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

Save this PDF as:

Size: px
Start display at page:

Download "Discuss the size of the instance for the minimum spanning tree problem."

## Transcription

1 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 be solved in 1 hour with algorithm A on a given computer, and n B 0 be the corresponding size for algorithm B. Give the values of n A 0 e nb 0 for a computer 100 times faster. 3.2 Size of the instances Discuss the size of the instance for the minimum spanning tree problem. 3.3 N P-complete and N P-hard problems Given a directed graph G = (N, A) with rational costs on the arcs, and a pair of nodes s, t, show that the problem of identifying a simple path (i.e., where each node occurs, at most, once) of minimum length from s to t is N P-hard. Show that the associated recognition problem, i.e., Max-SimplePath-r Given a directed graph G = (N, A) with rational costs on the arcs, a pair of nodes s, t and an integer K, does it contain a simple path from s to t of length at least K? is N P-complete. [Hint: propose a polynomial reduction of the problem Hamiltonian-Circuit-r (given a directed graph, does it contain a Hamiltonian circuit?] 3.4 Integer Linear Programming is N P-hard Show that Integer Linear Programming is N P-hard, by showing that the associated recognition problem, is N P-complete. ILP: Given a matrix A Z m n and two vectors b Z m and c Z n, find x {0, 1} n such that Ax b which minimizes c T x. ILP-d: Given a matrix A Z m n and a vector b Z m, is there a vector x {0, 1} n such that Ax b? [Hint: propose a polynomial reduction of SAT (satisfiability problem for boolean clauses) to ILP-r.] 3.5 Complexity and size of the formulation Give an integer linear programming formulation for the problem of finding a spanning tree of minimum cost in a graph G = (V, E). Is the number of constrants polynomial or exponential in Document prepared by L. Liberti, S. Bosio, S. Coniglio, and C. Iuliano. Translation to English by S. Coniglio 1

2 n = V? Is there a relationship between the size of a formulation (number of constraints and variables) and the difficilty of the associated problem? Document prepared by L. Liberti, S. Bosio, S. Coniglio, and C. Iuliano. Translation to English by S. Coniglio 2

3 Solution 3.1 Algorithm complexity Let n A, n B be the size of the largest instances which can be solved, in one hour, with algorithms A, B, on a computer 100 times faster that the original one. Since, by definition of n A 0, (na 0 )2 elementary operations are performed when using algorithm A on the original computer, 100(n A 0 )2 operations can be executed on the faster machine. Therefore, (n A ) 2 = 100(n A 0 )2 and hence n A = 10n A 0. Similarly, for algorithm B, we have that 2nB 0 operations are performed on the original computer, nb 0 are performed on the faster one, and hence n B = n B 0 + log 2(100) < n B To summarize, with algorithm A, which has quadratic complexity, we can solve instances 10 times larger, while, with algorithm B, which has exponential complexity, we can solve instances which are only larger by a tiny quantity of bits. 3.2 Size of the instances An instance for the minimum spanning tree problem amounts to a graph G = (V, E), with a weight c ij per edge {i, j} E. Recall that, given an integer i Z, log i + 1 bits are needed to code it in memory. The +1 addendum takes into account the sign bit. For nonnegative integers, log i bits suffice. For each edge {i, j} E we need to memorize the indices of the nodes where it is incident, using 2 log n bits per node, and the weights c ij, using log c max + 1 bits, where max := max ij E c ij. In total, we need log n + log m + m(2 log n + log c max + 1) bits, i.e., O(m(log n + log c max )). Note that we always suppose m > n, otherwise the graph is not connected and it admits no spanning tree. Usually, the number of bits needed to code a numerical value is taken as a constant. For instance, for a 64 bit machine, we can suppose that we d never deal with instances with more than 2 64 arcs or nodes. Similarly, we can suppose c max 2 64, so that c max can be stored in a single memory word. Under this assumption, the size of an instance is, asymptotically, O(m). Note that, for dense graphs (m n 2 ), the instance has a quadratic, O(n 2 ), size n. For sparse graphs (m < n 2 ), the size is less than quadratic, since n 2 < m 2, and more than linear, since n 2 > m. Document prepared by L. Liberti, S. Bosio, S. Coniglio, and C. Iuliano. Translation to English by S. Coniglio 3

4 3.3 N P-complete and N P-hard problems To prove that Max-SimplePath-r is N P-complete, we need to (i) show that the problem is in N P and (b) show that any other problem in N P can be reduced to our problem in polynomial time. Max-SimplePath-r is clearly in N P, since (a) it is a recognition problem and (b) given any solution to it (a path), we can verify in polynomial time whether (1) it is a path, (2) it is simple, 3) it has total cost > K. To verify (ii), we show that the problem Hamiltonian-Circuit-r, which is known to be N P-comlete, can be reduced in polynomial time to Max-SimplePath-r. Hamiltonian-Circuit-r: Given a directed graph G = (N, A ), does it contain a Hamiltonian circuit, i.e., a circuit where each node of G occurrs exactly once? We show that, given any instance of Hamiltonian-Circuit-r, we can create, in polynomial time, a particular instance of Max-SimplePath-r, such that the answer to Hamiltonian- Circuit-r is yes if and only if the answer to Max-SimplePath-r is yes. Consider, as an instance of Hamiltonian-Circuit-r, the directed graph G = (N, A), with unit arc costs. Let s = t be any node in N. G contains a Hamiltonian circuit if and only if it contains a simple path from s to t of length, at least, N. Indeed, any Hamiltonian circuit is a simple path from a node to itself, of cost N. For this pair of problems the reduction is trivial, as any instance of Hamiltonian-Circuit-r can directly be used as an instance of Max-SimplePath-r. By definition, since Max-SimplePath-r is N P-complete, Max-SimplePath, i.e., the problem of, given a graph and a pair of nodes s, t, finding a simple path of maximum length between s, t, is N P-hard, since it is at least as hard as Max-SimplePath-r (you can use the former to solve the latter) and is not in N P (since it s not a recognition problem). As a consequence, the problem Min-SimplePath, i.e., the problem of, given a directed graph with rational (nonrestricted in sign) arc weights and two nodes s, t, finding a simple path between s, t of minimum length, is N P-hard. Indeed, we can reduce in polynomial time the problem Max-SimplePath-r to the its recognition version Min-SimplePath-r Min-SimplePath-r: Given a directed graph G = (N, A), with rational arc weights, a pair of nodes s, t, and an integer k, does G contain a simple s t path of total length nonlarger than k? Indeed, it suffices to, given an instance of Max-SimplePath-r, composed of G = (V, A), weights c ij, s, t, k, build a new instance G = (V, A ), c ij, k, with V = V, A = A, c ij = c ij, k = k, and solve the Min-SimplePath-r problem on it. 3.4 ILP is N P-hard Consider the recognition problem ILP-r Document prepared by L. Liberti, S. Bosio, S. Coniglio, and C. Iuliano. Translation to English by S. Coniglio 4

5 ILP-r: Given a matrix A Z m n and a vector b Z m, is there a vector x {0, 1} n such that Ax b? Note that the case where we are also given an objective function with cost vector c and an integer k and are asked to find a vector x such that c T x k can be taken into account by adding the constraint c T x k to Ax b. To show that ILP-r is N P-complete, we need to (i) verify that ILP-r is in N P, and (ii) show that we can reduce a NP-complete problem, in polynomial time, to ILP-r. We will use the SAT problem (boolean satisfaibility). SAT: Given m boolean clauses C 1,...,C m (disjunction over 2n literals y 1,...,y n, y 1,...,y n ), is there a valuation which satisfies all the clauses? (ii) ILP-r is in N P since (a) it is a recognition problem and (b) given a vector x {0, 1} n, we can verify in polynomial time whether it satisfies the system Ax b. As to the reduction, it suffices to, given any SAT instance with variables y 1,...,y n and clauses C 1,...,C m, show that we can construct a particular instance of ILP-r with binary variables x 1,...,x n and m linear constraints given as inequalities, such that SAT has answer yes if and only if ILP-r has answer yer. We proceed as follows. We introduce a linear constraint per clause. If the k-th variable in clause C j, 1 j m, is y i, the k-th term of the j-th constraint is x i. If the k-th variable is ȳ i, the k-th term is (1 x i ). We replace the addition operator to the local disjunction one. For example, given the SAT instance we construct the ILP-r instance C 1 = (y 1 y 2 y 3 ) C 2 = (ȳ 1 ȳ 2 ) C 3 = (y 2 ȳ 3 ), x 1 + x 2 + x 3 1 (1 x 1 ) + (1 x 2 ) 1 x 2 + (1 x 3 ) 1 x 1, x 2, x 3 {0, 1}. Given a solution to ILP-r, we can construct the corresponding solution to SAT-r by setting to true all variables at value 1, and to false all those at value Complexity and size of the formulation Consider the graph G = (V, E), with costs c ij on the edges {i, j} E. Associate to each edge {i, j} a variables x ij, such that x ij = 1 if the edge is in the tree and 0 otherwise. The model is Variables x ij {0, 1}, {i, j} E Document prepared by L. Liberti, S. Bosio, S. Coniglio, and C. Iuliano. Translation to English by S. Coniglio 5

6 Formulation min c ij x ij s.t. x ij = V 1 {i,j} E {i,j} E x ij S 1 S V : 2 S < V (subtour elimination). i,j S (cardinality) We have a subtour elimination constraint for each proper subset S of V, of cardinality nonsmaller than 2. Indeed, for S = V, the corresponding constraint amounts to the cardinality constraint. As an exercise, discuss on why the constraints need not be imposed for S = i, i V. Observe that an instance of the problem, given as an ILP, has exponentially many constraints (exponentially w.r.t. n = V ), since we have exactly 2 n n 2 subtour elimination constraints. Any polynomial algorithm (supposing that there is any!) for solving ILP problems will not run in polynomial time, as it will we polynomial in the size of the formulation, which is exponential in that of the instance. Document prepared by L. Liberti, S. Bosio, S. Coniglio, and C. Iuliano. Translation to English by S. Coniglio 6

### 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 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

### 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

### One last point: we started off this book by introducing another famously hard search problem:

S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani 261 Factoring One last point: we started off this book by introducing another famously hard search problem: FACTORING, the task of finding all prime factors

### CHAPTER 9. Integer Programming

CHAPTER 9 Integer Programming An integer linear program (ILP) is, by definition, a linear program with the additional constraint that all variables take integer values: (9.1) max c T x s t Ax b and x integral

### 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

### Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits

Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique

### 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

### 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

### 1. Nondeterministically guess a solution (called a certificate) 2. Check whether the solution solves the problem (called verification)

Some N P problems Computer scientists have studied many N P problems, that is, problems that can be solved nondeterministically in polynomial time. Traditionally complexity question are studied as languages:

### 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

### 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

### 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

### 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

### 2.8 An application of Dynamic Programming to machine renewal

ex-.6-. Foundations of Operations Research Prof. E. Amaldi.6 Shortest paths with nonnegative costs Given the following graph, find a set of shortest paths from node to all the other nodes, using Dijkstra

### 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;

### Introduction to Algorithms. Part 3: P, NP Hard Problems

Introduction to Algorithms Part 3: P, NP Hard Problems 1) Polynomial Time: P and NP 2) NP-Completeness 3) Dealing with Hard Problems 4) Lower Bounds 5) Books c Wayne Goddard, Clemson University, 2004 Chapter

### 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

### 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

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

### 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

### Introduction to Algorithms Review information for Prelim 1 CS 4820, Spring 2010 Distributed Wednesday, February 24

Introduction to Algorithms Review information for Prelim 1 CS 4820, Spring 2010 Distributed Wednesday, February 24 The final exam will cover seven topics. 1. greedy algorithms 2. divide-and-conquer algorithms

### 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

### SIMS 255 Foundations of Software Design. Complexity and NP-completeness

SIMS 255 Foundations of Software Design Complexity and NP-completeness Matt Welsh November 29, 2001 mdw@cs.berkeley.edu 1 Outline Complexity of algorithms Space and time complexity ``Big O'' notation Complexity

### 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

### 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

### NP-Completeness. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University

NP-Completeness CptS 223 Advanced Data Structures Larry Holder School of Electrical Engineering and Computer Science Washington State University 1 Hard Graph Problems Hard means no known solutions with

### 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

### Notes on NP Completeness

Notes on NP Completeness Rich Schwartz November 10, 2013 1 Overview Here are some notes which I wrote to try to understand what NP completeness means. Most of these notes are taken from Appendix B in Douglas

### 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,

### RANDOMIZATION IN APPROXIMATION AND ONLINE ALGORITHMS. Manos Thanos Randomized Algorithms NTUA

RANDOMIZATION IN APPROXIMATION AND ONLINE ALGORITHMS 1 Manos Thanos Randomized Algorithms NTUA RANDOMIZED ROUNDING Linear Programming problems are problems for the optimization of a linear objective function,

### ! 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

### Approximated Probabilistic Answer Set Programming

Approximated Probabilistic Answer Set Programming Department of Computer Science Universidade de São Paulo São Paulo, Brazil 2014 Toy Example 4 6 2 5 1 3 Limit the percentage of time the edge (1, 3) is

### 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.

### Inverse Optimization by James Orlin

Inverse Optimization by James Orlin based on research that is joint with Ravi Ahuja Jeopardy 000 -- the Math Programming Edition The category is linear objective functions The answer: When you maximize

### 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:

### Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

### The Classes P and NP

The Classes P and NP We now shift gears slightly and restrict our attention to the examination of two families of problems which are very important to computer scientists. These families constitute the

### Efficiency of algorithms. Algorithms. Efficiency of algorithms. Binary search and linear search. Best, worst and average case.

Algorithms Efficiency of algorithms Computational resources: time and space Best, worst and average case performance How to compare algorithms: machine-independent measure of efficiency Growth rate Complexity

### Completely Positive Cone and its Dual

On the Computational Complexity of Membership Problems for the Completely Positive Cone and its Dual Peter J.C. Dickinson Luuk Gijben July 3, 2012 Abstract Copositive programming has become a useful tool

### Reductions & NP-completeness as part of Foundations of Computer Science undergraduate course

Reductions & NP-completeness as part of Foundations of Computer Science undergraduate course Alex Angelopoulos, NTUA January 22, 2015 Outline Alex Angelopoulos (NTUA) FoCS: Reductions & NP-completeness-

### 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

### ! 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

### 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

### Minimum Caterpillar Trees and Ring-Stars: a branch-and-cut algorithm

Minimum Caterpillar Trees and Ring-Stars: a branch-and-cut algorithm Luidi G. Simonetti Yuri A. M. Frota Cid C. de Souza Institute of Computing University of Campinas Brazil Aussois, January 2010 Cid de

### 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

### Graphs and Network Flows IE411 Lecture 1

Graphs and Network Flows IE411 Lecture 1 Dr. Ted Ralphs IE411 Lecture 1 1 References for Today s Lecture Required reading Sections 17.1, 19.1 References AMO Chapter 1 and Section 2.1 and 2.2 IE411 Lecture

### 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

### 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.

Answers to Homework 1 p. 13 1.2 3 What is the smallest value of n such that an algorithm whose running time is 100n 2 runs faster than an algorithm whose running time is 2 n on the same machine? Find the

### 8.1 Min Degree Spanning Tree

CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

### Polytope Examples (PolyComp Fukuda) Matching Polytope 1

Polytope Examples (PolyComp Fukuda) Matching Polytope 1 Matching Polytope Let G = (V,E) be a graph. A matching in G is a subset of edges M E such that every vertex meets at most one member of M. A matching

### 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

### 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

### Introduction to Flocking {Stochastic Matrices}

Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur - Yvette May 21, 2012 CRAIG REYNOLDS - 1987 BOIDS The Lion King CRAIG REYNOLDS

### Max Flow, Min Cut, and Matchings (Solution)

Max Flow, Min Cut, and Matchings (Solution) 1. The figure below shows a flow network on which an s-t flow is shown. The capacity of each edge appears as a label next to the edge, and the numbers in boxes

### 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

### Math 181 Handout 16. Rich Schwartz. March 9, 2010

Math 8 Handout 6 Rich Schwartz March 9, 200 The purpose of this handout is to describe continued fractions and their connection to hyperbolic geometry. The Gauss Map Given any x (0, ) we define γ(x) =

### 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,

### 6. Mixed Integer Linear Programming

6. Mixed Integer Linear Programming Javier Larrosa Albert Oliveras Enric Rodríguez-Carbonell Problem Solving and Constraint Programming (RPAR) Session 6 p.1/40 Mixed Integer Linear Programming A mixed

### Counting Falsifying Assignments of a 2-CF via Recurrence Equations

Counting Falsifying Assignments of a 2-CF via Recurrence Equations Guillermo De Ita Luna, J. Raymundo Marcial Romero, Fernando Zacarias Flores, Meliza Contreras González and Pedro Bello López Abstract

### Page 1. CSCE 310J Data Structures & Algorithms. CSCE 310J Data Structures & Algorithms. P, NP, and NP-Complete. Polynomial-Time Algorithms

CSCE 310J Data Structures & Algorithms P, NP, and NP-Complete Dr. Steve Goddard goddard@cse.unl.edu CSCE 310J Data Structures & Algorithms Giving credit where credit is due:» Most of the lecture notes

### 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

### Data Structures and Algorithms Written Examination

Data Structures and Algorithms Written Examination 22 February 2013 FIRST NAME STUDENT NUMBER LAST NAME SIGNATURE Instructions for students: Write First Name, Last Name, Student Number and Signature where

### 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

### 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

### Integer Programming Approach to Printed Circuit Board Assembly Time Optimization

Integer Programming Approach to Printed Circuit Board Assembly Time Optimization Ratnesh Kumar Haomin Li Department of Electrical Engineering University of Kentucky Lexington, KY 40506-0046 Abstract A

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### 4. Factor polynomials over complex numbers, describe geometrically, and apply to real-world situations. 5. Determine and apply relationships among syn

I The Real and Complex Number Systems 1. Identify subsets of complex numbers, and compare their structural characteristics. 2. Compare and contrast the properties of real numbers with the properties of

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

Measuring Rationality with the Minimum Cost of Revealed Preference Violations Mark Dean and Daniel Martin Online Appendices - Not for Publication 1 1 Algorithm for Solving the MASP In this online appendix

### 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

### 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

### Chapter. NP-Completeness. Contents

Chapter 13 NP-Completeness Contents 13.1 P and NP......................... 593 13.1.1 Defining the Complexity Classes P and NP...594 13.1.2 Some Interesting Problems in NP.......... 597 13.2 NP-Completeness....................

### P versus NP, and More

1 P versus NP, and More Great Ideas in Theoretical Computer Science Saarland University, Summer 2014 If you have tried to solve a crossword puzzle, you know that it is much harder to solve it than to verify

### 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.

### 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,

### 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 }

### The Transportation Problem: LP Formulations

The Transportation Problem: LP Formulations An LP Formulation Suppose a company has m warehouses and n retail outlets A single product is to be shipped from the warehouses to the outlets Each warehouse

### 2 The Euclidean algorithm

2 The Euclidean algorithm Do you understand the number 5? 6? 7? At some point our level of comfort with individual numbers goes down as the numbers get large For some it may be at 43, for others, 4 In

MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

### 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

### Bounded Treewidth in Knowledge Representation and Reasoning 1

Bounded Treewidth in Knowledge Representation and Reasoning 1 Reinhard Pichler Institut für Informationssysteme Arbeitsbereich DBAI Technische Universität Wien Luminy, October 2010 1 Joint work with G.

### Dynamic programming. Doctoral course Optimization on graphs - Lecture 4.1. Giovanni Righini. January 17 th, 2013

Dynamic programming Doctoral course Optimization on graphs - Lecture.1 Giovanni Righini January 1 th, 201 Implicit enumeration Combinatorial optimization problems are in general NP-hard and we usually

### Boolean Representations and Combinatorial Equivalence

Chapter 2 Boolean Representations and Combinatorial Equivalence This chapter introduces different representations of Boolean functions. It then discuss the applications of these representations for proving

### Solving systems of linear equations over finite fields

Solving systems of linear equations over finite fields June 1, 2007 1 Introduction 2 3 4 Basic problem Introduction x 1 + x 2 + x 3 = 1 x 1 + x 2 = 1 x 1 + x 2 + x 4 = 1 x 2 + x 4 = 0 x 1 + x 3 + x 4 =

### Lecture 2: The Complexity of Some Problems

IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 2: The Complexity of Some Problems David Mix Barrington and Alexis Maciel July 18, 2000

### 6.852: Distributed Algorithms Fall, 2009. Class 2

.8: Distributed Algorithms Fall, 009 Class Today s plan Leader election in a synchronous ring: Lower bound for comparison-based algorithms. Basic computation in general synchronous networks: Leader election

### Branch and Cut for TSP

Branch and Cut for TSP jla,jc@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark 1 Branch-and-Cut for TSP Branch-and-Cut is a general technique applicable e.g. to solve symmetric

### 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

### Analysis of Algorithms, I

Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadth-first search (BFS) 4 Applications

### A network flow algorithm for reconstructing. binary images from discrete X-rays

A network flow algorithm for reconstructing binary images from discrete X-rays Kees Joost Batenburg Leiden University and CWI, The Netherlands kbatenbu@math.leidenuniv.nl Abstract We present a new algorithm

### Bounded-width QBF is PSPACE-complete

Bounded-width QBF is PSPACE-complete Albert Atserias 1 and Sergi Oliva 2 1 Universitat Politècnica de Catalunya Barcelona, Spain atserias@lsi.upc.edu 2 Universitat Politècnica de Catalunya Barcelona, Spain

### 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

### INTEGER PROGRAMMING. Integer Programming. Prototype example. BIP model. BIP models

Integer Programming INTEGER PROGRAMMING In many problems the decision variables must have integer values. Example: assign people, machines, and vehicles to activities in integer quantities. If this is

### Fall 2015 Midterm 1 24/09/15 Time Limit: 80 Minutes

Math 340 Fall 2015 Midterm 1 24/09/15 Time Limit: 80 Minutes Name (Print): This exam contains 6 pages (including this cover page) and 5 problems. Enter all requested information on the top of this page,

### 1 Basic Definitions and Concepts in Graph Theory

CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered

### Informatique Fondamentale IMA S8

Informatique Fondamentale IMA S8 Cours 4 : graphs, problems and algorithms on graphs, (notions of) NP completeness Laure Gonnord http://laure.gonnord.org/pro/teaching/ Laure.Gonnord@polytech-lille.fr Université