# The multi-integer set cover and the facility terminal cover problem

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 The multi-integer set cover and the facility teral cover problem Dorit S. Hochbaum Asaf Levin December 5, 2007 Abstract The facility teral cover problem is a generalization of the vertex cover problem. The problem is to cover the edges of an undirected graph G = (V, E) where each edge e is associated with a non-negative demand d e. An edge e = [u, v] is covered if at least one of its endpoint vertices is allocated capacity of at least d e. Each vertex v is associated with a non-negative weight w v. The goal is to allocate capacity c v 0 to each vertex v so that all edges are covered and the total allocation cost, w v c v, is imized. A recent paper by Xu, Yang and v V Xu (Networks, 50, (2007), ), studied this problem, and presented a 2e- approximation algorithm for this problem for e the base of the natural logarithm. We generalize here the facility teral cover problem to the multi-integer set cover, and relate that problem to the set cover problem, which it generalizes, and the multi-cover problem. We present a -approximation algorithm for the multi-integer set cover problem, for the maximum coverage. This demonstrates that even though the multi-integer set cover problem generalizes the set cover problem, the same approximation ratio holds. In the special case of the facility teral cover problem this yields a 2-approximation algorithm, and with run time doated by the sorting of the edge demands. This approximation algorithm improves considerably on the result of Xu, Yang and Xu. Keywords: algorithms, approximation algorithms, vertex cover, set cover, network design. 1 Introduction The facility teral cover problem (ftc) is defined as follows. Given an undirected graph G = (V, E) where each vertex v is associated with a non-negative weight w v, and each edge e is associated with a non-negative demand d e, the goal is to allocate capacity c v 0 to each vertex v V so that for every edge e = [u, v] either c u d e or c v d e, and so that w v c v is imum. We note that it is possible to allocate no capacity to a vertex v by setting c v = 0. The problem ftc was introduced by Xu, Yang and Xu [11], and was motivated by several problems in networks and operations research. For an algorithm A, denote the objective value of a solution it delivers on an input I by A(I). An optimal solution is denoted by opt, and the optimal objective value is denoted by opt as well. The (absolute) approximation ratio of A is defined as the infimum ρ such that for any input I, A(I) ρ opt(i). We restrict ourselves to algorithms that run in polynomial time. Department of Industrial Engineering and Operations Research and Walter A. Haas School of Business, University of California, Berkeley California Supported by NSF award No. DMI Department of Statistics, The Hebrew University, Jerusalem 91905, Israel. v V 1

2 The vertex cover problem is a special case of ftc where the demand of each edge is one, i.e., for all e E d e = 1. For the vertex cover problem any optimal solution consists of capacity allocation to the vertices that is either zero or one. For the ftc problem the capacity can assume the value zero or any value in the set {d e e E}. Previous results for the weighted vertex cover problem include the first 2-approximation algorithms by [6] that was shown to run in the time of solving a imum cut on a bipartite graph in [7]. Various algorithms with the same approximation ratio, or slightly better for restricted types of graphs, were shown, for example, in [7, 1, 9]. In [7] it was conjectured that unless P=NP it is impossible to improve the approximation ratio to 2 ε for any ε > 0 for the vertex cover problem on general graphs. To date NP-hardness results confirmed that unless P=NP it is impossible to approximate the vertex cover problem within [3], and under the stronger assumption of the unique games conjecture it is impossible to approximate the vertex cover problem within an approximation ratio of 2 ε for all ε > 0 [10]. A survey of approximation algorithms for the vertex cover problem and the set cover problem is presented in [8]. Another related problem is vertex cover with hard capacities which is the variant of vertex cover where each selected vertex can be used to cover up to a pre-specified number of edges incident to this vertex (its capacity). Chuzhoy and Naor [2] proved that the weighted problem is at least as hard (to approximate) as the set cover problem, and for the unweighted version they presented a 3-approximation algorithm. This 3-approximation algorithm was improved by Gandhi et al. [4] to a 2-approximation algorithm. Xu et al. [11] presented a 2e-approximation algorithm for ftc where e is the base of the natural logarithm. Their algorithm is based on randomized geometric grouping of the edges into sets of roughly equal demand edges, and then applying a vertex cover 2-approximation algorithm for each resulting edge class. They also wrote that It also seems difficult to apply those LP-based methods to solve the ftc problem, as the natural formulation of the ftc problem is a quadratic programg. Our contributions here are in several respects: We present a 2-approximation algorithm for ftc. We present a formulation of ftc as a linear mixed integer programg problem. A linear programg relaxation of this formulation is used to obtain the 2-approximation algorithm. We devise an alternative O(m log n) time 2-approximation algorithm for ftc based on the primal-dual scheme. We then define a generalization of the set cover problem, the ftc problem and the multicover problem which we call the multi-integer set cover problem or misc. We provide a mixed integer programg formulation of misc and generate from the relaxation, or from a primal-dual approach, an efficient approximation algorithm for the problem that gives the same approximation bound as for the easier set cover problem. The multi-integer set cover problem (misc) is a generalization of ftc for hypergraphs and a generalization of the set cover problem. In a graph each edge is a set of 2 endpoint vertices, whereas in a hypergraph a hyperedge is an arbitrary sized set of vertices. A formal definition of misc, in which sets are to be covered and which is dual to another possible formalization, is as follows: Given a collection of sets (the hyperedges) S 1, S 2,..., S m over a ground set V, where each element v V is associated with a non-negative weight w v, and each set S i is associated with a non-negative demand d i. The goal is to allocate capacities to the elements, c v 0 is the capacity of v, such that for each set S i there is an element u S i with c u d i and so that w v c v is imized. Clearly, misc generalizes ftc where each set S i consists of the two end-vertices of the edge e i in the graph G. The following notation is used throughout: Vectors are denoted in boldface, so x is a vec- v V 2

3 tor and x i is the value of the i th entry in the vector. denotes the maximum coverage, = max i=1,2,...,m S i. For ftc = 2. A (1) denotes the total size of the hyperedges in the input, A (1) = m i=1 S i. Obviously A (1) m. We let the size of V be n = V. Paper outline. In Section 2 we present a -approximation algorithm for misc based on solving a linear program. In Section 3 we present a -approximation algorithm for misc using the primal-dual technique with an improved time complexity of O(m log n + A (1) ). Our approximation algorithm gives a 2-approximation algorithm for ftc thus improving the result of [11]. 2 Formulation and a linear programg approximation algorithm Let x v be a decision variable indicating the capacity assigned to v V, and z iv be a binary variable for each pair (i, v), i {1,..., m} and v S i, indicating whether element v S i is designated to cover i and thus assume a capacity level at least d i : v V w vx v (misc) v S i z iv 1 i = 1,... m x v d i z iv i = 1,... m v V 0 z iv integer for 1 = 1,..., m and v V. In this formulation it is not necessary to restrict the value of the variables z iv to be 1. It is easy to see that for any feasible misc problem there is always an optimal solution where the variables z iv are binary. A fractional feasible solution to (misc) is obtained by solving the linear programg (LP) relaxation of the problem: (LP-misc) v V w vx v v S i x v d i i = 1,... m 0 x v for all v V. Obviously an optimal solution of value opt to misc is a feasible solution to (LP-misc) and with the same objective value. To see that LP-misc is indeed a relaxation of misc notice that for an optimal solution x to LP-misc, setting z iv = x v /d i is a feasible fractional solution to misc. We conclude that the objective value of an optimal solution x to LP-misc is at most opt. We now argue that the solution x = ( x v ) v V resulting from multiplying by the capacity of each element with respect to its capacity in x, is a feasible solution to misc. This is so because for each set S i, v S i x v d i, and thus there must be an element u S i such that d i S i x u. Hence for this u, d i S i x u x u. Since this argument holds for all sets, we conclude that the solution ( x v ) v V is a feasible solution to misc. By the linearity (in the capacities) of the objective function of misc, we conclude that the cost of ( x v ) v V is exactly times the cost of x and hence at most times the cost of opt. Therefore, ( x v ) v V is a -approximate solution, and since (LP-misc) can be solved in polynomial time, we established the following theorem. Theorem 1 There is a polynomial time -approximation algorithm for misc. Therefore, there is a polynomial time 2-approximation algorithm for ftc. 3

4 3 The primal-dual algorithm We next improve the time complexity of our approximation algorithm for misc. To do so, we extend the approximation algorithm of Hall and Hochbaum [5] for the multi-set cover problem. The multi-set cover problem (MSC) is defined, for positive integer demands d i, by the following formulation: v V w vx v (MSC ) v S i x v d i i = 1,... m 0 x v 1 integer for v V. In contrast to misc, the cover for the multi-set cover problem is formed of elements v V each with capacity 1. Let A = (a ij ) be the 0 1 constraint matrix of (LP-misc). Then LP-misc is restated as follows. (LP-misc) n j=1 w jx j n j=1 a ijx j d i i = 1,... m 0 x j for j = 1,..., n. The dual to the linear programg relaxation LP-misc is: (misc-dual) max m i=1 d iy i m i=1 a ijy i w j j = 1,..., n y i 0 i = 1, 2,..., m. In the primal-dual algorithm, we generate, like the algorithm for multi-set cover, a feasible cover along with a feasible and maximal dual solution. (For definition of maximal and its importance in approximating covering problems see [8].) The algorithm works, like approximation algorithms for other covering problems, by choosing at each iteration any violated constraint, or uncovered set, and covering it with the cheapest (in terms of reduced cost) available element. The distinct feature needed in this algorithm is that the selected violated constraint is the one that has the largest demand level among all violated constraints. We use this feature to ensure that the dual solution remains feasible throughout the algorithm. The dual-feasible misc algorithm: Step 0: {initialize} I = {1, 2,..., m}, J = V = {1, 2,..., n}, x j = 0 for all j J, y i = 0 for all i I. Step 1: Let i I with d i = max p I d p. Let w k = {w j j J and a ij = 1}. (k is the imum reduced cost column covering row i.) If no such imum exists, stop - the problem is infeasible. Set x k = d i. Step 2: {update of dual costs and reduced costs} Set y i y i + w k. For all j J such that a ij = 1 set w j w j w k. Step 3: {remove all sets/rows covered by element/column k. Their demand value can only be smaller than d i } J J \ {k}. For all i such that a ki = 1, set I I {i }. If I = stop, else go to Step 1. The output of the algorithm is two vectors x and y and the set of indices of elements in the cover V \ J. From the choice of w k and the update in Step 2 all the reduced costs w j are non-negative throughout the algorithm, and the reduced costs of the elements in the cover are 0, w k = 0 for all k V \ J. 4

5 The properties that are satisfied by the output of the algorithm x and y are: (1) x is a feasible solution to misc. (2) For every j = 1,..., n for which x j > 0, its dual constraint is binding (i.e., satisfied with equality). That is because the reduced costs at the end of the algorithm are the slacks in the dual constraints (3) The dual solution y is feasible for misc-dual. For these properties to hold it is necessary that the violated constraint is chosen to be the one with the largest demand. We note that A (1), the number of nonzero entries in a 0 1 matrix A, is equal to n j=1 m i=1 a ij. Lemma 1 The time complexity of algorithm dual-feasible misc is O(m log n + A (1) ) which is doated by O(m log n + m). Proof The claim follows by noting that after sorting the demand values d i in O(m log n) time, the other operations of the algorithm take O(1) time per non-zero entry of the constraint matrix, for a total of A (1) extra operations. Since A (1) m) the stated complexity follows. Lemma 2 Algorithm dual-feasible misc is a -approximation algorithm. Proof Let i(j) be the unique row for which column j was selected and removed from J. Let the (primal) solution delivered by the algorithm be x = x H and the optimal solution be x = x. We denote the total cost of a solution x by w(x) = n j=1 w jx j. We denote by J the subset of J at the end of the algorithm corresponding to variables of value 0. We need to show that w(x H ) max i { j a ij}w(x ). Let y be the dual solution delivered by the algorithm. Then, w(x H ) = j V \J w jd i(j) = j V \J ( m i=1 a ijy i )d i(j). Using the weak duality theorem we get that for any solution to the linear programg relaxation, and in particular for an optimal solution x to the linear programg relaxation, j V \J ( m i=1 a ijd i(j) y i ) max i { j V\J a ij} m i=1 d i(j)y i max i { j V\J a ij} n j=1 w jx j = max i { j V\J a ij}w(x ). It follows that w(x H ) max i { j V\J a ij}w(x ) opt. With Lemma 1 and Lemma 2 we have thus established the following theorem. Theorem 2 There is an O(m log n + m) time -approximation algorithm for misc. Therefore, there is an O(m log n) time 2-approximation algorithm for ftc. References [1] R. Bar-Yehuda, S. Even, A linear time approximation algorithm for the weighted vertex cover algorithm, Journal of Algorithms, 2, (1981), [2] J. Chuzhoy and J. Naor, Covering Problems with Hard Capacities, SIAM Journal on Computing, 36, (2006), [3] I. Dinur, S. Safra, On the hardness of approximating imum vertex-cover, Annals of Mathematics, 162, (2005),

6 [4] R. Gandhi, E. Halperin, S. Khuller, G. Kortsarz and A. Srinivasan, An improved approximation algorithm for vertex cover with hard capacities, Journal of Computer and System Sciences, 72, (2006), [5] N. G. Hall and D. S. Hochbaum, A fast approximation algorithm for the multicovering problem, Discrete Applied Mathematics, 15, (1986), [6] D. S. Hochbaum, Approximation algorithms for the set covering and vertex cover problems, SIAM Journal on Computing, 11, (1982), An extended version: Approximation algorithms for the weighted set covering and node cover problems, GSIA Working Paper No , Carnegie Mellon University, Pittsburgh, PA, April [7] D. S. Hochbaum, Efficient bounds for the stable set, vertex cover and set packing problems, Discrete Applied Mathematics, 6, (1983), [8] D. S. Hochbaum, Approximating covering and packing problems: set cover, vertex cover, independent set and related problems, Chapter 3 in Approximation algorithms for NPhard problems edited by D. S. Hochbaum. PWS Boston. 1997, [9] G. Karakostas, A better approximation ratio for the vertex cover problem, Proceedings of International Colloquium on Automata, Languages and Programg, Lisbon, Portugal, 2005 (ICALP 2005), [10] S. Khot and O. Regev, Vertex Cover Might be Hard to Approximate to within 2 ε, 18th Annual IEEE Conference on Computational Complexity (CCC 03), [11] G. Xu, Y. Yang and J. Xu, Linear time algorithms for approximating the facility teral cover problem, Networks, 50, (2007),

### Lecture 6: Approximation via LP Rounding

Lecture 6: Approximation via LP Rounding Let G = (V, E) be an (undirected) graph. A subset C V is called a vertex cover for G if for every edge (v i, v j ) E we have v i C or v j C (or both). In other

### Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

### Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization

### princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.

Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three

### max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x

Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study

### 1 Unrelated Parallel Machine Scheduling

IIIS 014 Spring: ATCS - Selected Topics in Optimization Lecture date: Mar 13, 014 Instructor: Jian Li TA: Lingxiao Huang Scribe: Yifei Jin (Polishing not finished yet.) 1 Unrelated Parallel Machine Scheduling

### Proximal mapping via network optimization

L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:

### Lecture 7 - Linear Programming

COMPSCI 530: Design and Analysis of Algorithms DATE: 09/17/2013 Lecturer: Debmalya Panigrahi Lecture 7 - Linear Programming Scribe: Hieu Bui 1 Overview In this lecture, we cover basics of linear programming,

### Markov Chains, part I

Markov Chains, part I December 8, 2010 1 Introduction A Markov Chain is a sequence of random variables X 0, X 1,, where each X i S, such that P(X i+1 = s i+1 X i = s i, X i 1 = s i 1,, X 0 = s 0 ) = P(X

### Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.

Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of

Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one

### LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method

LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method Introduction to dual linear program Given a constraint matrix A, right

### Week 5 Integral Polyhedra

Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

### 10.1 Integer Programming and LP relaxation

CS787: Advanced Algorithms Lecture 10: LP Relaxation and Rounding In this lecture we will design approximation algorithms using linear programming. The key insight behind this approach is that the closely

### Section Notes 4. Duality, Sensitivity, Dual Simplex, Complementary Slackness. Applied Math 121. Week of February 28, 2011

Section Notes 4 Duality, Sensitivity, Dual Simplex, Complementary Slackness Applied Math 121 Week of February 28, 2011 Goals for the week understand the relationship between primal and dual programs. know

### THEORY OF SIMPLEX METHOD

Chapter THEORY OF SIMPLEX METHOD Mathematical Programming Problems A mathematical programming problem is an optimization problem of finding the values of the unknown variables x, x,, x n that maximize

### Definition of a Linear Program

Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

### Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

### The Conference Call Search Problem in Wireless Networks

The Conference Call Search Problem in Wireless Networks Leah Epstein 1, and Asaf Levin 2 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. lea@math.haifa.ac.il 2 Department of Statistics,

### Nan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA

A Factor 1 2 Approximation Algorithm for Two-Stage Stochastic Matching Problems Nan Kong, Andrew J. Schaefer Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA Abstract We introduce

### 4.6 Linear Programming duality

4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal

### Definition 11.1. Given a graph G on n vertices, we define the following quantities:

Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define

### Algorithm Design and Analysis

Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;

### Minimize subject to. x S R

Chapter 12 Lagrangian Relaxation This chapter is mostly inspired by Chapter 16 of [1]. In the previous chapters, we have succeeded to find efficient algorithms to solve several important problems such

### A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem

A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem John Karlof and Peter Hocking Mathematics and Statistics Department University of North Carolina Wilmington Wilmington,

### Permutation Betting Markets: Singleton Betting with Extra Information

Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu

### Spectral graph theory

Spectral graph theory Uri Feige January 2010 1 Background With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian

### Minimum Makespan Scheduling

Minimum Makespan Scheduling Minimum makespan scheduling: Definition and variants Factor 2 algorithm for identical machines PTAS for identical machines Factor 2 algorithm for unrelated machines Martin Zachariasen,

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### Primal-Dual Schema for Capacitated Covering Problems

Primal-Dual Schema for Capacitated Covering Problems Tim Carnes and David Shmoys Cornell University, Ithaca NY 14853, USA Abstract. Primal-dual algorithms have played an integral role in recent developments

### 1 Polyhedra and Linear Programming

CS 598CSC: Combinatorial Optimization Lecture date: January 21, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im 1 Polyhedra and Linear Programming In this lecture, we will cover some basic material

### Min-cost flow problems and network simplex algorithm

Min-cost flow problems and network simplex algorithm The particular structure of some LP problems can be sometimes used for the design of solution techniques more efficient than the simplex algorithm.

### Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,

### By W.E. Diewert. July, Linear programming problems are important for a number of reasons:

APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### Good luck, veel succes!

Final exam Advanced Linear Programming, May 7, 13.00-16.00 Switch off your mobile phone, PDA and any other mobile device and put it far away. No books or other reading materials are allowed. This exam

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### 5.1 Bipartite Matching

CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

### CHAPTER 17. Linear Programming: Simplex Method

CHAPTER 17 Linear Programming: Simplex Method CONTENTS 17.1 AN ALGEBRAIC OVERVIEW OF THE SIMPLEX METHOD Algebraic Properties of the Simplex Method Determining a Basic Solution Basic Feasible Solution 17.2

### Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal

Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal ABSTACT Mohit Singh Tepper School of Business Carnegie Mellon University Pittsburgh, PA USA mohits@andrew.cmu.edu In the MINIMUM

### 9th Max-Planck Advanced Course on the Foundations of Computer Science (ADFOCS) Primal-Dual Algorithms for Online Optimization: Lecture 1

9th Max-Planck Advanced Course on the Foundations of Computer Science (ADFOCS) Primal-Dual Algorithms for Online Optimization: Lecture 1 Seffi Naor Computer Science Dept. Technion Haifa, Israel Introduction

### A simpler and better derandomization of an approximation algorithm for Single Source Rent-or-Buy

A simpler and better derandomization of an approximation algorithm for Single Source Rent-or-Buy David P. Williamson Anke van Zuylen School of Operations Research and Industrial Engineering, Cornell University,

### ARTICLE IN PRESS. European Journal of Operational Research xxx (2004) xxx xxx. Discrete Optimization. Nan Kong, Andrew J.

A factor 1 European Journal of Operational Research xxx (00) xxx xxx Discrete Optimization approximation algorithm for two-stage stochastic matching problems Nan Kong, Andrew J. Schaefer * Department of

### Two General Methods to Reduce Delay and Change of Enumeration Algorithms

ISSN 1346-5597 NII Technical Report Two General Methods to Reduce Delay and Change of Enumeration Algorithms Takeaki Uno NII-2003-004E Apr.2003 Two General Methods to Reduce Delay and Change of Enumeration

### Stanford University CS261: Optimization Handout 6 Luca Trevisan January 20, In which we introduce the theory of duality in linear programming.

Stanford University CS261: Optimization Handout 6 Luca Trevisan January 20, 2011 Lecture 6 In which we introduce the theory of duality in linear programming 1 The Dual of Linear Program Suppose that we

### Institut für Informatik Lehrstuhl Theoretische Informatik I / Komplexitätstheorie. An Iterative Compression Algorithm for Vertex Cover

Friedrich-Schiller-Universität Jena Institut für Informatik Lehrstuhl Theoretische Informatik I / Komplexitätstheorie Studienarbeit An Iterative Compression Algorithm for Vertex Cover von Thomas Peiselt

Online Adwords Allocation Shoshana Neuburger May 6, 2009 1 Overview Many search engines auction the advertising space alongside search results. When Google interviewed Amin Saberi in 2004, their advertisement

### 3 Does the Simplex Algorithm Work?

Does the Simplex Algorithm Work? In this section we carefully examine the simplex algorithm introduced in the previous chapter. Our goal is to either prove that it works, or to determine those circumstances

### THE PROBLEM WORMS (1) WORMS (2) THE PROBLEM OF WORM PROPAGATION/PREVENTION THE MINIMUM VERTEX COVER PROBLEM

1 THE PROBLEM OF WORM PROPAGATION/PREVENTION I.E. THE MINIMUM VERTEX COVER PROBLEM Prof. Tiziana Calamoneri Network Algorithms A.y. 2014/15 2 THE PROBLEM WORMS (1)! A computer worm is a standalone malware

### CSL851: Algorithmic Graph Theory Semester I Lecture 4: August 5

CSL851: Algorithmic Graph Theory Semester I 2013-14 Lecture 4: August 5 Lecturer: Naveen Garg Scribes: Utkarsh Ohm Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have not

### Fairness in Routing and Load Balancing

Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria

### Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs

Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs Stavros Athanassopoulos, Ioannis Caragiannis, and Christos Kaklamanis Research Academic Computer Technology Institute

### The Approximability of the Binary Paintshop Problem

The Approximability of the Binary Paintshop Problem Anupam Gupta 1, Satyen Kale 2, Viswanath Nagarajan 2, Rishi Saket 2, and Baruch Schieber 2 1 Dept. of Computer Science, Carnegie Mellon University, Pittsburgh

### Class constrained bin covering

Class constrained bin covering Leah Epstein Csanád Imreh Asaf Levin Abstract We study the following variant of the bin covering problem. We are given a set of unit sized items, where each item has a color

### JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004

Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February

### 1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S

### Discrete Optimization

Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using

### Combinatorial Algorithms for Data Migration to Minimize Average Completion Time

Combinatorial Algorithms for Data Migration to Minimize Average Completion Time Rajiv Gandhi 1 and Julián Mestre 1 Department of Computer Science, Rutgers University-Camden, Camden, NJ 0810. Research partially

### I. Solving Linear Programs by the Simplex Method

Optimization Methods Draft of August 26, 2005 I. Solving Linear Programs by the Simplex Method Robert Fourer Department of Industrial Engineering and Management Sciences Northwestern University Evanston,

### Absolute Value Programming

Computational Optimization and Aplications,, 1 11 (2006) c 2006 Springer Verlag, Boston. Manufactured in The Netherlands. Absolute Value Programming O. L. MANGASARIAN olvi@cs.wisc.edu Computer Sciences

### Why? A central concept in Computer Science. Algorithms are ubiquitous.

Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### An Algorithm for Fractional Assignment Problems. Maiko Shigeno. Yasufumi Saruwatari y. and. Tomomi Matsui z. Department of Management Science

An Algorithm for Fractional Assignment Problems Maiko Shigeno Yasufumi Saruwatari y and Tomomi Matsui z Department of Management Science Science University of Tokyo 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162,

### Lecture 5: Conic Optimization: Overview

EE 227A: Conve Optimization and Applications January 31, 2012 Lecture 5: Conic Optimization: Overview Lecturer: Laurent El Ghaoui Reading assignment: Chapter 4 of BV. Sections 3.1-3.6 of WTB. 5.1 Linear

### Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs

Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs Leah Epstein Magnús M. Halldórsson Asaf Levin Hadas Shachnai Abstract Motivated by applications in batch scheduling of jobs in manufacturing

### Greedy Algorithm And Matroid Intersection Algorithm. Summer Term Talk Summary Paul Wilhelm

Greedy Algorithm And Matroid Intersection Algorithm Summer Term 2010 Talk Summary Paul Wilhelm December 20, 2011 1 Greedy Algorithm Abstract Many combinatorial optimization problems can be formulated in

### Steiner Tree Approximation via IRR. Randomized Rounding

Steiner Tree Approximation via Iterative Randomized Rounding Graduate Program in Logic, Algorithms and Computation μπλ Network Algorithms and Complexity June 18, 2013 Overview 1 Introduction Scope Related

### Fundamentals of Operations Research. Prof. G. Srinivasan. Department of Management Studies. Indian Institute of Technology, Madras. Lecture No.

Fundamentals of Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture No. # 22 Inventory Models - Discount Models, Constrained Inventory

### Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs

Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs These questions are from previous years and should you give you some idea of what to expect on Quiz 1. 1. Consider the

### Minimal Cost Reconfiguration of Data Placement in a Storage Area Network

Minimal Cost Reconfiguration of Data Placement in a Storage Area Network Hadas Shachnai Gal Tamir Tami Tamir Abstract Video-on-Demand (VoD) services require frequent updates in file configuration on the

### 9 On the Cover Time of Random Walks on Graphs

Journal of Theoretical Probability, Vol. 2, No. 1, 1989 9 On the Cover Time of Random Walks on Graphs Jeff D. Kahn, 1 Nathan Linial, 2'3 Noam Nisan, 4 and Michael E. Saks 1'5 Received June 28, 1988 This

### Linear Programming. March 14, 2014

Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1

### Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

### Scheduling Shop Scheduling. Tim Nieberg

Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations

### Transportation Polytopes: a Twenty year Update

Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,

### LINEAR PROGRAMMING P V Ram B. Sc., ACA, ACMA Hyderabad

LINEAR PROGRAMMING P V Ram B. Sc., ACA, ACMA 98481 85073 Hyderabad Page 1 of 19 Question: Explain LPP. Answer: Linear programming is a mathematical technique for determining the optimal allocation of resources

Adaptive Linear Programming Decoding Mohammad H. Taghavi and Paul H. Siegel ECE Department, University of California, San Diego Email: (mtaghavi, psiegel)@ucsd.edu ISIT 2006, Seattle, USA, July 9 14, 2006

### Linear Programming: Chapter 5 Duality

Linear Programming: Chapter 5 Duality Robert J. Vanderbei October 17, 2007 Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 http://www.princeton.edu/ rvdb Resource

### Discrete (and Continuous) Optimization Solutions of Exercises 1 WI4 131

Discrete (and Continuous) Optimization Solutions of Exercises 1 WI4 131 Kees Roos Technische Universiteit Delft Faculteit Informatietechnologie en Systemen Afdeling Informatie, Systemen en Algoritmiek

### Linear Programming is the branch of applied mathematics that deals with solving

Chapter 2 LINEAR PROGRAMMING PROBLEMS 2.1 Introduction Linear Programming is the branch of applied mathematics that deals with solving optimization problems of a particular functional form. A linear programming

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### Permutation Betting Markets: Singleton Betting with Extra Information

Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu

### Uniform Multicommodity Flow through the Complete Graph with Random Edge-capacities

Combinatorics, Probability and Computing (2005) 00, 000 000. c 2005 Cambridge University Press DOI: 10.1017/S0000000000000000 Printed in the United Kingdom Uniform Multicommodity Flow through the Complete

### Linear Programming I

Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

### 1 Solving LPs: The Simplex Algorithm of George Dantzig

Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

### Section Notes 3. The Simplex Algorithm. Applied Math 121. Week of February 14, 2011

Section Notes 3 The Simplex Algorithm Applied Math 121 Week of February 14, 2011 Goals for the week understand how to get from an LP to a simplex tableau. be familiar with reduced costs, optimal solutions,

### Duplicating and its Applications in Batch Scheduling

Duplicating and its Applications in Batch Scheduling Yuzhong Zhang 1 Chunsong Bai 1 Shouyang Wang 2 1 College of Operations Research and Management Sciences Qufu Normal University, Shandong 276826, China

### Chap 4 The Simplex Method

The Essence of the Simplex Method Recall the Wyndor problem Max Z = 3x 1 + 5x 2 S.T. x 1 4 2x 2 12 3x 1 + 2x 2 18 x 1, x 2 0 Chap 4 The Simplex Method 8 corner point solutions. 5 out of them are CPF solutions.

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### Every tree contains a large induced subgraph with all degrees odd

Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University

### Answers to some of the exercises.

Answers to some of the exercises. Chapter 2. Ex.2.1 (a) There are several ways to do this. Here is one possibility. The idea is to apply the k-center algorithm first to D and then for each center in D

### A short OPL tutorial

A short OPL tutorial Truls Flatberg January 17, 2009 1 Introduction This note is a short tutorial to the modeling language OPL. The tutorial is by no means complete with regard to all features of OPL.

### This exposition of linear programming

Linear Programming and the Simplex Method David Gale This exposition of linear programming and the simplex method is intended as a companion piece to the article in this issue on the life and work of George

### OPRE 6201 : 2. Simplex Method

OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2