.6.1 Branching on Original Variables . 87 5.6.2 Ryan and Foster Branching .
    [Show full text]
  • Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity∗
    Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity∗ Erik D. Demaine, Martin L. Demaine MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA, {edemaine,mdemaine}@mit.edu Dedicated to Jin Akiyama in honor of his 60th birthday. Abstract. We show that jigsaw puzzles, edge-matching puzzles, and polyomino packing puzzles are all NP-complete. Furthermore, we show direct equivalences between these three types of puzzles: any puzzle of one type can be converted into an equivalent puzzle of any other type. 1. Introduction Jigsaw puzzles [37,38] are perhaps the most popular form of puzzle. The original jigsaw puzzles of the 1760s were cut from wood sheets using a hand woodworking tool called a jig saw, which is where the puzzles get their name. The images on the puzzles were European maps, and the jigsaw puzzles were used as educational toys, an idea still used in some schools today. Handmade wooden jigsaw puzzles for adults took off around 1900. Today, jigsaw puzzles are usually cut from cardboard with a die, a technology that became practical in the 1930s. Nonetheless, true addicts can still find craftsmen who hand-make wooden jigsaw puzzles. Most jigsaw puzzles have a guiding image and each side of a piece has only one “mate”, although a few harder variations have blank pieces and/or pieces with ambiguous mates. Edge-matching puzzles [21] are another popular puzzle with a similar spirit to jigsaw puzzles, first appearing in the 1890s. In an edge-matching puzzle, the goal is to arrange a given collection of several identically shaped but differently patterned tiles (typically squares) so that the patterns match up along the edges of adjacent tiles.
    [Show full text]
  • A Branch and Price Approach to the K-Clustering Minimum Biclique Completion Problem
    A Branch and Price Approach to the k-Clustering Minimum Biclique Completion Problem Stefano Gualandia,1, Francesco Maffiolib, Claudio Magnic aDipartimento di Matematica, Universit`adegli Studi di Pavia, Via Ferrata 1, 27100, Pavia, Italy bPolitecnico di Milano, Dipartimento di Elettronica e Informazione, Piazza Leonardo da Vinci 32, 20133 Milano, Italy cMax Planck Institute for Computer Science, Department 1: Algorithms and Complexity, Campus E1 4, 66123 Saarbr¨ucken, Germany Abstract Given a bipartite graph G = (S, T, E), we consider the problem of finding k bipartite subgraphs, called ”clusters”, such that each vertex i of S appears in exactly one of them, every vertex j of T appears in each cluster in which at least one of its neighbors appears, and the total number of edges needed to make each cluster complete (i.e., to become a biclique) is minimized. This problem is known as k-clustering Minimum Biclique Completion Problem and has been shown strongly NP-hard. It has applications in bundling channels for multicast transmissions. Given a set of demands of services from clients, the application consists of finding k multicast sessions that partition the set of demands. Each service has to belong to a single multicast session, while each client can appear in more sessions. We extend previous work by developing a Branch and Price algorithm that embeds a new metaheuristic based on Variable Neighborhood Infeasible Search and a non-trivial branching rule. The metaheuristic is also adapted to solve efficiently the pricing subproblem. In addition to the random instances used in the literature, we present structured instances generated using the MovieLens data set collected by the GroupLens Research Project.
    [Show full text]