Bin Completion Algorithms for Multicontainer Packing, Knapsack, and Covering Problems
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Approximation and Online Algorithms for Multidimensional Bin Packing: a Survey✩
Computer Science Review 24 (2017) 63–79 Contents lists available at ScienceDirect Computer Science Review journal homepage: www.elsevier.com/locate/cosrev Survey Approximation and online algorithms for multidimensional bin packing: A surveyI Henrik I. Christensen a, Arindam Khan b,∗,1, Sebastian Pokutta c, Prasad Tetali c a University of California, San Diego, USA b Istituto Dalle Molle di studi sull'Intelligenza Artificiale (IDSIA), Scuola universitaria professionale della Svizzera italiana (SUPSI), Università della Svizzera italiana (USI), Switzerland c Georgia Institute of Technology, Atlanta, USA article info a b s t r a c t Article history: The bin packing problem is a well-studied problem in combinatorial optimization. In the classical bin Received 13 August 2016 packing problem, we are given a list of real numbers in .0; 1U and the goal is to place them in a minimum Received in revised form number of bins so that no bin holds numbers summing to more than 1. The problem is extremely important 23 November 2016 in practice and finds numerous applications in scheduling, routing and resource allocation problems. Accepted 20 December 2016 Theoretically the problem has rich connections with discrepancy theory, iterative methods, entropy Available online 16 January 2017 rounding and has led to the development of several algorithmic techniques. In this survey we consider approximation and online algorithms for several classical generalizations of bin packing problem such Keywords: Approximation algorithms as geometric bin packing, vector bin packing and various other related problems. There is also a vast Online algorithms literature on mathematical models and exact algorithms for bin packing. -
15.083 Lecture 21: Approximation Algorithsm I
15.083J Integer Programming and Combinatorial Optimization Fall 2009 Approximation Algorithms I The knapsack problem • Input: nonnegative numbers p1; : : : ; pn; a1; : : : ; an; b. n X max pj xj j=1 n X s.t. aj xj ≤ b j=1 n x 2 Z+ Additive performance guarantees Theorem 1. There is a polynomial-time algorithm A for the knapsack problem such that A(I) ≥ OP T (I) − K for all instances I (1) for some constant K if and only if P = NP. Proof: • Let A be a polynomial-time algorithm satisfying (1). • Let I = (p1; : : : ; pn; a1; : : : ; an; b) be an instance of the knapsack problem. 0 0 0 • Let I = (p1 := (K + 1)p1; : : : ; pn := (K + 1)pn; a1; : : : ; an; b) be a new instance. • Clearly, x∗ is optimal for I iff it is optimal for I0. • If we apply A to I0 we obtain a solution x0 such that p0x∗ − p0x0 ≤ K: • Hence, 1 K px∗ − px0 = (p0x∗ − p0x0) ≤ < 1: K + 1 K + 1 • Since px0 and px∗ are integer, it follows that px0 = px∗, that is x0 is optimal for I. • The other direction is trivial. • Note that this technique applies to any combinatorial optimization problem with linear ob jective function. 1 Approximation algorithms • There are few (known) NP-hard problems for which we can find in polynomial time solutions whose value is close to that of an optimal solution in an absolute sense. (Example: edge coloring.) • In general, an approximation algorithm for an optimization Π produces, in polynomial time, a feasible solution whose objective function value is within a guaranteed factor of that of an optimal solution. -
Rounding Algorithms for Covering Problems
Mathematical Programming 80 (1998) 63 89 Rounding algorithms for covering problems Dimitris Bertsimas a,,,1, Rakesh Vohra b,2 a Massachusetts Institute of Technology, Sloan School of Management, 50 Memorial Drive, Cambridge, MA 02142-1347, USA b Department of Management Science, Ohio State University, Ohio, USA Received 1 February 1994; received in revised form 1 January 1996 Abstract In the last 25 years approximation algorithms for discrete optimization problems have been in the center of research in the fields of mathematical programming and computer science. Re- cent results from computer science have identified barriers to the degree of approximability of discrete optimization problems unless P -- NP. As a result, as far as negative results are con- cerned a unifying picture is emerging. On the other hand, as far as particular approximation algorithms for different problems are concerned, the picture is not very clear. Different algo- rithms work for different problems and the insights gained from a successful analysis of a par- ticular problem rarely transfer to another. Our goal in this paper is to present a framework for the approximation of a class of integer programming problems (covering problems) through generic heuristics all based on rounding (deterministic using primal and dual information or randomized but with nonlinear rounding functions) of the optimal solution of a linear programming (LP) relaxation. We apply these generic heuristics to obtain in a systematic way many known as well as new results for the set covering, facility location, general covering, network design and cut covering problems. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V. -
Structural Graph Theory Meets Algorithms: Covering And
Structural Graph Theory Meets Algorithms: Covering and Connectivity Problems in Graphs Saeed Akhoondian Amiri Fakult¨atIV { Elektrotechnik und Informatik der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Rolf Niedermeier Gutachter: Prof. Dr. Stephan Kreutzer Gutachter: Prof. Dr. Marcin Pilipczuk Gutachter: Prof. Dr. Dimitrios Thilikos Tag der wissenschaftlichen Aussprache: 13. October 2017 Berlin 2017 2 This thesis is dedicated to my family, especially to my beautiful wife Atefe and my lovely son Shervin. 3 Contents Abstract iii Acknowledgementsv I. Introduction and Preliminaries1 1. Introduction2 1.0.1. General Techniques and Models......................3 1.1. Covering Problems.................................6 1.1.1. Covering Problems in Distributed Models: Case of Dominating Sets.6 1.1.2. Covering Problems in Directed Graphs: Finding Similar Patterns, the Case of Erd}os-P´osaproperty.......................9 1.2. Routing Problems in Directed Graphs...................... 11 1.2.1. Routing Problems............................. 11 1.2.2. Rerouting Problems............................ 12 1.3. Structure of the Thesis and Declaration of Authorship............. 14 2. Preliminaries and Notations 16 2.1. Basic Notations and Defnitions.......................... 16 2.1.1. Sets..................................... 16 2.1.2. Graphs................................... 16 2.2. Complexity Classes................................ -
3.1 Matchings and Factors: Matchings and Covers
1 3.1 Matchings and Factors: Matchings and Covers This copyrighted material is taken from Introduction to Graph Theory, 2nd Ed., by Doug West; and is not for further distribution beyond this course. These slides will be stored in a limited-access location on an IIT server and are not for distribution or use beyond Math 454/553. 2 Matchings 3.1.1 Definition A matching in a graph G is a set of non-loop edges with no shared endpoints. The vertices incident to the edges of a matching M are saturated by M (M-saturated); the others are unsaturated (M-unsaturated). A perfect matching in a graph is a matching that saturates every vertex. perfect matching M-unsaturated M-saturated M Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. 3 Perfect Matchings in Complete Bipartite Graphs a 1 The perfect matchings in a complete b 2 X,Y-bigraph with |X|=|Y| exactly c 3 correspond to the bijections d 4 f: X -> Y e 5 Therefore Kn,n has n! perfect f 6 matchings. g 7 Kn,n The complete graph Kn has a perfect matching iff… Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. 4 Perfect Matchings in Complete Graphs The complete graph Kn has a perfect matching iff n is even. So instead of Kn consider K2n. We count the perfect matchings in K2n by: (1) Selecting a vertex v (e.g., with the highest label) one choice u v (2) Selecting a vertex u to match to v K2n-2 2n-1 choices (3) Selecting a perfect matching on the rest of the vertices. -
Quadratic Multiple Knapsack Problem with Setups and a Solution Approach
Proceedings of the 2012 International Conference on Industrial Engineering and Operations Management Istanbul, Turkey, July 3 – 6, 2012 Quadratic Multiple Knapsack Problem with Setups and a Solution Approach Nilay Dogan, Kerem Bilgiçer and Tugba Saraç Industrial Engineering Department Eskisehir Osmangazi University Meselik 26480, Eskisehir Turkey Abstract In this study, the quadratic multiple knapsack problem that include setup costs is considered. In this problem, if any item is assigned to a knapsack, a setup cost is incurred for the class which it belongs. The objective is to assign each item to at most one of the knapsacks such that none of the capacity constraints are violated and the total profit is maximized. QMKP with setups is NP-hard. Therefore, a GA based solution algorithm is proposed to solve it. The performance of the developed algorithm compared with the instances taken from the literature. Keywords Quadratic Multiple Knapsack Problem with setups, Genetic algorithm, Combinatorial Optimization. 1. Introduction The knapsack problem (KP) is one of the well-known combinatorial optimization problems. There are different types of knapsack problems in the literature. A comprehensive survey which can be considered as a through introduction to knapsack problems and their variants was published by Kellerer et al. (2004). The classical KP seeks to select, from a finite set of items, the subset, which maximizes a linear function of the items chosen, subject to a single inequality constraint. In many real life applications it is important that the profit of a packing also should reflect how well the given items fit together. One formulation of such interdependence is the quadratic knapsack problem. -
Approximate Algorithms for the Knapsack Problem on Parallel Computers
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector INFORMATION AND COMPUTATION 91, 155-171 (1991) Approximate Algorithms for the Knapsack Problem on Parallel Computers P. S. GOPALAKRISHNAN IBM T. J. Watson Research Center, P. 0. Box 218, Yorktown Heights, New York 10598 I. V. RAMAKRISHNAN* Department of Computer Science, State University of New York, Stony Brook, New York 11794 AND L. N. KANAL Department of Computer Science, University of Maryland, College Park, Maryland 20742 Computingan optimal solution to the knapsack problem is known to be NP-hard. Consequently, fast parallel algorithms for finding such a solution without using an exponential number of processors appear unlikely. An attractive alter- native is to compute an approximate solution to this problem rapidly using a polynomial number of processors. In this paper, we present an efficient parallel algorithm for hnding approximate solutions to the O-l knapsack problem. Our algorithm takes an E, 0 < E < 1, as a parameter and computes a solution such that the ratio of its deviation from the optimal solution is at most a fraction E of the optimal solution. For a problem instance having n items, this computation uses O(n51*/&3’2) processors and requires O(log3 n + log* n log( l/s)) time. The upper bound on the processor requirement of our algorithm is established by reducing it to a problem on weighted bipartite graphs. This processor complexity is a signiti- cant improvement over that of other known parallel algorithms for this problem. 0 1991 Academic Press, Inc 1. -
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. -
CS-258 Solving Hard Problems in Combinatorial Optimization Pascal
CS-258 Solving Hard Problems in Combinatorial Optimization Pascal Van Hentenryck Course Overview Study of combinatorial optimization from a practitioner standpoint • the main techniques and tools • several representative problems What is expected from you: • 7 team assignments (2 persons team) • Short write-up on the assignments • Final presentation of the results What you should not do in this class? • read papers/books/web/... on combinatorial optimization • Read the next slide during class Course Overview January 31: Knapsack * Assignment: Coloring February 7: LS + CP + LP + IP February 14: Linear Programming * Assignment: Supply Chain February 28: Integer Programming * Assignment: Green Zone March 7: Travelling Salesman Problem * Assignment: TSP March 14: Advanced Mathematical Programming March 21: Constraint Programming * Assignment: Nuclear Reaction April 4: Constraint-Based Scheduling April 11: Approximation Algorithms (Claire) * Assignment: Operation Hurricane April 19: Vehicle Routing April 25: Advanced Constraint Programming * Assignment: Special Ops. Knapsack Problem n c x max ∑ i i i = 1 subject to: n ∑ wi xi ≤ l i = 1 xi ∈ [0, 1] NP-Complete Problems Decision problems Polynomial-time verifiability • We can verify if a guess is a solution in polynomial time Hardest problem in NP (NP-complete) • If I solve one NP-complete problem, I solve them all Approaches Exponential Algorithms (sacrificing time) • dynamic programming • branch and bound • branch and cut • constraint programming Approximation Algorithms (sacrificing quality) -
0784085 Patrickdeenen
Eindhoven University of Technology MASTER Optimizing wafer allocation for back-end semiconductor production Deenen, Patrick C. Award date: 2018 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain Manufacturing Systems Engineering Department of Mechanical Engineering De Rondom 70, 5612 AP Eindhoven P.O. Box 513, 5600 MB Eindhoven The Netherlands www.tue.nl Author P.C. Deenen Optimizing wafer allocation for 0784085 back-end semiconductor production Committee members I.J.B.F. Adan A.E. Akçay C.A.J. Hurkens DC 2018.101 Master Thesis Report Company supervisors J. Adan J. Stokkermans Date November 29, 2018 Where innovation starts Optimizing wafer allocation for back-end semiconductor production Patrick Deenena,b,∗ aEindhoven University of Technology, Department of Mechanical Engineering, Eindhoven, The Netherlands bNexperia, Industrial Technology and Engineering Centre, Nijmegen, The Netherlands Abstract This paper discusses the development of an efficient and practical algorithm that minimizes overproduction in the allocation of wafers to customer orders prior to assembly at a semiconductor production facility. -
Online 3D Bin Packing with Constrained Deep Reinforcement Learning
Online 3D Bin Packing with Constrained Deep Reinforcement Learning Hang Zhao1, Qijin She1, Chenyang Zhu1, Yin Yang2, Kai Xu1,3* 1National University of Defense Technology, 2Clemson University, 3SpeedBot Robotics Ltd. Abstract RGB image Depth image We solve a challenging yet practically useful variant of 3D Bin Packing Problem (3D-BPP). In our problem, the agent has limited information about the items to be packed into a single bin, and an item must be packed immediately after its arrival without buffering or readjusting. The item’s place- ment also subjects to the constraints of order dependence and physical stability. We formulate this online 3D-BPP as a constrained Markov decision process (CMDP). To solve the problem, we propose an effective and easy-to-implement constrained deep reinforcement learning (DRL) method un- Figure 1: Online 3D-BPP, where the agent observes only a der the actor-critic framework. In particular, we introduce a limited numbers of lookahead items (shaded in green), is prediction-and-projection scheme: The agent first predicts a feasibility mask for the placement actions as an auxiliary task widely useful in logistics, manufacture, warehousing etc. and then uses the mask to modulate the action probabilities output by the actor during training. Such supervision and pro- problem) as many real-world challenges could be much jection facilitate the agent to learn feasible policies very effi- more efficiently handled if we have a good solution to it. A ciently. Our method can be easily extended to handle looka- good example is large-scale parcel packaging in modern lo- head items, multi-bin packing, and item re-orienting. -
1 Load Balancing / Multiprocessor Scheduling
CS 598CSC: Approximation Algorithms Lecture date: February 4, 2009 Instructor: Chandra Chekuri Scribe: Rachit Agarwal Fixed minor errors in Spring 2018. In the last lecture, we studied the KNAPSACK problem. The KNAPSACK problem is an NP- hard problem but does admit a pseudo-polynomial time algorithm and can be solved efficiently if the input size is small. We used this pseudo-polynomial time algorithm to obtain an FPTAS for KNAPSACK. In this lecture, we study another class of problems, known as strongly NP-hard problems. Definition 1 (Strongly NP-hard Problems) An NPO problem π is said to be strongly NP-hard if it is NP-hard even if the inputs are polynomially bounded in combinatorial size of the problem 1. Many NP-hard problems are in fact strongly NP-hard. If a problem Π is strongly NP-hard, then Π does not admit a pseudo-polynomial time algorithm. For more discussion on this, refer to [1, Section 8.3] We study two such problems in this lecture, MULTIPROCESSOR SCHEDULING and BIN PACKING. 1 Load Balancing / MultiProcessor Scheduling A central problem in scheduling theory is to design a schedule such that the finishing time of the last jobs (also called makespan) is minimized. This problem is often referred to as the LOAD BAL- ANCING, the MINIMUM MAKESPAN SCHEDULING or MULTIPROCESSOR SCHEDULING problem. 1.1 Problem Description In the MULTIPROCESSOR SCHEDULING problem, we are given m identical machines M1;:::;Mm and n jobs J1;J2;:::;Jn. Job Ji has a processing time pi ≥ 0 and the goal is to assign jobs to the machines so as to minimize the maximum load2.