DOCSLIB.ORG

IEOR 266: Graph Algorithms and Network Flows

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
IEOR 266: Graph Algorithms and Network Flows Lecture Notes for IEOR 266: Graph Algorithms and Network Flows Professor Dorit S. Hochbaum Contents 1 Introduction 1 1.1 Assignment problem .................................... 1 1.2 Basic graph definitions ................................... 2 2 Totally unimodular matrices 4 3 Minimum cost network flow problem 5 3.1 Transportation problem .................................. 7 3.2 The maximum flow problem ................................ 8 3.3 The shortest path problem ................................. 8 3.4 The single source shortest paths ............................. 9 3.5 The bipartite matching problem ............................. 11 3.6 Summary of classes of network flow problems ...................... 12 3.7 Negative cost cycles in graphs ............................... 13 4 Other network problems 14 4.1 Eulerian tour ........................................ 14 4.1.1 Eulerian walk .................................... 15 4.2 Chinese postman problem ................................. 15 4.2.1 Undirected chinese postman problem ....................... 16 4.2.2 Directed chinese postman problem ........................ 16 4.2.3 Mixed chinese postman problem ......................... 16 4.3 Hamiltonian tour problem ................................. 16 4.4 Traveling salesman problem (TSP) ............................ 16 4.5 Vertex packing problems (Independent Set) ....................... 17 4.6 Maximum clique problem ................................. 17 4.7 Vertex cover problem .................................... 17 4.8 Edge cover problem .................................... 18 4.9 b-factor (b-matching) problem ............................... 18 4.10 Graph colorability problem ................................ 18 4.11 Minimum spanning tree problem ............................. 19 5 Complexity analysis 19 5.1 Measuring quality of an algorithm ............................ 19 5.1.1 Examples ...................................... 20 5.2 Growth of functions .................................... 23 i IEOR 266 notes: Updated 2018 ii 5.3 Definitions for asymptotic comparisons of functions ................... 23 5.4 Properties of asymptotic notation ............................. 23 5.5 Caveats of complexity analysis .............................. 24 5.6 A sketch of the ellipsoid method ............................. 24 6 Graph representations 25 6.1 Node-arc adjacency matrix ................................ 25 6.2 Node-node adjacency matrix ............................... 26 6.3 Node-arc adjacency list .................................. 26 6.4 Comparison of the graph representations ......................... 27 6.4.1 Storage efficiency comparison ........................... 27 6.4.2 Advantages and disadvantages comparison .................... 27 7 Graph search algorithms 27 7.1 Generic search algorithm .................................. 27 7.2 Breadth first search (BFS) ................................. 28 7.3 Depth first search (DFS) .................................. 29 7.4 Applications of BFS and DFS ............................... 30 7.4.1 Checking if a graph is strongly connected .................... 30 7.4.2 Checking if a graph is acyclic ........................... 31 7.4.3 Checking if a graph is bipartite .......................... 31 8 Shortest paths 31 8.1 Introduction ......................................... 31 8.2 Properties of DAGs ..................................... 32 8.2.1 Topological sort and directed acyclic graphs ................... 32 8.3 Properties of shortest paths ................................ 33 8.4 Alternative formulation for SP from s to t ........................ 34 8.5 Shortest paths on a directed acyclic graph (DAG) .................... 35 8.6 Applications of the shortest/longest path problem on a DAG . 35 8.7 Dijkstra's algorithm .................................... 39 8.8 Bellman-Ford algorithm for single source shortest paths . 41 8.9 Floyd-Warshall algorithm for all pairs shortest paths . 43 8.10 D.B. Johnson's algorithm for all pairs shortest paths . 44 8.11 Matrix multiplication algorithm for all pairs shortest paths . 45 8.12 Why finding shortest paths in the presence of negative cost cycles is difficult . 46 9 Maximum flow problem 47 9.1 Introduction ......................................... 47 9.2 Linear programming duality and max flow min cut ................... 48 9.3 Applications ......................................... 49 9.3.1 Hall's theorem ................................... 49 9.3.2 The selection problem ............................... 50 9.3.3 Vertex cover on bipartite graphs ......................... 53 9.3.4 Forest clearing ................................... 54 9.3.5 Producing memory chips (VLSI layout) ..................... 55 9.4 Flow Decomposition .................................... 56 9.5 Algorithms ......................................... 57 IEOR 266 notes: Updated 2018 iii 9.5.1 Ford-Fulkerson algorithm ............................. 58 9.5.2 Maximum capacity augmenting path algorithm . 60 9.5.3 Capacity scaling algorithm ............................ 61 9.5.4 Dinic's algorithm for maximum flow ....................... 63 10 Goldberg's Algorithm - the Push/Relabel Algorithm 68 10.1 Overview .......................................... 68 10.2 The Generic Algorithm .................................. 69 10.3 Variants of Push/Relabel Algorithm ........................... 72 10.4 Wave Implementation of Goldberg's Algorithm (Lift-to-front) . 73 11 The pseudoflow algorithm (Hochbaum's PseudoFlow HPF) 74 11.1 Initialization ........................................ 74 11.2 A labeling pseudoflow algorithm ............................. 75 11.3 The monotone pseudoflow algorithm ........................... 75 11.4 Complexity summary .................................... 76 12 The minimum spanning tree problem (MST) 79 12.1 IP formulation ....................................... 79 12.2 Properties of MST ..................................... 80 12.2.1 Cut Optimality Condition ............................. 80 12.2.2 Path Optimality Condition ............................ 80 12.3 Algorithms of MST ..................................... 81 12.3.1 Prim's algorithm .................................. 82 12.3.2 Kruskal's algorithm ................................ 83 12.4 Maximum spanning tree .................................. 83 13 Complexity classes and NP-completeness 85 13.1 Search vs. Decision ..................................... 85 13.2 The class NP ........................................ 86 13.2.1 Some Problems in NP ............................... 86 13.3 co-NP ............................................ 87 13.3.1 Some Problems in co-NP ............................. 87 13.4 NP and co-NP ....................................... 87 13.5 NP-completeness and reductions ............................. 88 13.5.1 Reducibility ..................................... 88 13.5.2 NP-Completeness .................................. 89 14 Approximation algorithms 93 14.1 Traveling salesperson problem (TSP) ........................... 94 15 Integer programs with two variables per inequality (IP2) 95 15.1 A practical example: open-pit mining .......................... 95 15.2 The maximum closure problem .............................. 97 15.3 Monotone IP2 ........................................100 15.4 Non{monotone IP2 .....................................102 15.5 2-approximations for integer programs with two variables per inequality . 103 15.5.1 The quality of the bound provided by the monotonized system . 103 15.5.2 Computing an approximate solution . 104 IEOR 266 notes: Updated 2018 iv 15.6 Vertex cover problem ....................................105 15.6.1 Vertex cover on bipartite graphs . 105 15.6.2 Vertex cover on general graphs . 106 15.7 The s-excess problem ....................................108 16 Necessary and Sufficient Condition for Feasible Flow in a Network 109 16.1 For a network with zero lower bounds . 109 16.2 In the presence of positive lower bounds . 110 16.3 For a circulation problem .................................110 16.4 For the transportation problem ..............................110 17 Planar Graphs 111 17.1 Properties of Planar Graphs ................................112 17.2 Geometric Dual of Planar Graph .............................113 17.3 s-t Planar Flow Network ..................................114 17.4 Min Cut in an Undirected Weighted s-t Planar Flow Network . 114 17.5 Max Flow in a Undirected Weighted s-t Planar Flow Network . 114 18 Cut Problems and Algorithms 115 18.1 Network Connectivity ...................................115 18.2 Matula's Algorithm .....................................116 18.3 Brief Review of Additional Results ............................117 19 Algorithms for MCNF 118 19.1 Network Simplex ......................................118 19.2 (T,L,U) structure of an optimal solution . 119 19.3 Simplex' basic solutions and the corresponding spanning trees . 119 19.4 Optimality Conditions ...................................120 19.5 Implied algorithms { Cycle cancelling . 121 19.6 Solving maximum flow as MCNF .............................122 19.7 The gap between the primal and dual . 123 These notes are based on \scribe" notes taken by students attending Professor Hochbaum's course IEOR 266. The current version has been updated and edited by Professor Hochbaum in fall 2017. The text book used for the course, and mentioned in the notes, is Network Flows: theory, algo- rithms and applications by Ravindra K. Ahuja, Thomas L. Magnanti and James B. Orlin. Published by Prentice-Hall,
Load more

Recommended publications
  • Linear Sum Assignment with Edition Sébastien Bougleux, Luc Brun
    Linear Sum Assignment with Edition Sébastien Bougleux, Luc Brun To cite this version: Sébastien Bougleux, Luc Brun. Linear Sum Assignment with Edition. [Research Report] Normandie Université; GREYC CNRS UMR 6072. 2016. hal-01288288v3 HAL Id: hal-01288288 https://hal.archives-ouvertes.fr/hal-01288288v3 Submitted on 21 Mar 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NoDerivatives| 4.0 International License Linear Sum Assignment with Edition S´ebastienBougleux and Luc Brun Normandie Universit´e GREYC UMR 6072 CNRS - Universit´ede Caen Normandie - ENSICAEN Caen, France March 21, 2016 Abstract We consider the problem of transforming a set of elements into another by a sequence of elementary edit operations, namely substitutions, removals and insertions of elements. Each possible edit operation is penalized by a non-negative cost and the cost of a transformation is measured by summing the costs of its operations. A solution to this problem consists in defining a transformation having a minimal cost, among all possible transformations. To compute such a solution, the classical approach consists in representing removal and insertion operations by augmenting the two sets so that they get the same size.
    [Show full text]
  • Asymptotic Moments of the Bottleneck Assignment Problem
    Asymptotic Moments of the Bottleneck Assignment Problem Michael Z. Spivey Department of Mathematics and Computer Science, University of Puget Sound, Tacoma, WA 98416-1043 email: [email protected] http://math.pugetsound.edu/~mspivey/ One of the most important variants of the standard linear assignment problem is the bottleneck assignment problem. In this paper we give a method by which one can find all of the asymptotic moments of a random bottleneck assignment problem in which costs (independent and identically distributed) are chosen from a wide variety of continuous distributions. Our method is obtained by determining the asymptotic moments of the time to first complete matching in a random bipartite graph process and then transforming those, via a Maclaurin series expansion for the inverse cumulative distribution function, into the desired moments for the bottleneck assignment problem. Our results improve on the previous best-known expression for the expected value of a random bottleneck assignment problem, yield the first results on moments other than the expected value, and produce the first results on the moments for the time to first complete matching in a random bipartite graph process. Key words: bottleneck assignment problem; random assignment problem; asymptotic analysis; probabilistic anal- ysis; complete matching; degree 1; random bipartite graph MSC2000 Subject Classification: Primary: 90C47, 90C27, 41A60; Secondary: 60C05, 41A58 OR/MS subject classification: Primary: networks/graphs { theory, programming { integer { theory; Secondary: mathematics { combinatorics, mathematics { matrices 1. Introduction. The behavior of random assignment problems has been the subject of much study ∗ in the last few years. One of the most well-known results, due to Aldous [2], is that if Zn is the opti- mal value of an n × n random linear assignment problem with independent and identically distributed ∗ 2 (iid) exponential(1) costs then limn!1 E[Zn] = (2) = =6.
    [Show full text]
  • The Optimal Assignment Problem: an Investigation Into Current Solutions, New Approaches and the Doubly Stochastic Polytope
    The Optimal Assignment Problem: An Investigation Into Current Solutions, New Approaches and the Doubly Stochastic Polytope Frans-Willem Vermaak A Dissertation submitted to the Faculty of Engineering, University of the Witwatersand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering Johannesburg 2010 Declaration I declare that this research dissertation is my own, unaided work. It is being submitted for the Degree of Master of Science in Engineering in the University of the Witwatersrand, Johannesburg. It has not been submitted before for any degree or examination in any other University. signed this day of 2010 ii Abstract This dissertation presents two important results: a novel algorithm that approximately solves the optimal assignment problem as well as a novel method of projecting matrices into the doubly stochastic polytope while preserving the optimal assignment. The optimal assignment problem is a classical combinatorial optimisation problem that has fuelled extensive research in the last century. The problem is concerned with a matching or assignment of elements in one set to those in another set in an optimal manner. It finds typical application in logistical optimisation such as the matching of operators and machines but there are numerous other applications. In this document a process of iterative weighted normalization applied to the benefit matrix associated with the Assignment problem is considered. This process is derived from the application of the Computational Ecology Model to the assignment problem and referred to as the OACE (Optimal Assignment by Computational Ecology) algorithm. This simple process of iterative weighted normalisation converges towards a matrix that is easily converted to a permutation matrix corresponding to the optimal assignment or an assignment close to optimality.
    [Show full text]
  • He Stable Marriage Problem, and LATEX Dark Magic
    UCLA UCLA Electronic Theses and Dissertations Title Distributed Almost Stable Matchings Permalink https://escholarship.org/uc/item/6pr8d66m Author Rosenbaum, William Bailey Publication Date 2016 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles Distributed Almost Stable Matchings A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by William Bailey Rosenbaum óþÕä © Copyright by William Bailey Rosenbaum óþÕä ABSTRACT OF THE DISSERTATION Distributed Almost Stable Matchings by William Bailey Rosenbaum Doctor of Philosophy in Mathematics University of California, Los Angeles, óþÕä Professor Rafail Ostrovsky, Chair e Stable Marriage Problem (SMP) is concerned with the follow scenario: suppose we have two disjoint sets of agents—for example prospective students and colleges, medical residents and hospitals, or potential romantic partners—who wish to be matched into pairs. Each agent has preferences in the form of a ranking of her potential matches. How should we match agents based on their preferences? We say that a matching is stable if no unmatched pair of agents mutually prefer each other to their assigned partners. In their seminal work on the SMP, Gale and Shapley [ÕÕ] prove that a stable matching exists for any preferences. ey further describe an ecient algorithm for nding a stable matching. In this dissertation, we consider the computational complexity of the SMP in the distributed setting, and the complexity of nding “almost sta- ble” matchings. Highlights include (Õ) communication lower bounds for nding stable and almost stable matchings, (ó) a distributed algorithm which nds an almost stable matching in polylog time, and (ì) hardness of approximation results for three dimensional analogues of the SMP.
    [Show full text]
  • The Assignment Problem and Totally Unimodular Matrices
    Methods and Models for Combinatorial Optimization The assignment problem and totally unimodular matrices L.DeGiovanni M.DiSumma G.Zambelli 1 The assignment problem Let G = (V,E) be a bipartite graph, where V is the vertex set and E is the edge set. Recall that “bipartite” means that V can be partitioned into two disjoint subsets V1, V2 such that, for every edge uv ∈ E, one of u and v is in V1 and the other is in V2. In the assignment problem we have |V1| = |V2| and there is a cost cuv for every edge uv ∈ E. We want to select a subset of edges such that every node is the end of exactly one selected edge, and the sum of the costs of the selected edges is minimized. This problem is called “assignment problem” because selecting edges with the above property can be interpreted as assigning each node in V1 to exactly one adjacent node in V2 and vice versa. To model this problem, we define a binary variable xuv for every u ∈ V1 and v ∈ V2 such that uv ∈ E, where 1 if u is assigned to v (i.e., edge uv is selected), x = uv 0 otherwise. The total cost of the assignment is given by cuvxuv. uvX∈E The condition that for every node u in V1 exactly one adjacent node v ∈ V2 is assigned to v1 can be modeled with the constraint xuv =1, u ∈ V1, v∈VX2:uv∈E while the condition that for every node v in V2 exactly one adjacent node u ∈ V1 is assigned to v2 can be modeled with the constraint xuv =1, v ∈ V2.
    [Show full text]
  • Linear Sum Assignment with Edition
    Linear Sum Assignment with Edition S´ebastienBougleux and Luc Brun Normandie Universit´e GREYC UMR 6072 CNRS - Universit´ede Caen Normandie - ENSICAEN Caen, France October 10, 2018 Abstract We consider the problem of transforming a set of elements into another by a sequence of elementary edit operations, namely substitutions, removals and insertions of elements. Each possible edit operation is penalized by a non-negative cost and the cost of a transformation is measured by summing the costs of its operations. A solution to this problem consists in defining a transformation having a minimal cost, among all possible transformations. To compute such a solution, the classical approach consists in representing removal and insertion operations by augmenting the two sets so that they get the same size. This allows to express the problem as a linear sum assignment problem (LSAP), which thus finds an optimal bijection (or permutation, perfect matching) between the two augmented sets. While the LSAP is known to be efficiently solvable in polynomial time complexity, for instance with the Hungarian algorithm, useless time and memory are spent to treat the elements which have been added to the initial sets. In this report, we show that the problem can be formalized as an extension of the LSAP which considers only one additional element in each set to represent removal and insertion operations. A solution to the problem is no longer represented as a bijection between the two augmented sets. We show that the considered problem is a binary linear program (BLP) very close to the LSAP. While it can be solved by any BLP solver, we propose an adaptation of the Hungarian algorithm which improves the time and memory complexities previously obtained by the approach based on the LSAP.
    [Show full text]
  • Assignment Problems
    Assignment Problems y Eranda C ela Abstract Assignment problems arise in dierent situations where we have to nd an optimal way to assign n ob jects to m other ob jects in an injective fashion Dep ending on the ob jective we want to optimize we obtain dierent problems ranging from linear assignment problems to quadratic and higher dimensional assignment problems The assignment problems are a well studied topic in combinatorial opti mization These problems nd numerous application in pro duction planning telecommunication VLSI design economics etc We intro duce the basic problems classied into three groups linear as signment problems three and higher dimensional assignment problems and quadratic assignment problems and problems related to it For each group of problems we mention some applications show some basic prop erties and de scrib e briey some of the most successful algorithms used to solve these prob lems Intro duction Assignment problems deal with the question how to assign n ob jects to m other ob jects in an injective fashion in the b est p ossible way An assignment problem is completely sp ecied by its two comp onents the assignments which represent the underlying combinatorial structure and the ob jective function to b e optimized which mo dels the b est p ossible way In the classical assignment problem one has m n and most of the problems with m n can b e transformed or are strongly related to analogous problems with m n Therefore we will consider m n through the rest of this chapter unless otherwise sp ecied This research
    [Show full text]
  • Revised Note on Learning Algorithms for Quadratic Assignment With
    REVISED NOTE ON LEARNING QUADRATIC ASSIGNMENT WITH GRAPH NEURAL NETWORKS Alex Nowak†,§, Soledad Villar‡,∗,§, Afonso S Bandeira‡,∗ and Joan Bruna‡,∗ † Sierra Team, INRIA and Ecole Normale Superieure, Paris ‡ Courant Institute, New York University, New York ∗ Center for Data Science, New York University, New York ABSTRACT Instead of investigating a designed algorithm for the problem in question, we consider a data-driven approach to learn algorithms Inverse problems correspond to a certain type of optimization from solved instances of the problem. In other words, given a collec- problems formulated over appropriate input distributions. Recently, tion (xi,yi)i≤L of problem instances drawn from a certain distribu- there has been a growing interest in understanding the computational tion, we ask whether one can learn an algorithm that achieves good hardness of these optimization problems, not only in the worst case, accuracy at solving new instances of the same problem – also being but in an average-complexity sense under this same input distribu- drawn from the same distribution, and to what extent the resulting tion. algorithm can reach those statistical/computational thresholds. In this revised note, we are interested in studying another aspect The general approach is to cast an ensemble of algorithms as of hardness, related to the ability to learn how to solve a problem by neural networks yˆ = Φ(x; θ) with specific architectures that en- simply observing a collection of previously solved instances. These code prior knowledge on the algorithmic class, parameterized by ‘planted solutions’ are used to supervise the training of an appropri- θ ∈ RS . The network is trained to minimize the empirical loss ate predictive model that parametrizes a broad class of algorithms, −1 L(θ)= L i ℓ(yi, Φ(xi; θ)), for a given measure of error ℓ, us- with the hope that the resulting model will provide good accuracy- ing stochastic gradient descent.
    [Show full text]
  • Integer Linear Programming Modeling
    DM545 Linear and Integer Programming Lecture 8 Integer Linear Programming Modeling Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Integer Programming Outline Modeling 1. Integer Programming 2. Modeling Assignment Problem Knapsack Problem Set Covering Graph Problems 2 Integer Programming LP: Rational Solutions Modeling I A precise analysis of running time for an algorithm includes the number of bit operations together with number of arithmetic operations I Strongly polynomial algorithms: the running time of the algorithm is independent on the number of bit operations. Eg: same running time for input numbers with 10 bits as for inputs with a million bits. I No strongly polynomial-time algorithm for LP is known. I Running time depends on the sizes of numbers. We have to restrict attention to rational instances when analyzing the running time of algorithms and assume they are coded in binary. Theorem Optimal feasible solutions to LP problems are always rational as long as all coefficient and constants are rational. Proof: derives from the fact that in the simplex we only perform multiplications, divisions and sums of rational numbers 3 Integer Programming Outline Modeling 1. Integer Programming 2. Modeling Assignment Problem Knapsack Problem Set Covering Graph Problems 4 Integer Programming Discrete Optimization Modeling I Often need to deal with integral inseparable quantities. I Sometimes rounding can go. I Other times rounding not feasible: eg, presence of a bus on a line is 0.3... 5 Integer
    [Show full text]
  • Global Sensitivity Analysis for the Linear Assignment Problem
    Global Sensitivity Analysis for the Linear Assignment Problem Elad Michael, Tony A. Wood, Chris Manzie, and Iman Shames Abstract— In this paper, the following question is addressed: finding maximum flow within a network [6], lower bound- given a linear assignment problem, how much can the all of the ing solutions to the quadratic assignment problem [7], and individual assignment weights be perturbed without changing maximum a-posteriori data association for object tracking the optimal assignment? The extension of results involving perturbations in just one edge or one row/column are presented. [8]. Algorithms for the derivation of these bounds are provided. We In this paper we focus on understanding which perturba- also show how these bounds may be used to prevent assignment tions the optimal assignment is robust to. However, in some churning in a multi-vehicle guidance scenario. applications, it is preferable to find a potentially sub-optimal assignment which is less sensitive to perturbation. For linear I. INTRODUCTION assignment, [10] provides an algorithm with guaranteed Task assignment is a fundamental part of multi-agent robustness to uncertainty in a subset of the parameters. control. Here we represent task assignment optimization Similarly, but for the bottleneck assignment problem, [11] over a weighted bipartite graph, with agents and tasks as provides robustness to weights within predefined intervals, vertices and assignments as edges. The objective function as well as complexity improvements on similar bottleneck may be minimum completion time of all tasks, minimum assignment algorithms. However, these papers necessarily total fuel usage, balanced completion times, or any of a sacrifice optimality for robustness.
    [Show full text]
  • Lecture Notes on Bipartite Matching 1 Maximum Cardinality Matching
    Massachusetts Institute of Technology Handout 3 18.433: Combinatorial Optimization February 8th, 2007 Michel X. Goemans Lecture notes on bipartite matching We start by introducing some basic graph terminology. A graph G = (V; E) consists of a set V of vertices and a set E of pairs of vertices called edges. For an edge e = (u; v), we say that the endpoints of e are u and v; we also say that e is incident to u and v. A graph G = (V; E) is bipartite if the vertex set V can be partitioned into two sets A and B (the bipartition) such that no edge in E has both endpoints in the same set of the bipartition. A matching M E is a collection of edges such that every vertex of V is incident to at most one edge of M⊆. If a vertex v has no edge of M incident to it then v is said to be exposed (or unmatched). A matching is perfect if no vertex is exposed; in other words, a matching is perfect if its cardinality is equal to A = B . j j j j 1 6 2 7 exposed 3 8 matching 4 9 5 10 Figure 1: Example. The edges (1; 6), (2; 7) and (3; 8) form a matching. Vertices 4, 5, 9 and 10 are exposed. We are interested in the following two problems: Maximum cardinality matching problem: Find a matching M of maximum size. Minimum weight perfect matching problem: Given a cost cij for all (i; j) E, find a perfect matching of minimum cost where the cost of a matching M is given by2c(M) = (i;j)2M cij.
    [Show full text]
  • The Assignment Problem and Primal-Dual Algorithms 1
    Summer 2011 Optimization I Lecture 11 The Assignment Problem and Primal-Dual Algorithms 1 Assignment Problem Suppose we want to solve the following problem: We are given • a set of people I, and a set of jobs J, with jIj = jJj = n • a cost cij ≥ 0 for assigning job j to person i. We would like to assign jobs to people, so that each job is assigned to one person and each person is assigned one job, and we minimize the cost of the assignment. This is called the assignment problem. Example input: Jobs 90 75 75 80 People 35 85 55 65 125 95 90 105 45 110 95 115 The assignment problem is related to another problem, the maximum cardinality bipartite matching problem. In the maximum cardinality bipartite matching problem, you are given a bipartite graph G = (V; E), and you want to find a matching, i.e., a subset of the edges F such that each node is incident on at most one edge in F , that maximizes jF j. Suppose you have an algorithm for finding a maximum cardinality bipartite matching. If all of the costs in the assignment problem were either 0 or 1, then we could solve the assignment problem using the algorithm for the maximum cardinality bipartite matching problem: We construct a bipartite graph which has a node for every person i and every job j, and an edge from i to j if cij = 0. A matching in this graph of size k corresponds to a solution to the assignment problem of cost at most n − k and vice versa.
    [Show full text]