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A Learning-Based Iterative Method for Solving Vehicle
Published as a conference paper at ICLR 2020 ALEARNING-BASED ITERATIVE METHOD FOR SOLVING VEHICLE ROUTING PROBLEMS Hao Lu ∗ Princeton University, Princeton, NJ 08540 [email protected] Xingwen Zhang ∗ & Shuang Yang Ant Financial Services Group, San Mateo, CA 94402 fxingwen.zhang,[email protected] ABSTRACT This paper is concerned with solving combinatorial optimization problems, in par- ticular, the capacitated vehicle routing problems (CVRP). Classical Operations Research (OR) algorithms such as LKH3 (Helsgaun, 2017) are inefficient and dif- ficult to scale to larger-size problems. Machine learning based approaches have recently shown to be promising, partly because of their efficiency (once trained, they can perform solving within minutes or even seconds). However, there is still a considerable gap between the quality of a machine learned solution and what OR methods can offer (e.g., on CVRP-100, the best result of learned solutions is between 16.10-16.80, significantly worse than LKH3’s 15.65). In this paper, we present “Learn to Improve” (L2I), the first learning based approach for CVRP that is efficient in solving speed and at the same time outperforms OR methods. Start- ing with a random initial solution, L2I learns to iteratively refine the solution with an improvement operator, selected by a reinforcement learning based controller. The improvement operator is selected from a pool of powerful operators that are customized for routing problems. By combining the strengths of the two worlds, our approach achieves the new state-of-the-art results on CVRP, e.g., an average cost of 15.57 on CVRP-100. 1 INTRODUCTION In this paper, we focus on an important class of combinatorial optimization, vehicle routing problems (VRP), which have a wide range of applications in logistics. -
7. NETWORK FLOW II ‣ Bipartite Matching ‣ Disjoint Paths
Soviet rail network (1950s) "Free world" goal. Cut supplies (if cold war turns into real war). 7. NETWORK FLOW II ‣ bipartite matching ‣ disjoint paths ‣ extensions to max flow ‣ survey design ‣ airline scheduling ‣ image segmentation ‣ project selection Lecture slides by Kevin Wayne ‣ baseball elimination Copyright © 2005 Pearson-Addison Wesley http://www.cs.princeton.edu/~wayne/kleinberg-tardos Reference: On the history of the transportation and maximum flow problems. Alexander Schrijver in Math Programming, 91: 3, 2002. Last updated on Mar 31, 2013 3:25 PM Figure 2 2 From Harris and Ross [1955]: Schematic diagram of the railway network of the Western So- viet Union and Eastern European countries, with a maximum flow of value 163,000 tons from Russia to Eastern Europe, and a cut of capacity 163,000 tons indicated as ‘The bottleneck’. Max-flow and min-cut applications Max-flow and min-cut are widely applicable problem-solving model. ・Data mining. 7. NETWORK FLOW II ・Open-pit mining. ・Bipartite matching. ‣ bipartite matching ・Network reliability. ‣ disjoint paths ・Baseball elimination. ‣ extensions to max flow ・Image segmentation. ・Network connectivity. ‣ survey design liver and hepatic vascularization segmentation ・Distributed computing. ‣ airline scheduling ・Security of statistical data. ‣ image segmentation ・Egalitarian stable matching. ・Network intrusion detection. ‣ project selection ・Multi-camera scene reconstruction. ‣ baseball elimination ・Sensor placement for homeland security. ・Many, many, more. 3 Matching Bipartite matching Def. Given an undirected graph G = (V, E) a subset of edges M ⊆ E is Def. A graph G is bipartite if the nodes can be partitioned into two subsets a matching if each node appears in at most one edge in M. -
Models for Networks with Consumable Resources: Applications to Smart Cities Hayato Montezuma Ushijima-Mwesigwa Clemson University, [email protected]
Clemson University TigerPrints All Dissertations Dissertations 12-2018 Models for Networks with Consumable Resources: Applications to Smart Cities Hayato Montezuma Ushijima-Mwesigwa Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_dissertations Recommended Citation Ushijima-Mwesigwa, Hayato Montezuma, "Models for Networks with Consumable Resources: Applications to Smart Cities" (2018). All Dissertations. 2284. https://tigerprints.clemson.edu/all_dissertations/2284 This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact [email protected]. Models for Networks with Consumable Resources: Applications to Smart Cities A Dissertation Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Computer Science by Hayato Ushijima-Mwesigwa December 2018 Accepted by: Dr. Ilya Safro, Committee Chair Dr. Mashrur Chowdhury Dr. Brian Dean Dr. Feng Luo Abstract In this dissertation, we introduce different models for understanding and controlling the spreading dynamics of a network with a consumable resource. In particular, we consider a spreading process where a resource necessary for transit is partially consumed along the way while being refilled at special nodes on the network. Examples include fuel consumption of vehicles together with refueling stations, information loss during dissemination with error correcting nodes, consumption of ammunition of military troops while moving, and migration of wild animals in a network with a limited number of water-holes. We undertake this study from two different perspectives. First, we consider a network science perspective where we are interested in identifying the influential nodes, and estimating a nodes’ relative spreading influence in the network. -
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................................ -
The Generalized Simplex Method for Minimizing a Linear Form Under Linear Inequality Restraints
Pacific Journal of Mathematics THE GENERALIZED SIMPLEX METHOD FOR MINIMIZING A LINEAR FORM UNDER LINEAR INEQUALITY RESTRAINTS GEORGE BERNARD DANTZIG,ALEXANDER ORDEN AND PHILIP WOLFE Vol. 5, No. 2 October 1955 THE GENERALIZED SIMPLEX METHOD FOR MINIMIZING A LINEAR FORM UNDER LINEAR INEQUALITY RESTRAINTS GEORGE B. DANTZIG, ALEX ORDEN, PHILIP WOLFE 1. Background and summary. The determination of "optimum" solutions of systems of linear inequalities is assuming increasing importance as a tool for mathematical analysis of certain problems in economics, logistics, and the theory of games [l;5] The solution of large systems is becoming more feasible with the advent of high-speed digital computers; however, as in the related problem of inversion of large matrices, there are difficulties which remain to be resolved connected with rank. This paper develops a theory for avoiding as- sumptions regarding rank of underlying matrices which has import in applica- tions where little or nothing is known about the rank of the linear inequality system under consideration. The simplex procedure is a finite iterative method which deals with problems involving linear inequalities in a manner closely analogous to the solution of linear equations or matrix inversion by Gaussian elimination. Like the latter it is useful in proving fundamental theorems on linear algebraic systems. For example, one form of the fundamental duality theorem associated with linear inequalities is easily shown as a direct consequence of solving the main prob- lem. Other forms can be obtained by trivial manipulations (for a fuller discus- sion of these interrelations, see [13]); in particular, the duality theorem [8; 10; 11; 12] leads directly to the Minmax theorem for zero-sum two-person games [id] and to a computational method (pointed out informally by Herman Rubin and demonstrated by Robert Dorfman [la]) which simultaneously yields optimal strategies for both players and also the value of the game. -
Network Flow - Theory and Applications with Practical Impact
3 Network flow - Theory and applications with practical impact Masao Iri Department of Information and System Engineering Faculty of Science and Engineering, Chuo University 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112, Japan. Tel: +81-3-3817-1690. Fax: +81-3-3817-1681. e-mail: [email protected] Keywords Network flow, history, applications, duality, future 1 INTRODUCTION The research in this subject has now a long history and a large variety of related fields in science, engineering and operations research, and is still developing fast, especially in its algorithmic aspects, attracting many excellent senior as well as young researchers. And the space allotted to this paper is limited. So, in this paper I will not lay so much stress on particular frontiers of research in the subject as on the wider and historical viewpoint. Network-flow theory is one of the best studied and developed fields of optimization, and has important relations to quite different fields of science and technology such as com binatorial mathematics, algebraic topology, electric circuit theory, nonlinear continuum theory including plasticity theory, geographic information systems, VLSI design, etc., etc., besides standard applications to transportation, scheduling, etc. in operations research. An important point of view from which we should look at network flow theory is the "duality" viewpoint, to which due attention was paid at the earlier stages of development of the theory but which tends to be gradually ignored as time goes. Since the papers and books published on network flow are too many to cite here, I do not intend to compile even an incomplete list of references but only to cite a few which I directly mention and which I think tend to become obscured during the long history in spite of their ever lasting importance. -
Inequity in Searching for Knowledge in Organizations
The World is Not Small for Everyone: Inequity in Searching for Knowledge in Organizations _______________ Jasjit SINGH Morten HANSEN Joel PODOLNY 2010/11/ST/EFE (Revised version of 2009/49/ST/EFE) The World is Not Small for Everyone: Inequity in Searching for Knowledge in Organizations By Jasjit Singh * Morten T. Hansen** Joel M. Podolny*** ** February 2010 Revised version of INSEAD Working Paper 2009/49/ST/EFE * Assistant Professor of Strategy at INSEAD, 1 Ayer Rajah Avenue, Singapore 138676 Ph: +65 6799 5341 E-Mail: [email protected] ** Professor of Entrepreneurship at INSEAD, Boulevard de Constance, Fontainebleau 77305 France Ph: + 33 (0) 1 60 72 40 20 E-Mail: [email protected] *** Professor at Apple University, 1 Infinite Loop, MS 301-4AU, Cupertino, CA 95014, USA Email: [email protected] A working paper in the INSEAD Working Paper Series is intended as a means whereby a faculty researcher's thoughts and findings may be communicated to interested readers. The paper should be considered preliminary in nature and may require revision. Printed at INSEAD, Fontainebleau, France. Kindly do not reproduce or circulate without permission. ABSTRACT We explore why some employees may be at a disadvantage in searching for information in large complex organizations. The “small world” argument in social network theory emphasizes that people are on an average only a few connections away from the information they seek. However, we argue that such a network structure may benefit some people more than others. Specifically, some employees may have longer search paths in locating knowledge in an organization—their world may be large. -
An Efficient Three-Term Iterative Method for Estimating Linear
mathematics Article An Efficient Three-Term Iterative Method for Estimating Linear Approximation Models in Regression Analysis Siti Farhana Husin 1,*, Mustafa Mamat 1, Mohd Asrul Hery Ibrahim 2 and Mohd Rivaie 3 1 Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Terengganu 21300, Malaysia; [email protected] 2 Faculty of Entrepreneurship and Business, Universiti Malaysia Kelantan, Kelantan 16100, Malaysia; [email protected] 3 Department of Computer Sciences and Mathematics, Universiti Teknologi Mara, Terengganu 54000, Malaysia; [email protected] * Correspondence: [email protected] Received: 15 April 2020; Accepted: 20 May 2020; Published: 15 June 2020 Abstract: This study employs exact line search iterative algorithms for solving large scale unconstrained optimization problems in which the direction is a three-term modification of iterative method with two different scaled parameters. The objective of this research is to identify the effectiveness of the new directions both theoretically and numerically. Sufficient descent property and global convergence analysis of the suggested methods are established. For numerical experiment purposes, the methods are compared with the previous well-known three-term iterative method and each method is evaluated over the same set of test problems with different initial points. Numerical results show that the performances of the proposed three-term methods are more efficient and superior to the existing method. These methods could also produce an approximate linear regression equation to solve the regression model. The findings of this study can help better understanding of the applicability of numerical algorithms that can be used in estimating the regression model. Keywords: steepest descent method; large-scale unconstrained optimization; regression model 1. -
Network Flow-Based Refinement for Multilevel Hypergraph Partitioning
Network Flow-Based Refinement for Multilevel Hypergraph Partitioning Tobias Heuer Karlsruhe Institute of Technology, Germany [email protected] Peter Sanders Karlsruhe Institute of Technology, Germany [email protected] Sebastian Schlag Karlsruhe Institute of Technology, Germany [email protected] Abstract We present a refinement framework for multilevel hypergraph partitioning that uses max-flow computations on pairs of blocks to improve the solution quality of a k-way partition. The frame- work generalizes the flow-based improvement algorithm of KaFFPa from graphs to hypergraphs and is integrated into the hypergraph partitioner KaHyPar. By reducing the size of hypergraph flow networks, improving the flow model used in KaFFPa, and developing techniques to improve the running time of our algorithm, we obtain a partitioner that computes the best solutions for a wide range of benchmark hypergraphs from different application areas while still having a running time comparable to that of hMetis. 2012 ACM Subject Classification Mathematics of computing → Graph algorithms Keywords and phrases Multilevel Hypergraph Partitioning, Network Flows, Refinement Digital Object Identifier 10.4230/LIPIcs.SEA.2018.1 Related Version A full version of the paper is available at https://arxiv.org/abs/1802. 03587. 1 Introduction Given an undirected hypergraph H = (V, E), the k-way hypergraph partitioning problem is to partition the vertex set into k disjoint blocks of bounded size (at most 1 + ε times the average block size) such that an objective function involving the cut hyperedges is minimized. Hypergraph partitioning (HGP) has many important applications in practice such as scientific computing [10] or VLSI design [40]. Particularly VLSI design is a field where small improvements can lead to significant savings [53]. -
Chemical Reaction Optimization for Max Flow Problem
(IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 7, No. 8, 2016 Chemical Reaction Optimization for Max Flow Problem Reham Barham Ahmad Sharieh Azzam Sliet Department of Computer Science Department of Computer Science Department of Computer Science King Abdulla II School for King Abdulla II School for King Abdulla II School for Information and Technology Information and Technology Information and Technology The University of Jordan The University of Jordan The University of Jordan Amman, Jordan Amman, Jordan Amman, Jordan Abstract—This study presents an algorithm for MaxFlow represents the flow capacity of the edge. Under these problem using "Chemical Reaction Optimization algorithm constraints, we want to maximize the total flow from the (CRO)". CRO is a recently established meta-heuristics algorithm source to the sink". In [3] they define it as " In deterministic for optimization, inspired by the nature of chemical reactions. networks, the maximum-flow problem asks to send as much The main concern is to find the best maximum flow value at flow (information or goods) from a source to a destination, which the flow can be shipped from the source node to the sink without exceeding the capacity of any of the used links. node in a flow network without violating any capacity constraints Solving maximum-flow problems is for instance important to in which the flow of each edge remains within the upper bound avoid congestion and improve network utilization in computer value of the capacity. The proposed MaxFlow-CRO algorithm is networks or data centers or to improve fault tolerance". presented, analyzed asymptotically and experimental test is conducted. -
Revised Simplex Algorithm - 1
Lecture 9: Linear Programming: Revised Simplex and Interior Point Methods (a nice application of what we have learned so far) Prof. Krishna R. Pattipati Dept. of Electrical and Computer Engineering University of Connecticut Contact: [email protected] (860) 486-2890 ECE 6435 Fall 2008 Adv Numerical Methods in Sci Comp October 22, 2008 Copyright ©2004 by K. Pattipati Lecture Outline What is Linear Programming (LP)? Why do we need to solve Linear-Programming problems? • L1 and L∞ curve fitting (i.e., parameter estimation using 1-and ∞-norms) • Sample LP applications • Transportation Problems, Shortest Path Problems, Optimal Control, Diet Problem Methods for solving LP problems • Revised Simplex method • Ellipsoid method….not practical • Karmarkar’s projective scaling (interior point method) Implementation issues of the Least-Squares subproblem of Karmarkar’s method ….. More in Linear Programming and Network Flows course Comparison of Simplex and projective methods References 1. Dimitris Bertsimas and John N. Tsisiklis, Introduction to Linear Optimization, Athena Scientific, Belmont, MA, 1997. 2. I. Adler, M. G. C. Resende, G. Vega, and N. Karmarkar, “An Implementation of Karmarkar’s Algorithm for Linear Programming,” Mathematical Programming, Vol. 44, 1989, pp. 297-335. 3. I. Adler, N. Karmarkar, M. G. C. Resende, and G. Vega, “Data Structures and Programming Techniques for the Implementation of Karmarkar’s Algorithm,” ORSA Journal on Computing, Vol. 1, No. 2, 1989. 2 Copyright ©2004 by K. Pattipati What is Linear Programming? One of the most celebrated problems since 1951 • Major breakthroughs: • Dantzig: Simplex method (1947-1949) • Khachian: Ellipsoid method (1979) - Polynomial complexity, but not competitive with the Simplex → not practical. -
An Introduction
Index G(n; m) model, 107 behavior analysis methodology, G(n; p) model, 106 320 k-clan, 183 betweenness centrality, 78 k-clique, 182 BFS, see breadth-first search k-club, 183 big data, 13 k-means, 156 big data paradox, 13 k-plex, 180 bipartite graph, 44 botnet, 171 actor, see node breadth-first search, 47 bridge, 45 Adamic and Adar measure, 324 bridge detection, 60 adjacency list, 32 adjacency matrix, 31 cascade maximization, 225 adopter types, 229 causality testing, 321 adoption of hybrid seed corn, 229 centrality, 70 affiliation network, 44 betweenness centrality, 78 Anderson, Lisa R., 218 computing, 78 Asch conformity experiment, 216 closeness centrality, 80 Asch, Solomon, 216 degree centrality, 70 assortativity, 255 gregariousness, 70 measuring, 257 normalized, 71 nominal attributes, 258 prestige, 70 ordinal attributes, 261 eigenvector centrality, 71 significance, 258 group centrality, 81 attribute, see feature betweenness, 82 average path length, 102, 105 closeness, 82 373 degree, 81 evolution, 193 Katz centrality, 74 explicit, 173 divergence in computation, implicit, 174 75 community detection, 175 PageRank, 76 group-based, 175, 184 Christakis, Nicholas A., 245 member-based, 175, 177 citizen journalist, 11 node degree, 178 class, see class attribute node reachability, 181 class attribute, 133 node similarity, 183 clique identification, 178 community evaluation, 200 clique percolation method, 180 community membership behav- closeness centrality, 80 ior, 317 cluster centroid, 156 commute time, 327 clustering, 155, 156 confounding, 255 k-means,