Save this PDF as:

Size: px
Start display at page:

## Transcription

2 server nodes is known, while the set of requests arrive one at a time. As a node arrives, the weights of all edges incident on the node, i.e., servers that can fulfill the request, are divulged. Each server can service a single request. The task is to find a minimum weight matching of requests to servers online as each node arrives. [3] The effectiveness of an online algorithm is measured by competitive analysis. We say an algorithm ALG is α-competitive if the ratio between its performance and the optimal offline performance, OPT, is bounded by α. In other words, for any instance of the problem I. ALG(I) OPT(I) α Karp et. al. describe an optimal randomized algorithm for online bipartite matching in [4]. The input is represented as adjacency matrix B for G(U,V,E), a bipartite graph on 2n vertices that contains a perfect matching. We can let the rows represent the nodes in U, the boys, and the columns represent the nodes in V, the girls. Then, an entry in the adjacency matrix B, b ij, is a boolean value. b ij = 1, if boy i is acceptable to girl j, 0, otherwise. The online matching is constructed while the matrix is revealed column-by-column. We can consider a simple greedy algorithm. greedy matches a girl whenever possible. greedy achieves a maximal matching of size at least n/2. An adaptive adversary can limit any deterministic algorithm to a matching of size n/2. As an example, let the first n/2 columns contain all ones and the last n/2 columns contain ones only in those rows which are matched by the deterministic algorithm in the first n/2 steps. Thus, the competitive ratio of greedy is 1 2. The randomized algorithm ranking of [4] achieves a competitive ratio of 1 1 e. The algorithm fixes a random permutation of the first set of vertices, assigning a rank to each boy. As each girl arrives, she is matched to the eligible boy (if any) of highest rank. Karp et. al. proved that this algorithm is optimal for online bipartite matching. Online bipartite matching is a special case of the Adwords allocation problem in which advertisers are restricted to unit bids and unit daily budgets. For this special case, the ranking algorithm matches ads as they arrive with a competitive factor of 1 1 e. 3 Online b-matching The work of Karp, Vazirani, and Vazirani was extended to b-matching by Bala Kalyanasundaram and Kirk Pruhs in [2]. In a b-matching prolem, each server can be matched to several requests within a predetermined service limit. This is unlike the assignment problem in which a server vertex is matched only once. A b-matching problem seeks to allocate requests to servers so that the maximum number of requests are serviced. In effect, the unused service is kept to a minimum. Kalyanasundaram and Pruhs developed a deterministic algorithm called balance. Their 2

5 Figure 1: Tradeoff Function Figure 2: Discretization of Budget 5

6 the new algorithm also has a competitive ratio of 1 1 e. The tradeoff function is the basis of the new algorithm. In the special case when all bids are equal, the new algorithm is precisely balance provided by [2]. The budget of each advertiser is discretized into k equal parts numbered 1 through k. k can be any large integer. A bidder spends money in slab j before proceeding to slab j + 1. A bidder is classified by the fraction of his budget that is spent at the end of the algorithm. A bidder is of type j if the fraction of his budget spent at the end of the algorithm is in ( j 1 k, j k ]. Then, we can define x i as the number of bidders of type i. Alternatively, x i is the number of bidders who spend i/k of their budget at the end of balance. x i is defined for 1 i k. The slabs are depicted in the accompanying image. Let ψ k : [1 k] R + be the following (monotonically decreasing) function: ψ k (i) = 1 e (i i/k) Note: ψ k ψ as k. Revenue = k i=1 x i i k The factor revealing LP bounds the number of bidders of type j for small j. We can examine the area of each rectange in the figure to identify the constraints. The first constraint is: x 1 N k. Similarly, the second constraint is: x 1 +x 2 2 N k x 1 k. Rearranging the terms reveals the following generalization of x i : i,1 i k 1: i j=1 (1 + i j k )x j i k N. The following LP, L, gives a lower bound on balance. i ranges from 1 to k 1. k 1 k i Maximize k x i i=1 i such that i : j=1 i : x i 0 The dual LP, D, follows. Minimize k 1 i k Ny i i=1 k 1 such that i : j=i i : y i 0 (1 + i j k )x j i k N (1 + j i k )y j k i k The primal LP, L, can be written as Max c x s.t. Ax b x 0. The dual LP, D, can be written as Min b y s.t. A T y c y 0. 6

9 Consistent Tie-Breaking is another property that is taken for granted in online bipartite matching but needs to be formalized for the Adwords problem. The monotonicity of the allocation algorithm is an important property. If a query is moved to the front of the permutation, each bidder would spend at least as much money as he did before. This property holds in the greedy algorithm, provided we allow a bidder to overspend his budget in the last bid he wins. The overspending is not counted towards revenue. This variant of greedy is monotonic and thus easier to analyze. The tagging procedure is also used to develop the Partition property of the Adwords problem. Thus, the lemmas of the online bipartite matching greedy algorithm have direct counterparts in Adwords allocation. The relationship between the sets miss, bad, and good remains the same and the same inequalities hold. After some rearrangement, the same factor revealing LP maximizes the loss of revenue over the inequalities. The same factor of 1 1 e is achieved in the random input model of the Adwords problem. Although greedy was modified to allow overspending, the revenue is almost the same as that of pure greedy. This relies on the assumption that bids are small compared to budgets. The competitive factor of Greedy in both the random-permutation input model and independent distribution input model (i.i.d.) is exactly 1 1 e. This bound is tight. 6 Open Problems Independent of the Adwords motivation, the problem is a general combinatorial allocation problem. The algorithms in [5] and [1] assume that bids are small compared to an advertiser s budget. Can we generalize their work to problems in which this assumption does not hold? A search engine must publicize the algorithm that is used for Adwords allocation. How would an advertiser react to this knowledge under the algorithms we have considered? Would they change their bids in any way? From my analysis, it seems that the bidders would devalue the keywords in all their bids. References [1] Gagan Goel and Aranyak Mehta. Online budgeted matching in random input models with applications to adwords. In Shang-Hua Teng, editor, SODA, pages SIAM, [2] Kalyanasundaram and Pruhs. An optimal deterministic algorithm for online b-matching. TCS: Theoretical Computer Science, 233, [3] B. Kalyanasundaram and K. Pruhs. Online weighted matching. Journal of Algorithms, 14(3): , May [4] Karp, Vazirani, and Vazirani. An optimal algorithm for on-line bipartite matching. In STOC: ACM Symposium on Theory of Computing (STOC),

10 [5] Mehta, Saberi, Vazirani, and Vazirani. Adwords and generalized on-line matching. In FOCS: IEEE Symposium on Foundations of Computer Science (FOCS), [6] Mehta, Saberi, Vazirani, and Vazirani. Adwords and generalized online matching. JACM: Journal of the ACM, 54, [7] Sara Robinson. Computer scientists optimize innovative ad auction. SIAM News, 38, April

### AdWords and Generalized On-line Matching

AdWords and Generalized On-line Matching Aranya Mehta Amin Saberi Umesh Vazirani Vijay Vazirani Abstract How does a search engine company decide what ads to display with each query so as to maximize its

### Adwords Auctions with Decreasing Valuation Bids

Adwords Auctions with Decreasing Valuation Bids Aranyak Mehta Gagan Goel Abstract The choice of a bidding language is crucial in auction design in order to correctly capture bidder utilities. We propose

### CSC2420 Fall 2012: Algorithm Design, Analysis and Theory

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory Allan Borodin November 15, 2012; Lecture 10 1 / 27 Randomized online bipartite matching and the adwords problem. We briefly return to online algorithms

### AdWords and Generalized On-line Matching

AdWords and Generalized On-line Matching Aranya Mehta Amin Saberi Umesh Vazirani Vijay Vazirani Abstract How does a search engine company decide what ads to display with each query so as to maximize its

### Online Primal-Dual Algorithms for Maximizing Ad-Auctions Revenue

Online Primal-Dual Algorithms for Maximizing Ad-Auctions Revenue Niv Buchbinder 1, Kamal Jain 2, and Joseph (Seffi) Naor 1 1 Computer Science Department, Technion, Haifa, Israel. 2 Microsoft Research,

### Issues and Challenges in Advertising on the Web

International Journal of Electrical and Computer Engineering (IJECE) Vol. 4, No. 5, October 2014, pp. 810~816 ISSN: 2088-8708 810 Issues and Challenges in Advertising on the Web Deepthi Gurram, B. Vijaya

### AdWords and Generalized On-line Matching

AdWords and Generalized On-line Matching Aranyak Mehta Amin Saberi Umesh Vazirani Vijay Vazirani Abstract How does a search engine company decide what ads to display with each query so as to maximize its

### Frequency Capping in Online Advertising

Frequency Capping in Online Advertising (Extended Abstract) Niv Buchbinder 1, Moran Feldman 2, Arpita Ghosh 3, and Joseph (Seffi) Naor 2 1 Open University, Israel niv.buchbinder@gmail.com 2 Computer Science

### AdWords and Generalized On-line Matching

AdWords and Generalized On-line Matching ARANYAK MEHTA Google, Inc. AMIN SABERI Stanford University UMESH VAZIRANI University of California, Bereley and VIJAY VAZIRANI Georgia Institute of Technology How

### Online Matching and Ad Allocation. Contents

Foundations and Trends R in Theoretical Computer Science Vol. 8, No. 4 (2012) 265 368 c 2013 A. Mehta DOI: 10.1561/0400000057 Online Matching and Ad Allocation By Aranyak Mehta Contents 1 Introduction

### Frequency Capping in Online Advertising

Frequency Capping in Online Advertising Niv Buchbinder Moran Feldman Arpita Ghosh Joseph (Seffi) Naor July 2, 2014 Abstract We study the following online problem. There are n advertisers. Each advertiser

Advertisement Allocation for Generalized Second Pricing Schemes Ashish Goel Mohammad Mahdian Hamid Nazerzadeh Amin Saberi Abstract Recently, there has been a surge of interest in algorithms that allocate

### 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

### 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

### Online Bipartite Perfect Matching With Augmentations

Online Bipartite Perfect Matching With Augmentations Kamalika Chaudhuri, Constantinos Daskalakis, Robert D. Kleinberg, and Henry Lin Information Theory and Applications Center, U.C. San Diego Email: kamalika@soe.ucsd.edu

### Online Optimization with Uncertain Information

Online Optimization with Uncertain Information Mohammad Mahdian Yahoo! Research mahdian@yahoo-inc.com Hamid Nazerzadeh Stanford University hamidnz@stanford.edu Amin Saberi Stanford University saberi@stanford.edu

### ! 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

AdCell: Ad Allocation in Cellular Networks Saeed Alaei 1, Mohammad T. Hajiaghayi 12, Vahid Liaghat 1, Dan Pei 2, and Barna Saha 1 {saeed, hajiagha, vliaghat, barna}@cs.umd.edu, peidan@research.att.com

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

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

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.

### 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

### 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

### 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

### 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

### 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 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

### 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

### 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

### 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

### Mathematical finance and linear programming (optimization)

Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may

CS787: Advanced Algorithms Lecture 14: Online algorithms We now shift focus to a different kind of algorithmic problem where we need to perform some optimization without knowing the input in advance. Algorithms

### 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

### 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

### 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 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

### 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;

### 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

### Asymptotically Optimal Algorithm for Stochastic Adwords

Asymptotically Optimal Algorithm for Stochastic Adwords Nikhil R. Devanur, Microsoft Research alasubramanian Sivan, University of Wisconsin-Madison Yossi Azar, Tel-Aviv University In this paper we consider

### On Competitiveness in Uniform Utility Allocation Markets

On Competitiveness in Uniform Utility Allocation Markets Deeparnab Chakrabarty Department of Combinatorics and Optimization University of Waterloo Nikhil Devanur Microsoft Research, Redmond, Washington,

### 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

### 5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition

### 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

### 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

Handling Advertisements of Unknown Quality in Search Advertising Sandeep Pandey Carnegie Mellon University spandey@cs.cmu.edu Christopher Olston Yahoo! Research olston@yahoo-inc.com Abstract We consider

Chapter 8 Advertising on the Web One of the big surprises of the 21st century has been the ability of all sorts of interesting Web applications to support themselves through advertising, rather than subscription.

### A Simple Characterization for Truth-Revealing Single-Item Auctions

A Simple Characterization for Truth-Revealing Single-Item Auctions Kamal Jain 1, Aranyak Mehta 2, Kunal Talwar 3, and Vijay Vazirani 2 1 Microsoft Research, Redmond, WA 2 College of Computing, Georgia

### Offline sorting buffers on Line

Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com

### 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

### A o(n)-competitive Deterministic Algorithm for Online Matching on a Line

A o(n)-competitive Deterministic Algorithm for Online Matching on a Line Antonios Antoniadis 2, Neal Barcelo 1, Michael Nugent 1, Kirk Pruhs 1, and Michele Scquizzato 3 1 Department of Computer Science,

### Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method

Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming

### Ecient approximation algorithm for minimizing makespan. on uniformly related machines. Chandra Chekuri. November 25, 1997.

Ecient approximation algorithm for minimizing makespan on uniformly related machines Chandra Chekuri November 25, 1997 Abstract We obtain a new ecient approximation algorithm for scheduling precedence

### Pricing Digital Goods via Auctions: A Survey

Pricing Digital Goods via Auctions: A Survey Mahdi Tayarani Najaran tayarani@cs.ubc.ca April 24, 2008 Abstract The cost of reproducing a digital good is upper bounded by some low cost, providing the seller

### Online Degree-Bounded Steiner Network Design

Online Degree-Bounded Steiner Network Design The problem of satisfying connectivity demands on a graph while respecting given constraints has been a pillar of the area of network design since the early

### Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range

THEORY OF COMPUTING, Volume 1 (2005), pp. 37 46 http://theoryofcomputing.org Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range Andris Ambainis

### 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

### 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

### 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,

### Online Ad Assignment with Free Disposal

Online Ad Assignment with Free Disposal Jon Feldman 1, Nitish Korula 2, Vahab Mirrokni 1, S. Muthukrishnan 1, and Martin Pál 1 1 Google Inc., 76 9th Avenue, New York, NY 10011 {jonfeld,mirrokni,muthu,mpal}@google.com

### 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,

### Optimal Online Assignment with Forecasts

Optimal Online Assignment with Forecasts Erik Vee Yahoo! Research Santa Clara, CA erikvee@yahoo-inc.com Sergei Vassilvitskii Yahoo! Research New York, NY sergei@yahoo-inc.com Jayavel Shanmugasundaram Yahoo!

### Competitive Analysis of On line Randomized Call Control in Cellular Networks

Competitive Analysis of On line Randomized Call Control in Cellular Networks Ioannis Caragiannis Christos Kaklamanis Evi Papaioannou Abstract In this paper we address an important communication issue arising

### COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

### A Truthful Mechanism for Offline Ad Slot Scheduling

A Truthful Mechanism for Offline Ad Slot Scheduling Jon Feldman 1, S. Muthukrishnan 1, Evdokia Nikolova 2,andMartinPál 1 1 Google, Inc. {jonfeld,muthu,mpal}@google.com 2 Massachusetts Institute of Technology

### 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.

### 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:

### 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

### 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.

### 17.6.1 Introduction to Auction Design

CS787: Advanced Algorithms Topic: Sponsored Search Auction Design Presenter(s): Nilay, Srikrishna, Taedong 17.6.1 Introduction to Auction Design The Internet, which started of as a research project in

### 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

### 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

### ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN. Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015

ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015 ONLINE STEINER FOREST PROBLEM An initially given graph G. s 1 s 2 A sequence of demands (s i, t i ) arriving

### 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

### 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

### CITY UNIVERSITY OF HONG KONG. Revenue Optimization in Internet Advertising Auctions

CITY UNIVERSITY OF HONG KONG l ½ŒA Revenue Optimization in Internet Advertising Auctions p ]zwû ÂÃÙz Submitted to Department of Computer Science õò AX in Partial Fulfillment of the Requirements for the

### 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,

### . P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2

4. Basic feasible solutions and vertices of polyhedra Due to the fundamental theorem of Linear Programming, to solve any LP it suffices to consider the vertices (finitely many) of the polyhedron P of the

### Advice Complexity of Online Coloring for Paths

Advice Complexity of Online Coloring for Paths Michal Forišek 1, Lucia Keller 2, and Monika Steinová 2 1 Comenius University, Bratislava, Slovakia 2 ETH Zürich, Switzerland LATA 2012, A Coruña, Spain Definition

### ..not just for email anymore. Let s do the math.. w Assume 500M searches/day on the web w All search engines combined w Assume 5% commercially viable

Web Spam University of Koblenz Landau..not just for email anymore Users follow search results w Money follows users Spam follows money There is value in getting ranked high w Funnel traffic from SEs to

### Online Ad Auctions. By Hal R. Varian. Draft: February 16, 2009

Online Ad Auctions By Hal R. Varian Draft: February 16, 2009 I describe how search engines sell ad space using an auction. I analyze advertiser behavior in this context using elementary price theory and

### 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

### Keyword Optimization in Search-Based Advertising Markets

Tel Aviv University Raymond and Beverly Sackler Faculty of Exact Sciences The Blavatnik School of Computer Science Keyword Optimization in Search-Based Advertising Markets Thesis submitted in partial fulfillment

### AM 221: Advanced Optimization Spring Prof. Yaron Singer Lecture 7 February 19th, 2014

AM 22: Advanced Optimization Spring 204 Prof Yaron Singer Lecture 7 February 9th, 204 Overview In our previous lecture we saw the application of the strong duality theorem to game theory, and then saw

### 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

### Lecture Notes 6: Approximations for MAX-SAT. 2 Notes about Weighted Vertex Cover Approximation

Algorithmic Methods //00 Lecture Notes 6: Approximations for MAX-SAT Professor: Yossi Azar Scribe:Alon Ardenboim Introduction Although solving SAT is nown to be NP-Complete, in this lecture we will cover

### Dynamic TCP Acknowledgement: Penalizing Long Delays

Dynamic TCP Acknowledgement: Penalizing Long Delays Karousatou Christina Network Algorithms June 8, 2010 Karousatou Christina (Network Algorithms) Dynamic TCP Acknowledgement June 8, 2010 1 / 63 Layout

### MapReduce and Distributed Data Analysis. Sergei Vassilvitskii Google Research

MapReduce and Distributed Data Analysis Google Research 1 Dealing With Massive Data 2 2 Dealing With Massive Data Polynomial Memory Sublinear RAM Sketches External Memory Property Testing 3 3 Dealing With

### Guessing Game: NP-Complete?

Guessing Game: NP-Complete? 1. LONGEST-PATH: Given a graph G = (V, E), does there exists a simple path of length at least k edges? YES 2. SHORTEST-PATH: Given a graph G = (V, E), does there exists a simple

### 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,

### 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

### LINEAR PROGRAMMING PROBLEM: A GEOMETRIC APPROACH

59 LINEAR PRGRAMMING PRBLEM: A GEMETRIC APPRACH 59.1 INTRDUCTIN Let us consider a simple problem in two variables x and y. Find x and y which satisfy the following equations x + y = 4 3x + 4y = 14 Solving

### 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

### CSE 101: Design and Analysis of Algorithms Homework 4 Winter 2016 Due: February 25, 2016

CSE 101: Design and Analysis of Algorithms Homework 4 Winter 2016 Due: February 25, 2016 Exercises (strongly recommended, do not turn in) 1. (DPV textbook, Problem 7.11.) Find the optimal solution to the

### Lecture 21: Competitive Analysis Steven Skiena. skiena

Lecture 21: Competitive Analysis Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Motivation: Online Problems Many problems

### 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.

### 4.4 The recursion-tree method

4.4 The recursion-tree method Let us see how a recursion tree would provide a good guess for the recurrence = 3 4 ) Start by nding an upper bound Floors and ceilings usually do not matter when solving