# Measuring Rationality with the Minimum Cost of Revealed Preference Violations. Mark Dean and Daniel Martin. Online Appendices - Not for Publication

Save this PDF as:

Size: px
Start display at page:

Download "Measuring Rationality with the Minimum Cost of Revealed Preference Violations. Mark Dean and Daniel Martin. Online Appendices - Not for Publication"

## Transcription

1 Measuring Rationality with the Minimum Cost of Revealed Preference Violations Mark Dean and Daniel Martin Online Appendices - Not for Publication 1

2 1 Algorithm for Solving the MASP In this online appendix we formally de ne the maximal acyclical set problem (MASP) and describe our algorithm for solving the problem. A MASP can be solved by identifying and solving a related minimum set covering problem (MSCP) a class of problems which is well studied within the eld of operations research. This is both good news and bad news for solving incidences of MASP. The bad news is that MSCP (and so MASP) are NP-hard. The implication is that there is no known method that can guarantee solution times will increase only as a polynomial function of the number of inputs to MASP. In other words, one cannot guarantee that solution times will not get very large, very quickly as the size of the input data grows. However, the good news is that researchers in operations research have developed a number of tools that in practice solve MSCP quickly and exactly for many data sets. The equivalence between MSCP and MASP means that these techniques can be adopted wholesale for solving MASP. Solving MASP is equivalent to calculating the Houtman-Max Index (HMI). To adjust the algorithm to calculate Minimum Cost Index (MCI), all that is required is to (1) remove relations rather than observations, (2) minimize cost instead of size, and (3) divide the solution by total expenditure. 1.1 De nitions MASP is the problem of nding the size of the largest subset of a set of choice data that generates acyclical revealed preference relations. The primitives of the problem are a grand set of alternatives Z, a set of observations X and a relation function D : X! 2 ZZ that characterizes a set of binary relations on Z generated by each observation in X. We call the triple fz; X; Dg a data set. 1 1 We assume that X is nite. In this case, we can solve MASP whatever the cardinality of Z. Moreover, acyclicality is enough to guarantee that choices can be rationalized by utility maximization even if Z is uncountable. This is because we can concentrate on the ( nite) set of objects Z that are chosen in any observation X. If the data is acyclic, we can generate a utility function u : Z! R that rationalizes choice 2

3 As an example, consider the case of a laboratory experiment in which we observe a subject making choices from subsets of Z. Furthermore, assume that we are prepared to say that the chosen object in any set is strictly preferred to all the other available alternatives. 2 In this case we could think of each observation in X as consisting of a tuple (z; A) with z 2 A and A 2 2 Z =;., implying that alternative z has been observed as being chosen from the set of alternatives A. The function D : X! 2 ZZ would then be de ned as the revealed preference relations generated by X: 8 (z; A) 2 X; D(z; A) = f(z; y) j y 2 A= fzgg We denote by the binary relation x Z Z the relations generated by the observation x 2 X, so that x = D (x). For any B X, we de ne the binary relation B on Z as z B w if, for some x 2 B, z x w: For an arbitrary binary relation, a cycle refers to a set of alternatives z 1 ; z 2 ; :::; z n 2 Z such that z 1 z 2 :::: z n z 1. We say that a set of observations B X is acyclic if the binary relation B generated by B contains no cycles. Thus, we de ne MASP as the problem of nding the size of the largest subset B X such that the resulting binary relation B is acyclic. De nition 1 The maximal acyclical set problem (MASP) for a data set fz; X; Dg is the problem of nding the size of a set B X such that (i) B is acyclic (ii) if B 0 X and jb 0 j > jbj, then B 0 is not acyclic between these objects. All remaining alternatives can be assumed to have a utility equal to min z2z u(z) 1. 2 Note that this assumption is not central to our methodology, which is exible enough to cope with almost any de nition of revealed preference. For example, if we are observing choices from budget sets, we could say that x is revealed preferred to y only if x was chosen when y was available at strictly lower cost. The precise assumption is de ned by the nature of the mapping D. 3

4 In other words, MASP is the problem of nding the size of the largest acyclical subsets of X. Note that the maximal acyclical set may not be unique. Next, we de ne a MSCP. In order to explain the idea behind this class of problems, we illustrate it with the following example: Example 1 Imagine you are setting up a cellphone network and need to buy rights to bandwidth in all 50 states. However, bandwidth is being sold in packages of di erent states (e.g. package 1 includes Alabama, Rhode Island and Wyoming, package 2 includes South Dakota, Minnesota and Wyoming and so on). Each package has a particular cost. The problem you face as a cellphone provider is: What collection of packages should I buy to ensure some bandwidth in all 50 states at the lowest possible cost? In other words, what is the minimum cost way of covering all 50 states? A formal statement of this class of problems is as follows: 3 De nition 2 Let S be a ( nite) set, 2 S be a collection of subsets of S and k : S! R be a cost function which attaches a cost to each element of S. A covering of is a subset T S such that \ T 6=? 8 2. In other words, every set in contains at least one element of T. A minimum set covering problem (MSCP) is the problem of nding the minimum cost of covering of, or X min T 22 S s2t k(s) subject to \ T 6=? 8 2 Again, note that the minimum covering set may not be unique. In the bandwidth example above, we can let S be the set of packages and be a collection of 50 sets, one for each state, containing the packages which cover each state (e.g. if Alabama was covered by packages 1, 7 and 9 then 1 = f1; 7; 9g, if Alaska was covered by packages 3, 3 Note that some people call the problem stated in this way as the minimum hitting problem. However, Ausiello et al. [1980] show that this is equivalent to other statements of the minimum set covering problem. 4

5 14, 19 and 23 then 2 = f3; 14; 19; 23g and so on). k would contain information on the cost of each package. 1.2 Equivalence of MASP and MSCP In order to show the equivalence of MASP and MSCP, we need to formalize the concept of the set of cycles generated by a data set. De nition 3 A cycle generated by a data set fx; Z; Dg consists of a non-repeating sequence z 1 ; :::; z n in Z and a sequence x 1 ; :::; x n in X such that z 1 x1 : : : xn 1 z n xn z 1 Let C denote the set of all cycles generated by X. We will say that observation x 2 X breaks a cycle c 2 C if x appears in the sequence x 1 ; :::; x n. Note that if a subset B of X breaks all cycles in C, then the complement of that subset X=B is acyclic. Next, we will de ne the components of a complimentary MSCP for a particular MASP in the following way. De nition 4 For the MASP associated with a data set fx; Z; Dg, we de ne the complimentary minimum set covering problem by the following elements S, and k: 1. S = X 2. k(x) = 1 8 x 2 S 3. = (c) j c 2 C, where (c) = fx 2 X j x breaks cg With this structure, the problem of nding the size of the smallest subset of X which breaks all cycles in C is the same as nding the cost of the minimum covering of. Further, a smallest subset of X which breaks all cycles in C is the complement of a largest subset of X that is acyclic. 5

6 Theorem 1 For the MASP associated with a data set fx; Z; Dg, if some number s is the solution to the complimentary MSCP ( S,, k), then jxj s is the solution to the MASP. Proof. See Appendix 3. Thus any MASP can be solved by solving the equivalent MSCP. While the MSCP is NPhard, these problems have been studied exhaustively in the operations research literature because they can be applied to many real world situations, such as train scheduling and city planning. As a result, algorithms have been developed to solve or approximate solutions to MSCP quickly for larger and larger data sets. 1.3 A Description of the Algorithm Using the result of Theorem 1 to solve MASP associated with a data set fx; Z; Dg requires two algorithmic components. First, in order to construct the set, we need an algorithm that identi es the set of all cycles C generated by MASP, as well as the observations x 2 X that break each cycle c 2 C. Second, we need an algorithm to solve the complimentary MSCP. Here we describe brie y how we implement each of these stages. In order to nd the set of cycles of C, we use a modi cation of Johnson s algorithm (Johnson [1975]) a computationally e cient graph theoretic algorithm. Johnson s algorithm is based on depth- rst search, a standard approach to nding cycles, which looks at the objects preferred to an initial object, then looks for the objects that are preferred to the rst of those preferred objects and so on until a cycle is found or the process terminates. At that point, the algorithm goes back one level and proceeds from the second preferred object until all possibilities are exhausted. To gain e ciency, Johnson adds a blocking function to prevent redundant searching on the tree, which gives it a computation time upper bound of O ((n + e) (c + 1)), where n is the number of nodes (in this case jzj), e is the number of edges (jd(x)j) and c is the number of cycles (jcj). In order to increase e ciency, we rst apply the elementary rules of Guardabassi [1971] to absorb dominated nodes and remove singular edges, which are inconsequential for nding 6

7 a solution to MASP. Note that this technique cannot be employed when calculating the Minimum Cost Index (MCI). Next, we need a method for solving the companion MSCP, which is NP-hard. These problems have been studied exhaustively in the operations research literature because they can be applied to many real world situations, such as train scheduling and city planning. As a result, algorithms have been developed to solve or approximate solutions to MSCP quickly for larger and larger data sets. The most widely used solution methods for MSCP are a class of branch and bound algorithms that nd an exact solution by iteratively relaxing the corresponding binary integer programming problem so that linear programming techniques can be used to create bounds on the problem. When MSCP is written as a binary integer programming problem, each cover is given a value of 0 or 1, where 1 signi es that a cover is included in the minimum cover. With this speci cation, MSCP can be written in the form: min x2f0;1g n f 0 x subject to Ax b where n is the number of covers, x is a vector of cover values, f is a vector that includes the cost of each cover, b is a vector of 1 s and a ij = 1 if cover j covers node i. This problem is relaxed by allowing the value assigned to each cover to be any real number between 0 and 1, which gives a lower bound to the solution of the binary integer problem. Now imagine the binary integer problem as a tree, where each fork is a decision whether or not to include a single cover in the minimum cover. The bound given by the relaxed problem allows the algorithm to remove many branches of the tree from consideration. These algorithms have been integrated into standard Integer Programming (IP) software packages. Many programming languages include optimization functions that use an internal IP solver (including Matlab). There are also a variety of specialist solvers available on a commercial (e.g. CPLEX) and noncommercial (e.g. SCIP and MINTO) basis. For our analysis, we use the GLPK callable C library ( which acts as an internal solver for our C++ program. By using an internal solver, we can easily pass data from Johnson s algorithm to the IP solver. 7

8 2 MASP and NP Completeness One advantage of connecting MASP with an associated MSCP is that it allows us to relate MASP to the concept of NP-completeness. Introduced by Cook [1971], NP-completeness is a property of decision problems. The set of NP-complete problems is a subset of the class of problems that are NP (which stands for Nondeterministic Polynomial time). A problem is NP if its solution can be veri ed in polynomial time 4 ; though importantly, a solution cannot necessarily be found in polynomial time. A decision problem M is described as NP-complete if it satis es two properties: 1. M is NP 2. Every problem in NP is reducible 5 to M 1. A problem is called NP-hard if it satis es condition 2, whether or not it satis es condition The class of NP-complete problems is interesting because, if any such problem can be solved in polynomial time (denoted as the class P of decision problems), then any problem in NP can be solved in polynomial time. In other words, this would imply that any problem whose solution can be veri ed in polynomial time can also be solved in polynomial time. Whether or not this is true is one of the big open problems in mathematics. 6 Thus, the quest to nd an algorithm that will solve an NP-complete problem in polynomial time is taking on one of the more di cult problems of mathematics. MSCP as we have described it above is NP-hard rather than NP-complete, because a solution cannot be veri ed in polynomial time. The decision version of MSCP, which asks whether there is a cover of with total cost less than some value V, is NP-complete (Karp 4 The run time of the veri er is no greater than a polynomial of the problem size. 5 Meaning there is a (polynomial time) transformation of any problem in NP to M. Thus, M can be used once as a subroutine to solve any problem in NP. 6 The Clay Mathematics Institute is o ering a prize of \$1 million for anyone who can prove either P=NP or P6=NP. 8

9 [1972]). However, the property of NP-hardness alone means that nding a polynomial time solution algorithm for the optimization version of MSCP would be enough to show that all NP-complete problems could be solved in polynomial time, making such a algorithm beyond the current state of mathematical knowledge. Does this mean that there is no known way of solving MASP in polynomial time? To prove this, we have to show that MASP is NP-hard. The standard way of doing this is to show that a known NP-complete problem is reducible to MASP, which implies that all NP problems are reducible to MASP. We show this to be the case by showing that the decision version of MSCP is reducible to MASP. This implies that the search for an algorithm that is guaranteed to solve MASP in polynomial time is essentially futile. 9

10 3 Proofs Theorem: For the MASP associated with a data set fx; Z; Dg, if some number s is the solution to the complimentary MSCP ( S,, k), then jxj s is the solution to the MASP. Proof. Let T be a minimum cost covering of S. In order to prove the result, we need to show two things about the set A = X=T : First, that A itself is acyclic. Second, that any set A 0 X such that ja 0 j > jaj contains a cycle. We prove both of these claims by contradiction. 1. Assume A is not acyclic. Then, there exists some cycle z 1 A ::: A z n A z 1. But this implies there exists some cycle c 2 C such that (c) A, contradicting the claim that T \ 6=? for all. 2. Let A 0 be an acyclic set such that ja 0 j > jaj, and consider the set T 0 = X=A 0. It must be true that T 0 \ 6=? 8 2. If not, then there exists some 2 such that A 0. This in turn implies that 9 z 1 x1 : : : xn 1 z n xn z 1 ) z 1 A0 ::: A0 z n A0 z 1 and so A 0 would not be acyclic. But, as ja 0 j > jaj, then jt j > jt 0 j, in turn implying that X k(s) = jt j > jt 0 j = X k(s) s2t 0 s2t which contradicts the fact that T is a minimum cost covering of S. Theorem: MASP is NP-hard. Proof. To show that MASP is NP-hard, we need to show that a known NP-complete problem can be reduced to MASP. We will use the decision version of MSCP, which was shown to be NP-complete by Garey and Johnson [1979]: Given a ( nite) set S, a collection of subsets 2 S, and an integer k does there exist a covering T of of size k or less? 10

11 Let X = S and Z = S, and apply an arbitrary index I to the elements of S. Next, de ne a function c () = f(s i ; s j ) j s i ; s j 2 and i < j s.t. i < k < jg [ f(s j ; s i ) j s i ; s j 2 s.t. k < i and j > kg : This creates a cycle from each cover that begins with the lowest indexed object, moves to the highest indexed object, and ends with the lowest indexed object again. For example, c (fs 2 ; s 3 ; s 5 g) = f(s 2 ; s 3 ) ; (s 3 ; s 5 ) ; (s 5 ; s 2 )g : Finally, let R = [ 2 c () and D (s i ) = fr 2 R j r 1 = s i g : Clearly, this simple structure fx; Z; Dg can be added in polynomial time. Further, this structure can be used to formulate a MASP, which returns M, the size of a maximal acyclical set A. We claim that jxj jaj jt j if and only if there exists a covering of of size k or less. First, assume that jxj jaj jt j and that there does not exist a covering of of size k or less. Let T = X=A. It must be that T covers all 2 because T breaks the cycles in D by including a member of each cycle, and thus, it contains a member of each 2. But then T is a covering of size k or less, contradicting the latter assumption. Second, assume that there exists a covering of of size k or less and that jxj jaj > jt j. Let A = X=T. It must be that A is acyclic because T contains an element from every cycle created by the c function. But then by the second assumption jxj jaj > jxj ja j jaj < ja j which contradicts that A is a maximal acyclical set. 11

### Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

### Computer Algorithms. NP-Complete Problems. CISC 4080 Yanjun Li

Computer Algorithms NP-Complete Problems NP-completeness The quest for efficient algorithms is about finding clever ways to bypass the process of exhaustive search, using clues from the input in order

### Discrete Mathematics (2009 Spring) Induction and Recursion (Chapter 4, 3 hours)

Discrete Mathematics (2009 Spring) Induction and Recursion (Chapter 4, 3 hours) Chih-Wei Yi Dept. of Computer Science National Chiao Tung University April 17, 2009 4.1 Mathematical Induction 4.1 Mathematical

### 160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

### ! 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

### ON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu

ON THE COMPLEXITY OF THE GAME OF SET KAMALIKA CHAUDHURI, BRIGHTEN GODFREY, DAVID RATAJCZAK, AND HOETECK WEE {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ABSTRACT. Set R is a card game played with a

### 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

### Representation of functions as power series

Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

### 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

### Math 443/543 Graph Theory Notes 4: Connector Problems

Math 443/543 Graph Theory Notes 4: Connector Problems David Glickenstein September 19, 2012 1 Trees and the Minimal Connector Problem Here is the problem: Suppose we have a collection of cities which we

Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions

### M. S. DASKIN², O. MAIMON, A. SHTUB and D. BRAHA *

int. j. prod. res., 1997, vol. 35, no. 6, 1617± 1638 Grouping components in printed circuit board assembly with limited component staging capacity and single card setup: problem characteristics and solution

### Models and Methods for the Microdata Protection Problem

Journal of Of cial Statistics, Vol. 14, No. 4, 1998, pp. 437±447 Models and Methods for the Microdata rotection roblem C.A.J. Hurkens and S.R. Tiourine 1 This article focuses on the mathematical modeling

### 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

### ! 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

### Pricing Cloud Computing: Inelasticity and Demand Discovery

Pricing Cloud Computing: Inelasticity and Demand Discovery June 7, 203 Abstract The recent growth of the cloud computing market has convinced many businesses and policy makers that cloud-based technologies

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

### Section 2.4: Equations of Lines and Planes

Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y

Reihe Informatik 11 / 1996 Constructing Optimal Bushy Processing Trees for Join Queries is NP-hard Wolfgang Scheufele Guido Moerkotte 1 Constructing Optimal Bushy Processing Trees for Join Queries is NP-hard

### 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

### Discuss the size of the instance for the minimum spanning tree problem.

3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can

### 4.6 Null Space, Column Space, Row Space

NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear

### 1.2 Solving a System of Linear Equations

1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

### Discrete Mathematics

Discrete Mathematics Chih-Wei Yi Dept. of Computer Science National Chiao Tung University March 16, 2009 2.1 Sets 2.1 Sets 2.1 Sets Basic Notations for Sets For sets, we ll use variables S, T, U,. We can

### Single machine parallel batch scheduling with unbounded capacity

Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University

### Tutorial 8. NP-Complete Problems

Tutorial 8 NP-Complete Problems Decision Problem Statement of a decision problem Part 1: instance description defining the input Part 2: question stating the actual yesor-no question A decision problem

### 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

### An Exact Algorithm for Steiner Tree Problem on Graphs

International Journal of Computers, Communications & Control Vol. I (2006), No. 1, pp. 41-46 An Exact Algorithm for Steiner Tree Problem on Graphs Milan Stanojević, Mirko Vujošević Abstract: The paper

### Notes on Chapter 1, Section 2 Arithmetic and Divisibility

Notes on Chapter 1, Section 2 Arithmetic and Divisibility August 16, 2006 1 Arithmetic Properties of the Integers Recall that the set of integers is the set Z = f0; 1; 1; 2; 2; 3; 3; : : :g. The integers

### CSC 373: Algorithm Design and Analysis Lecture 16

CSC 373: Algorithm Design and Analysis Lecture 16 Allan Borodin February 25, 2013 Some materials are from Stephen Cook s IIT talk and Keven Wayne s slides. 1 / 17 Announcements and Outline Announcements

### 14.451 Lecture Notes 10

14.451 Lecture Notes 1 Guido Lorenzoni Fall 29 1 Continuous time: nite horizon Time goes from to T. Instantaneous payo : f (t; x (t) ; y (t)) ; (the time dependence includes discounting), where x (t) 2

### Introduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by

Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes

### 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

### NP-complete? NP-hard? Some Foundations of Complexity. Prof. Sven Hartmann Clausthal University of Technology Department of Informatics

NP-complete? NP-hard? Some Foundations of Complexity Prof. Sven Hartmann Clausthal University of Technology Department of Informatics Tractability of Problems Some problems are undecidable: no computer

### NP-Completeness I. Lecture 19. 19.1 Overview. 19.2 Introduction: Reduction and Expressiveness

Lecture 19 NP-Completeness I 19.1 Overview In the past few lectures we have looked at increasingly more expressive problems that we were able to solve using efficient algorithms. In this lecture we introduce

### Portfolio selection based on upper and lower exponential possibility distributions

European Journal of Operational Research 114 (1999) 115±126 Theory and Methodology Portfolio selection based on upper and lower exponential possibility distributions Hideo Tanaka *, Peijun Guo Department

### Sigma Field Notes for Economics 770 : Econometric Theory

1 Sigma Field Notes for Economics 770 : Econometric Theory Jonathan B. Hill Dept. of Economics, University of North Carolina 0. DEFINITIONS AND CONVENTIONS The following is a small list of repeatedly used

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Elements of probability theory

2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted

### A Working Knowledge of Computational Complexity for an Optimizer

A Working Knowledge of Computational Complexity for an Optimizer ORF 363/COS 323 Instructor: Amir Ali Ahmadi TAs: Y. Chen, G. Hall, J. Ye Fall 2014 1 Why computational complexity? What is computational

### Generating models of a matched formula with a polynomial delay

Generating models of a matched formula with a polynomial delay Petr Savicky Institute of Computer Science, Academy of Sciences of Czech Republic, Pod Vodárenskou Věží 2, 182 07 Praha 8, Czech Republic

### NP-completeness and the real world. NP completeness. NP-completeness and the real world (2) NP-completeness and the real world

-completeness and the real world completeness Course Discrete Biological Models (Modelli Biologici Discreti) Zsuzsanna Lipták Imagine you are working for a biotech company. One day your boss calls you

### 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;

### 36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?

36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this

### 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

### Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows

TECHNISCHE UNIVERSITEIT EINDHOVEN Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows Lloyd A. Fasting May 2014 Supervisors: dr. M. Firat dr.ir. M.A.A. Boon J. van Twist MSc. Contents

### 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 }

### COMS4236: Introduction to Computational Complexity. Summer 2014

COMS4236: Introduction to Computational Complexity Summer 2014 Mihalis Yannakakis Lecture 17 Outline conp NP conp Factoring Total NP Search Problems Class conp Definition of NP is nonsymmetric with respect

### OHJ-2306 Introduction to Theoretical Computer Science, Fall 2012 8.11.2012

276 The P vs. NP problem is a major unsolved problem in computer science It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a \$ 1,000,000 prize for the

### Introduction to Logic in Computer Science: Autumn 2006

Introduction to Logic in Computer Science: Autumn 2006 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today Now that we have a basic understanding

### 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

### 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

### 11. APPROXIMATION ALGORITHMS

11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Lecture slides by Kevin Wayne Copyright 2005

### Chapter 3: Section 3-3 Solutions of Linear Programming Problems

Chapter 3: Section 3-3 Solutions of Linear Programming Problems D. S. Malik Creighton University, Omaha, NE D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming

### 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

### Lecture 6. Inverse of Matrix

Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

### Chapter 7. Sealed-bid Auctions

Chapter 7 Sealed-bid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)

### Exact Nonparametric Tests for Comparing Means - A Personal Summary

Exact Nonparametric Tests for Comparing Means - A Personal Summary Karl H. Schlag European University Institute 1 December 14, 2006 1 Economics Department, European University Institute. Via della Piazzuola

### varies, if the classication is performed with a certain accuracy, and if a complex input-output association. There exist theoretical guarantees

Training a Sigmoidal Network is Dicult Barbara Hammer, University of Osnabruck, Dept. of Mathematics/Comp. Science, Albrechtstrae 28, 49069 Osnabruck, Germany Abstract. In this paper we show that the loading

### On the Relationship between Classes P and NP

Journal of Computer Science 8 (7): 1036-1040, 2012 ISSN 1549-3636 2012 Science Publications On the Relationship between Classes P and NP Anatoly D. Plotnikov Department of Computer Systems and Networks,

### Bargaining Solutions in a Social Network

Bargaining Solutions in a Social Network Tanmoy Chakraborty and Michael Kearns Department of Computer and Information Science University of Pennsylvania Abstract. We study the concept of bargaining solutions,

### 24. The Branch and Bound Method

24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

### Lecture 7: NP-Complete Problems

IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NP-Complete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit

### Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

### 5. Factoring by the QF method

5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the

### Local periods and binary partial words: An algorithm

Local periods and binary partial words: An algorithm F. Blanchet-Sadri and Ajay Chriscoe Department of Mathematical Sciences University of North Carolina P.O. Box 26170 Greensboro, NC 27402 6170, USA E-mail:

### Similarity-Based Mistakes in Choice

INSTITUTO TECNOLOGICO AUTONOMO DE MEXICO CENTRO DE INVESTIGACIÓN ECONÓMICA Working Papers Series Similarity-Based Mistakes in Choice Fernando Payro Centro de Investigación Económica, ITAM Levent Ülkü Centro

### A Note on Maximum Independent Sets in Rectangle Intersection Graphs

A Note on Maximum Independent Sets in Rectangle Intersection Graphs Timothy M. Chan School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.ca September 12,

### 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

### Integrating Benders decomposition within Constraint Programming

Integrating Benders decomposition within Constraint Programming Hadrien Cambazard, Narendra Jussien email: {hcambaza,jussien}@emn.fr École des Mines de Nantes, LINA CNRS FRE 2729 4 rue Alfred Kastler BP

### Biinterpretability up to double jump in the degrees

Biinterpretability up to double jump in the degrees below 0 0 Richard A. Shore Department of Mathematics Cornell University Ithaca NY 14853 July 29, 2013 Abstract We prove that, for every z 0 0 with z

### Scheduling. Open shop, job shop, flow shop scheduling. Related problems. Open shop, job shop, flow shop scheduling

Scheduling Basic scheduling problems: open shop, job shop, flow job The disjunctive graph representation Algorithms for solving the job shop problem Computational complexity of the job shop problem Open

### 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

### 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

### Bicolored Shortest Paths in Graphs with Applications to Network Overlay Design

Bicolored Shortest Paths in Graphs with Applications to Network Overlay Design Hongsik Choi and Hyeong-Ah Choi Department of Electrical Engineering and Computer Science George Washington University Washington,

### CAPM, Arbitrage, and Linear Factor Models

CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose mean-variance e cient portfolios. By equating these investors

### Finding the critical path in an activity network with time-switch constraints

European Journal of Operational Research 120 (2000) 603±613 www.elsevier.com/locate/orms Theory and Methodology Finding the critical path in an activity network with time-switch constraints Hsu-Hao Yang

### Normalization and Mixed Degrees of Integration in Cointegrated Time Series Systems

Normalization and Mixed Degrees of Integration in Cointegrated Time Series Systems Robert J. Rossana Department of Economics, 04 F/AB, Wayne State University, Detroit MI 480 E-Mail: r.j.rossana@wayne.edu

### Security-Aware Beacon Based Network Monitoring

Security-Aware Beacon Based Network Monitoring Masahiro Sasaki, Liang Zhao, Hiroshi Nagamochi Graduate School of Informatics, Kyoto University, Kyoto, Japan Email: {sasaki, liang, nag}@amp.i.kyoto-u.ac.jp

### The Binomial Distribution

The Binomial Distribution James H. Steiger November 10, 00 1 Topics for this Module 1. The Binomial Process. The Binomial Random Variable. The Binomial Distribution (a) Computing the Binomial pdf (b) Computing

### Classification - Examples

Lecture 2 Scheduling 1 Classification - Examples 1 r j C max given: n jobs with processing times p 1,...,p n and release dates r 1,...,r n jobs have to be scheduled without preemption on one machine taking

### 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,

### Lecture 1: Systems of Linear Equations

MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables

### Complexity Theory. IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar

Complexity Theory IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar Outline Goals Computation of Problems Concepts and Definitions Complexity Classes and Problems Polynomial Time Reductions Examples

### Maximum Likelihood Estimation of an ARMA(p,q) Model

Maximum Likelihood Estimation of an ARMA(p,q) Model Constantino Hevia The World Bank. DECRG. October 8 This note describes the Matlab function arma_mle.m that computes the maximum likelihood estimates

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### IDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS. Steven T. Berry and Philip A. Haile. March 2011 Revised April 2011

IDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS By Steven T. Berry and Philip A. Haile March 2011 Revised April 2011 COWLES FOUNDATION DISCUSSION PAPER NO. 1787R COWLES FOUNDATION

### A Constraint Programming based Column Generation Approach to Nurse Rostering Problems

Abstract A Constraint Programming based Column Generation Approach to Nurse Rostering Problems Fang He and Rong Qu The Automated Scheduling, Optimisation and Planning (ASAP) Group School of Computer Science,

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Problem Set 7 Solutions

8 8 Introduction to Algorithms May 7, 2004 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik Demaine and Shafi Goldwasser Handout 25 Problem Set 7 Solutions This problem set is due in

### Chapter 3: The Multiple Linear Regression Model

Chapter 3: The Multiple Linear Regression Model Advanced Econometrics - HEC Lausanne Christophe Hurlin University of Orléans November 23, 2013 Christophe Hurlin (University of Orléans) Advanced Econometrics

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### 2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8]

Code No: R05220502 Set No. 1 1. (a) Describe the performance analysis in detail. (b) Show that f 1 (n)+f 2 (n) = 0(max(g 1 (n), g 2 (n)) where f 1 (n) = 0(g 1 (n)) and f 2 (n) = 0(g 2 (n)). [8+8] 2. (a)

### 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,

### Key words. multi-objective optimization, approximate Pareto set, bi-objective shortest path

SMALL APPROXIMATE PARETO SETS FOR BI OBJECTIVE SHORTEST PATHS AND OTHER PROBLEMS ILIAS DIAKONIKOLAS AND MIHALIS YANNAKAKIS Abstract. We investigate the problem of computing a minimum set of solutions that

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### Lecture notes on the Turing degrees, AII Graduate Summer School in Logic Singapore, 28 June - 23 July 2010

Lecture notes on the Turing degrees, AII Graduate Summer School in Logic Singapore, 28 June - 23 July 2010 Richard A. Shore 1 Department of Mathematics Cornell University Ithaca NY 14853 January 3, 2012

### Chapter 1. NP Completeness I. 1.1. Introduction. By Sariel Har-Peled, December 30, 2014 1 Version: 1.05

Chapter 1 NP Completeness I By Sariel Har-Peled, December 30, 2014 1 Version: 1.05 "Then you must begin a reading program immediately so that you man understand the crises of our age," Ignatius said solemnly.