# Lecture 6: Approximation via LP Rounding

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Lecture 6: Approximation via LP Rounding Let G = (V, E) be an (undirected) graph. A subset C V is called a vertex cover for G if for every edge (v i, v j ) E we have v i C or v j C (or both). In other words, for every edge in E at least one of its endpoints is in C. 6.1 Unweighted vertex cover The unweighted version of the Vertex Cover problem is to find a vertex cover of minimum size for a given graph G. This problem is NP-complete. Let s try to come up with an approximation algorithm. A natural greedy approach would be the following. Initialize the cover C as the empty set, and set E := E; the set E will contain the edges that are not yet covered by C. Now take an edge (v i, v j ) E, put one of its two vertices, say v i, into C, and remove from E all edges incident to v i. Repeat the process until E =. Clearly C is a vertex cover after the algorithm has finished. Unfortunately the algorithm has a very bad approximation ratio: there are instances where it can produce a vertex cover of size V 1 even though a vertex cover of size 1 exists. A small change in the algorithm leads to a 2-approximation algorithm. The change is based on the following lower bound. Call two edges e, e E adjacent if they share an endpoint. Lemma 6.1 Let G = (V, E) be a graph and let opt denote the minimum size of a vertex cover for G. Let E E be any subset of edges such that no two edges in E are adjacent. Then opt E. Proof. Let C be an optimal vertex cover for G. By definition, any edge e E must be covered by a vertex in C, and since the edges in E are non-adjacent any vertex in C can cover at most one edge in E. This lemma suggests the following greedy algorithm. Algorithm ApproxVertexCover(V, E) 1. C ; E E 2. Invariant: C is a vertex cover for the graph G = (V, E \ E ) 3. while E 4. do Take an arbitrary edge (v i, v j ) E. 5. C C {v i, v j } 6. Remove all edges from E that are adjacent to (v i, v j ). 7. return C It is easy to check that the while-loop indeed maintains the invariant. After the while-loop has terminated the loop must terminate since at every step we remove at least one edge from E we have E \ E = E \ = E. Together with the invariant this implies that the algorithm indeed returns a vertex cover. Next we show that the algorithm gives a 2-approximation. Theorem 6.2 Algorithm ApproxVertexCover produces a vertex cover C such that C 2 opt, where opt is the minimum size of a vertex cover. Proof. Let E be the set of edges selected in line 4 over the course of the algorithm. Then C consists of the endpoints of the edges in E, and so C 2 E. Moreover, no two edges in 1

2 E are adjacent because as soon as an edge (v i, v j ) is selected from E all edges in E adjacent to (v i, v j ) are removed from E. The theorem now follows from Lemma Weighted Vertex Cover Now let s consider a generalization of Vertex Cover, where each vertex v i V has a weight w i and we want to find a vertex cover of minimum total weight. We call the new problem Weighted Vertex Cover. The first idea that comes to mind to get an approximation algorithm for Weighted Vertex Cover is to generalize ApproxVertexCover as follows: instead of selecting an arbitrary edge (v i, v j ) from E in line 4, we select the edge of minimum weight (where the weight of an edge is defined as the sum of the weights of its endpoints). Unfortunately this doesn t work: the weight of the resulting cover can be arbitrarily much larger than the weight of an optimal cover. Our new approach will be based on linear programming. (Integer) linear programming. In a linear-programming problem we are given a linear cost function of d real variables x 1,..., x d and a set of n linear constraints on these variables. The goal is to assign values to the variables so that the cost function is minimized (or: maximized) and all constraints are satisfied. In other words, the Linear Programming problem can be stated as follows: Minimize c 1 x c d x d Subject to a 1,1 x a 1,d x d b 1 a 2,1 x a 2,d x d b 2. a n,1 x a n,d x d b n There are algorithms for example the so-called interior-point methods that can solve linear programs in time polynomial in the input size. 1 In practice linear programming is often done with with famous simplex method, which is exponential in the worst case but works quite well in most practical applications. Hence, if we can formulate a problem as a linear-programming problem then we can solve it efficiently, both in theory and in practice. There are several problems that can be formulated as a linear-programming problem but with one twist: the variables x 1,..., x d can not take on real values but only integer values. This is called Integer Linear Programming. (When the variables can only take the values 0 or 1, the problem is called 0/1 Linear Programming.) Unfortunately, Integer Linear Programming and 0/1 Linear Programming are considerably harder than Linear Programming. In fact, Integer Linear Programming and 0/1 Linear Programming are NP-complete. However, formulating a problem as an integer linear program can still be useful, as shall see next. Approximating via LP relaxation and rounding. Let s go back to Weighted Vertex Cover. Thus we are given a weighted graph G = (V, E) with V = {v 1,..., v n }, and we 1 Here the input size is measured in terms of the number of bits needed to describe the input, so this is different from the usual notion of input size. 2

3 want to find a minimum-weight vertex cover. To formulate this as a 0/1 linear program, we introduce a variable x i for each vertex v i V ; the idea is to have x i = 1 if v i is taken into the vertex cover, and x 1 = 0 if v i is not taken into the cover. When is a subset C V is vertex cover? Then C must contain at least one endpoint for every edge (v i, v j ). This means we must have x i = 1 or x j = 1. We can enforce this by introducing for every edge (v i, v j ) E the constraint x i + x j 1. Finally, we wish to minimize the total weight of the cover, so we get as a cost function n i=1 w ix i. To summarize, solving the weighted vertex-cover problem corresponds to solving the following 0/1 linear-programming problem. Minimize w 1 x w n x n Subject to x i + x j 1 for all edges (v i, v j ) E x i {0, 1} for 1 i n (1) As noted earlier, solving 0/1 linear programs is hard. Therefore we perform relaxation: we drop the restriction that the variables can only take integer values and we replace the integrality constraints (1) by 0 x i 1 for 1 i n (2) This linear program can be solved in polynomial time. But what good is a solution where the variables can take on any real number in the interval [0, 1]? A solution with x i = 1/3, for instance, would suggest that we put 1/3 of the vertex v i into the cover something that does not make sense. First we note that the solution to our new relaxed linear program provides us with a lower bound. Lemma 6.3 Let W denote the value of an optimal solution to the relaxed linear program described above, and let opt denote the minimum weight of a vertex cover. Then opt W. Proof. Any vertex cover corresponds to a feasible solution of the linear program, by setting the variables of the vertices in the cover to 1 and the other variables to 0. Hence, the optimal solution of the linear program is at least as good as this solution. (Stated differently: we already argued that an optimal solution of the 0/1-version of the linear program corresponds to an optimal solution of the vertex-cover problem. Relaxing the integrality constraints clearly cannot make the solution worse.) The next step is to derive a valid vertex cover or, equivalently, a feasible solution to the 0/1 linear program from the optimal solution to the relaxed linear program. We want to do this in such a way that the total weight of the solution does not increase by much. This can simply be done by rounding: variables whose value is at least 1/2 are rounded to 1, variables whose value is less than 1/2 are rounded to 0. We thus obtain the following algorithm for Weighted Vertex Cover. 3

4 Algorithm ApproxWeightedVertexCover(V, E) 1. G = (V, E) is a graph where each vertex v i V (1 i n) has weight w i. 2. Solve the relaxed linear program corresponding to the given problem: 3. C {v i V : x i 1/2} 4. return C Minimize w 1 x w n x n Subject to x i + x j 1 for all edges (v i, v j ) E 0 x i 1 for 1 i n Theorem 6.4 Algorithm ApproxWeightedVertexCover is a 2-approximation algorithm. Proof. We first argue that the set C returned by the algorithm is a vertex cover. Consider an edge (v i, v j ) E. Then x i + x j 1 is one of the constraints of the linear program. Hence, the reported solution to the linear program note that a solution will be reported, since the program is obviously feasible by setting all variables to 1 has max(x i, x j ) 1/2. It follows that at least one of v i and v j will be put into C. Let W := n i=1 w ix i be the total weight of the optimal solution to the relaxed linear program. By Lemma 6.5 we have opt W. Using that x i 1/2 for all v i C, we can now bound the total weight of C as follows: v i C w i v i C w i 2x i 2 v i C w i x i 2 n w i x i = 2W 2 opt Note that, as always, the approximation ratio of our algorithm is obtained by comparing the obtained solution to a certain lower bound in this case the solution to the LP relaxation. The worst-case ratio between the solution to the integer linear program (which models the problem exactly) and its relaxed version is called the integrality gap. For approximation algorithms based on rounding the relaxation of an integer linear program, one typically cannot prove a better approximation ratio than the integrality gap. 6.3 Set Cover Let Z := {z 1,..., z m } be a finite set. A set cover for Z is a collection of subsets of Z whose union is Z. The Set Cover problem is, given a set Z and a collection S = S 1,..., S n of subsets of Z, to select a minimum number of subsets from S that together form a set cover for Z. Set Cover is a generalization of Vertex Cover. This can be seen as follows. Let G = (V, E) be the graph for which we want to obtain a vertex cover. We can construct an instance of Set Cover from G as follows: The set Z is the set of edges of G, and every vertex v i V defines a subset S i consisting of those edges of which v i is an endpoint. Then a set cover for the input Z, S 1,..., S n corresponds to a vertex cover for G. (Note that Set Cover is more general than Vertex Cover, because in the instance of Set Cover defined by a Vertex Cover instance, every element occurs in exactly two sets in the general problem an element from Z can occur in many subsets.) In Weighted Set Cover every subset S i has a weight w i and we want to find a set cover of minimum total weight. i=1 4

5 In this section we will develop an approximation algorithm for Weighted Set Cover. To this end we first formulate the problem as a 0/1 linear program: we introduce a variable x i that indicates whether S i is in the cover (x i = 1) or not (x i = 0), and we introduce a constraint for each element z j Z that guarantees that z j will be in at least one of the chosen sets. The constraint for z j is defined as follows. Let S(j) := {i : 1 i n and z j S i }. Then one of the chosen sets contains z j if and only if i S(j) x i 1. following 0/1 linear program. This leads to the Minimize Subject to w 1 x w n x n i S(j) x i 1 for all 1 j m x i {0, 1} for 1 i n (3) We relax this 0/1 linear program by replacing the integrality constraints in (3) by the following constraints: 0 x i 1 for 1 i n (4) We obtain a linear program that we can solve in polynomial time. As in the case of Weighted Vertex Cover, the value of an optimal solution to this linear program is a lower bound on the value of an optimal solution to the 0/1 linear program and, hence, a lower bound on the minimum total weight of a set cover for the given instance: Lemma 6.5 Let W denote the value of an optimal solution to the relaxed linear program described above, and let opt denote the minimum weight of a set cover. Then opt W. The next step is to use the solution to the linear program to obtain a solution to the 0/1 linear program (or, in other words, to the set cover problem). Rounding in the same way as for the vertex cover problem rounding variables that are at least 1/2 to 1, and the other variables to 0 does not work: such a rounding scheme will not give a set cover. Instead we use the following randomized rounding strategy: For each S i independently, put S i into the cover C with probability x i. Lemma 6.6 The expected total weight of C is at most opt. Proof. By definition, the total weight of C is the sum of the weights of its subsets. Let s define an indicator random variable Y i that tells us whether a set S i is in the cover C: { 1 if Si C Y i = 0 otherwise We have E[ weight of C ] = E[ n i=1 w iy i ] = n i=1 w i E[Y i ] (by linearity of expectation) = n i=1 w i Pr[ S i is put into C ] = n i=1 w ix i opt (by Lemma 6.5) 5

6 So the total weight of C is very good. Is C is valid set cover? To answer this question, let s look at the probability that an element z j Z is not covered. Recall that i S(j) x i 1. Suppose that z j is present in l subsets, that is, S(j) = l. To simplify the notation, let s renumber the sets such that S(j) = {1,..., l}. Then we have Pr[ z j is not covered ] = (1 x 1 ) (1 x l ) (1 1 l )l, where the last inequality follows from the fact that (1 x 1 ) (1 x l ) is maximized when the x i s sum up to exactly 1 and are evenly distributed, that is, when x i = 1/l for all i. Since (1 (1/l)) l 1/e, where e is the base of the natural logarithm, we conclude that Pr[ z j is not covered ] 1 e So the probability that any element z j is covered is fairly high. But this is not good enough: there are many elements z j and even though each one of them has a good chance of being covered, we cannot expect all of them to be covered simultaneously. (This is only to be expected, of course, since Weighted Set Cover is NP-complete, so we shouldn t hope to find an optimal solution in polynomial time.) What we need is that each element z j is covered with high probability. To this end we simply repeat the above procedure t times, for a suitable value of t: we generate covers C 1,..., C t where each C s is obtained using the randomized rounding strategy, and we take C := C 1 C t as our cover. Our final algorithm is thus as follows. Algorithm ApproxWeightedSetCover(X, S) 1. X = {z 1,..., z m }, and S = {S 1,..., S n }, and set S i S (1 i n) has weight w i. 2. Solve the relaxed linear program corresponding to the given problem: Minimize w 1 x w n x n Subject to i S(j) x i 1 for all 1 j m 0 x i 1 for 1 i n 3. t 2 ln m 4. for s 1 to t 5. do Compute C s by randomized rounding 6. for i 1 to n 7. do Put S i into C s with probability x i 8. C C 1 C t 9. return C Theorem 6.7 Algorithm ApproxWeightedSetCover computes a collection C that is a set cover with probability at least 1 1/m and whose expected total weight is O(opt log m). Proof. The expected weight of each C s is at most opt, so the expected total weight of C is at most t opt = O(opt log m). What is the probability that some fixed element z j is not covered by any of the covers C s? Since the covers C s are generated independently, and each C s fails to cover z j with probability at most 1/e, we have Pr[ z j is not covered by any C s ] (1/e) t. 6

7 Since t = 2 ln m we conclude that z j is not covered with probability at most 1/m 2. Hence, Pr[ all elements z j are covered by C ] = 1 Pr[ at least one element z j is not covered by C ] 1 m j=1 Pr[ z j is not covered by C ] 1 1/m 7

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

### 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;

### 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

### 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

### 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

### 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

### ! 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

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

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

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

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

### Near Optimal Solutions

Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.

### 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,

### 8.1 Makespan Scheduling

600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Dynamic Programing: Min-Makespan and Bin Packing Date: 2/19/15 Scribe: Gabriel Kaptchuk 8.1 Makespan Scheduling Consider an instance

### 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

### 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

### CMPSCI611: Approximating MAX-CUT Lecture 20

CMPSCI611: Approximating MAX-CUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NP-hard problems. Today we consider MAX-CUT, which we proved to

### The Graphical Method: An Example

The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

### 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

### Distributed Computing over Communication Networks: Maximal Independent Set

Distributed Computing over Communication Networks: Maximal Independent Set What is a MIS? MIS An independent set (IS) of an undirected graph is a subset U of nodes such that no two nodes in U are adjacent.

### CMSC 858T: Randomized Algorithms Spring 2003 Handout 8: The Local Lemma

CMSC 858T: Randomized Algorithms Spring 2003 Handout 8: The Local Lemma Please Note: The references at the end are given for extra reading if you are interested in exploring these ideas further. You are

### 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

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### 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

### 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

### 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

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

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

### Small Maximal Independent Sets and Faster Exact Graph Coloring

Small Maximal Independent Sets and Faster Exact Graph Coloring David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science The Exact Graph Coloring Problem: Given an undirected

### 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

### Single-Link Failure Detection in All-Optical Networks Using Monitoring Cycles and Paths

Single-Link Failure Detection in All-Optical Networks Using Monitoring Cycles and Paths Satyajeet S. Ahuja, Srinivasan Ramasubramanian, and Marwan Krunz Department of ECE, University of Arizona, Tucson,

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

### On the Unique Games Conjecture

On the Unique Games Conjecture Antonios Angelakis National Technical University of Athens June 16, 2015 Antonios Angelakis (NTUA) Theory of Computation June 16, 2015 1 / 20 Overview 1 Introduction 2 Preliminary

### GENERATING LOW-DEGREE 2-SPANNERS

SIAM J. COMPUT. c 1998 Society for Industrial and Applied Mathematics Vol. 27, No. 5, pp. 1438 1456, October 1998 013 GENERATING LOW-DEGREE 2-SPANNERS GUY KORTSARZ AND DAVID PELEG Abstract. A k-spanner

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

### 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

### Steiner Tree Approximation via IRR. Randomized Rounding

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

### An Approximation Algorithm for Bounded Degree Deletion

An Approximation Algorithm for Bounded Degree Deletion Tomáš Ebenlendr Petr Kolman Jiří Sgall Abstract Bounded Degree Deletion is the following generalization of Vertex Cover. Given an undirected graph

### Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.

Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.

### 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

### 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

### 1 Approximating Set Cover

CS 05: Algorithms (Grad) Feb 2-24, 2005 Approximating Set Cover. Definition An Instance (X, F ) of the set-covering problem consists of a finite set X and a family F of subset of X, such that every elemennt

### Absolute Value Equations and Inequalities

Key Concepts: Compound Inequalities Absolute Value Equations and Inequalities Intersections and unions Suppose that A and B are two sets of numbers. The intersection of A and B is the set of all numbers

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

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

### CMPS 102 Solutions to Homework 1

CMPS 0 Solutions to Homework Lindsay Brown, lbrown@soe.ucsc.edu September 9, 005 Problem..- p. 3 For inputs of size n insertion sort runs in 8n steps, while merge sort runs in 64n lg n steps. For which

### 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

### 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

### 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

### 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].

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### Triangle deletion. Ernie Croot. February 3, 2010

Triangle deletion Ernie Croot February 3, 2010 1 Introduction The purpose of this note is to give an intuitive outline of the triangle deletion theorem of Ruzsa and Szemerédi, which says that if G = (V,

### Private Approximation of Clustering and Vertex Cover

Private Approximation of Clustering and Vertex Cover Amos Beimel, Renen Hallak, and Kobbi Nissim Department of Computer Science, Ben-Gurion University of the Negev Abstract. Private approximation of search

### 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

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### 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 }

### 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.,

### 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

### 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

### THE SCHEDULING OF MAINTENANCE SERVICE

THE SCHEDULING OF MAINTENANCE SERVICE Shoshana Anily Celia A. Glass Refael Hassin Abstract We study a discrete problem of scheduling activities of several types under the constraint that at most a single

### arxiv:1203.1525v1 [math.co] 7 Mar 2012

Constructing subset partition graphs with strong adjacency and end-point count properties Nicolai Hähnle haehnle@math.tu-berlin.de arxiv:1203.1525v1 [math.co] 7 Mar 2012 March 8, 2012 Abstract Kim defined

### 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

### Scheduling Real-time Tasks: Algorithms and Complexity

Scheduling Real-time Tasks: Algorithms and Complexity Sanjoy Baruah The University of North Carolina at Chapel Hill Email: baruah@cs.unc.edu Joël Goossens Université Libre de Bruxelles Email: joel.goossens@ulb.ac.be

### 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

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### 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

### 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

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### 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

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

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

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

### 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

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

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

### On the independence number of graphs with maximum degree 3

On the independence number of graphs with maximum degree 3 Iyad A. Kanj Fenghui Zhang Abstract Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs

### Determination of the normalization level of database schemas through equivalence classes of attributes

Computer Science Journal of Moldova, vol.17, no.2(50), 2009 Determination of the normalization level of database schemas through equivalence classes of attributes Cotelea Vitalie Abstract In this paper,

### Notes on Complexity Theory Last updated: August, 2011. Lecture 1

Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look

### 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

### 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

### Examples and Proofs of Inference in Junction Trees

Examples and Proofs of Inference in Junction Trees Peter Lucas LIAC, Leiden University February 4, 2016 1 Representation and notation Let P(V) be a joint probability distribution, where V stands for a

### 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,

### 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

### Linear Programming. Solving LP Models Using MS Excel, 18

SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting

### Induction Problems. Tom Davis November 7, 2005

Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily

### Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

### (x + a) n = x n + a Z n [x]. Proof. If n is prime then the map

22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find

### 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

### Factoring & Primality

Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount

### Topic: Greedy Approximations: Set Cover and Min Makespan Date: 1/30/06

CS880: Approximations Algorithms Scribe: Matt Elder Lecturer: Shuchi Chawla Topic: Greedy Approximations: Set Cover and Min Makespan Date: 1/30/06 3.1 Set Cover The Set Cover problem is: Given a set of

### 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,

### 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

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

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### 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,

### 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

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

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

### Minimizing the Number of Machines in a Unit-Time Scheduling Problem

Minimizing the Number of Machines in a Unit-Time Scheduling Problem Svetlana A. Kravchenko 1 United Institute of Informatics Problems, Surganova St. 6, 220012 Minsk, Belarus kravch@newman.bas-net.by Frank

### Resource Allocation with Time Intervals

Resource Allocation with Time Intervals Andreas Darmann Ulrich Pferschy Joachim Schauer Abstract We study a resource allocation problem where jobs have the following characteristics: Each job consumes

### 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

### Linear Programming I

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

### WRITING PROOFS. Christopher Heil Georgia Institute of Technology

WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this

### 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