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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

1 CSE 101: Design and Analysis of Algorithms Homework 4 Winter 2016 Due: February 25, 2016 Exercises (strongly recommended, do not turn in) 1. (DPV textbook, Problem 7.11.) Find the optimal solution to the following linear program. max x + y 2x + y 3 x + 3y 5 x, y 0 Solution: the extreme points in the feasible region are (0, 0), (0, 5/3), (4/5, 7/5), and (3/2, 0), of which (4/5, 7/5) has the highest objective function value at 11/5. 2. (DPV textbook, Problem 7.21.) An edge of a flow network is called critical if decreasing the capacity of this edge results in a decrease in the maximum flow. Give an efficient algorithm that finds a critical edge in a network. Solution: (a) Algorithm Description in English First, run the max flow algorithm. If an edge is critical, it must be filled to capacity, otherwise it would be possible to reduce its capacity to the amount of flow going through it without affecting the maximum flow. Then, for each edge (u, v) filled to capacity in the residual graph, run DFS and see if there is a path from u to v. If there isn t, then that edge is a critical edge. (b) Pseudocode define getcriticaledge(g) residualgraph = FordFulkerson(G) for edge (u, v) in residualgraph such that (v, u) is not in residualgraph and (u, v) is not in G: DFS(residualGraph, u) if u has no path to v: return (u, v) return null (c) Proof of Correctness If an edge is not full when max flow is achieved, it is possible to reduce its capacity to the amount of flow going through it without affecting the maximum flow. By running the Ford-Fulkerson algorithm, we can determine which edges are full when the maximum flow is achieved. It is not enough for the edge to be full, however. We must check that the flow cannot be re-routed along another path in the residual graph. Suppose (u, v) is an edge that is full after the Ford- Fulkerson algorithm is run. If there is a path from u to v not using (u, v), then if the amount of flow allowed through (u, v) is decreased by 1, then the flow can just be routed from u to v through the other path, without affecting the max flow. If there is

2 no path from u to v, then there is no way to re-route the flow that would have gone through (u, v), so the maximum flow must decrease. Therefore, full edges are only critical edges if there does not exist any path from u to v. Running DFS finds if there exists a path between u and v. (d) Running Time Analysis To run this algorithm, first you must run the Ford-Fulkerson algorithm, which takes O(V E 2 ). Next, you must run DFS on at most O(V ) vertices, taking O(V (V + E)) time. This term is dominated by the running time of the Ford-Fulkerson algorithm, so the overall running time is O(V E 2 ). 3. (DPV textbook, Problem 7.23.) The vertex cover of an undirected graph G = (V, E) is a subset of the vertices which touches every edge- that is, a subset S V such that for each edge u, v E, one or both of of u, v are in S. Show that the problem of finding the minimum vertex cover in a bipartite graph reduces to maximum flow. (Hint : Can you relate this to the minimum cut in a related network?) Solution: Call the two partitions of the bipartite graph S 1 and S 2. Add vertices s and t. Draw an edge from s to every vertex in S 1 and from every vertex in S 2 to t. By definition of bipartite graph, we know all edges in E will connect some vertex u in S 1 with some vertex v in S 2. For every edge in E, draw an edge into the flow graph with weight 1 from the vertex in S 1 to the vertex in S 2. Set the capacity of all edges in the graph to be 1. The maximum matching of a graph is the largest set of edges that can be selected so that no two edges in the set touch the same vertex. This problem can be solved by using max flow, by setting up the same graph described above and running the Ford-Fulkerson algorithm. The maximum matching will be the set of all edges between S 1 and S 2. The minimum vertex cover can be solved the same way. Solve for the maximum matching. Consider all of the vertices in the graph besides s and t. Call the set of these vertices that are an endpoint of an edge in the maximum matching A, and call the set of vertices that are not endpoints of any of the maximum matching edges B. There can be no edges between distinct elements in B. Assume for contradiction that there was. That edge could then have been added to increase the size of the maximum matching, because it runs between two vertices that were untouched by the maximum matching edges. This contradicts our assumption that the matching found was the maximum matching, so it is impossible to have edges between distinct elements of B. Therefore, all edges in the graph must either be included in the maximum matching or must be be- tween a vertex that is an endpoint of a maximum matching edge and a vertex in B. Note that it is also impossible for a maximum matching edge to have both endpoints have edges to elements in B, otherwise the size of the matching could have been increased by removing the edge from the maximum matching and adding the two edges from its endpoints to the vertices in B. Therefore, at most one endpoint vertex of every maximum matching edge can have an edge to a vertex in B. Therefore, when constructing the minimum vertex cover, it is sufficient to take every maximum match- ing edge and choose one of its endpoints for the minimum vertex cover. Specifically, if one of the endpoints has and edge to a vertex in B, choose that vertex. Otherwise, it doesnt matter which one you choose.

3 Because an endpoint was selected from every edge in the maximum matching, then the vertex cover must cover all of the maximum matching edges. Furthermore, if a vertex had an edge to an element in B, then that vertex was chosen. There can be no edges between elements in B, as we proved earlier. Therefore, the vertex cover chosen in this way must cover every edge in the graph. Furthermore, it must be minimal, as removing any of the vertices would mean that neither endpoint vertex of one of the maximum matching edges was in the set of vertices, because only one endpoint vertex was chosen per maximum matching edge. This means that the resultant set of vertices would not cover every edge in the graph. Therefore, if we can solve maximum matching, we can solve minimum vertex cover. Since maximum matching is solved by max flow, then minimum vertex cover reduces to max flow.

4 Problems (must be written up and turned in) 1. DPV 6.2: You are going on a long trip. You start on the road at mile post 0. Along the way there are n hotels, at mile posts a 1 < a 2 <... < a n, where each a i is measured from the starting point. The only places you are allowed to stop are at these hotels, but you can choose which of the hotels you stop at. You must stop at the final hotel (at distance a n ), which is your destination. You d ideally like to travel 200 miles a day, but this may not be possible (depending on the spacing of the hotels). If you travel x miles during a day, the penalty for that day is (200 x) 2. You want to plan your trip so as to minimize the total penalty- that is, the sum, over all travel days, of the daily penalties. Give an efficient algorithm that determines the optimal sequence of hotels at which to stop. Solution: Suproblems definition Let OP T (i) be the minimum total penalty to get to hotel i. Recursive formulation To get OP T (i), we consider all possible hotels j we can stay at the night before reaching hotel i. For each of these possibilities, the minimum penalty to reach i is the sum of: the minimum penalty OP T (j) to reach j, and the cost (200 (a j a i )) 2 of a one-day trip from j to i. Because we are interested in the minimum penalty to reach i, we take the minimum of these values over all the j: OP T (i) = min 0 j<i {OP T (j) + (200 (a j a i )) 2 } And the base case is OP T (0) = 0. Pseudo-code // base case OPT[0] = 0 // main loop for i = 1...n: OPT[i] = min([opt[j] +(200 (a j a i )) 2 // final result return OPT[n] for j=0...i-1]) Complexity - We have n subproblems, - each subproblems i takes time O(i) The overall complexity is n n n(n 1) O(i) = O( )i = O = O(n 2 ) 2 i=1 i=1

5 2. DPV 6.13: Consider the following game. A dealer produces a sequence s 1... s n of cards, face up, where each card s i has a value v i. Then two players take turns picking a card from the sequence, but can only pick the first or the last card of the (remaining) sequence. The goal is to collect cards of largest total value. (For example, you can think of the cards as bills of different denominations.) Assume n is even. (a) Show a sequence of cards such that it is not optimal for the first player to start by picking up the available card of larger value. That is, the natural greedy strategy is suboptimal. Solution: Consider the sequence (2, 10, 1, 1). Then the best available card at first is 2. Then the second player can pick the card 10 and will win. If the first player pick the rightmost 1 first, then the first will be able to pick the 10 and will win. (b) Give an O(n 2 ) algorithm to compute an optimal strategy for the first player. Given the initial sequence, your algorithm should precompute in O(n 2 ) time some information, and then the first player should be able to make each move optimally in O(1) time by looking up the precomputed information. Solution: Suproblems definition Let OP T (i, j) be the difference between: - the largest total the first player can obtain, - and the corresponding score of the second player when playing on sequence s i,..., s j. Recursive formulation At any stage of the game, there are 2 possible moves for the first player: - either choose the first card, in which case he will gain v i and score OP T (i + 1, j) in the rest of the game, - or the last card, gaining v j and OP T (i, j 1) from the remaining cards. Because we are interested in the largest total the first player can gain over the second, we take the maximum of these 2 values (for i < j): OP T (i, j) = max {v i OP T (i + 1, j), v j OP T (i, j 1)} And the base cases are: i {1,..., n}, OP T (i, i) = v i. Pseudo-code Precomputation // base cases for i = 1...n: OPT[i, i] = v[i] // main loop for j = 1...n: for i = j...1: OPT[i, j] = max (v[i] - OPT(i+1,j), v[j] - OPT(i,j-1)) // final result return OPT[1, n] Look up Given sequence s i,..., s j OP T move[i, j] = { s i, if OP T [i, j] = v[i] OP T (i + 1, j) s j, otherwise

6 Complexity Precomputation - We have O(n 2 ) subproblems, - each update takes time O(1). Therefore, the overall complexity is O(n 2 ).

7 3. Every morning you purchase the daily donut from the exotic bakery Dynamic Pastries. You have a schedule over a period of n days for which the donut on day i costs p i. They also offer a special where, on any given day, you can prepay for the next 13 days for a fixed price C. You don t want to have any prepaid days left over at the end of the n days, so you can t purchase the special if there are less than 13 days remaining. Given a set of n prices p 1,..., p n, the special cost C, and your insane donut addiction, you want to figure out the minimum amount of money that you need to cover the n days. (a) You first try a greedy approach. For any day i, you compare special price C versus the cost of purchasing individually for the next 13 days (p i + p i p i+12 ). If the special is cheaper, you buy it; otherwise you buy the donut individually for that day. Give an example where this strategy leads you to spending more than needed. Solution: For the sequence 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2 and C = 3. The greedy solution gives a cost of 5 but waiting a day to buy the special would yield a cost of 4. (b) Give an efficient dynamic programming algorithm to find the optimal solution. Be sure to provide a recursive formulation and analyze the time complexity. Solution: One possible subproblem: P(i) = minimum total price of donuts ending on day i. An important thing to note is that this subproblem assumes that day i is the last day and thus there can be no prepays left over at that point. Given that, we can then consider on that last day, is the best solution to buy the donut that day, or have already bought a donut 13 days earlier. if i < 0 P(i) = 0 if i = 0 min{p(i 1) + p 1, P(i 13) + C} otherwise Pseudo-code: P(i) min-price(p 1,..., p n, C) if i < 0: return for i = 1,..., n do: if P[i] = then: P[i] = P[i] = min{p(i 1) + p 1,P(i 13) + C} P[0] = 0 return P[i] return P(n) There are n subproblems which require O(1) work to solve at each step. The total running time then is O(n).

8 4. DPV 7.5: The Canine Products company offers two dog foods, Frisky Pup and Husky Hound, that are made from a blend of cereal and meat. A package of Frisky Pup requires 1 pound of cereal and 1.5 pounds of meat, and sells for \$7. A package of Husky Hound uses 2 pounds of cereal and 1 pound of meat, and sells for \$6. Raw cereal costs \$1 per pound and raw meat costs \$2 per pound. It also costs \$1.40 to package the Frisky Pup and \$0.60 to package the Husky Hound. A total of 240,000 pounds of cereal and 180,000 pounds of meat are available each month. The only production bottleneck is that the factory can only package 110,000 bags of Frisky Pup per month. Needless to say, management would like to maximize profit. (a) Formulate the problem as a linear program in two variables. Solution: Maximize 1.6F + 1.4H subject to: F + 2H 240, 000 (Cereal) 1.5F + H 180, 000 (Meat) F 110, 000 (Production) H, F 0 (b) Graph the feasible region, give the coordinates of every vertex, and circle the vertex maximizing profit. What is the maximum profit possible? Solution: The maximum profit possible is \$222,000 with units of F and units of H.

9 5. DPV 7.18: There are many common variations of the maximum flow problem. Here are four of them. (a) There are many sources and many sinks, and we wish to maximize the total flow from all sources to all sinks. Solution: - Let S be the set of source nodes. - Let T be the set of sink nodes. - Let I be the set of interior nodes, that are neither sources nor sinks. - Let G = (V, E) be the original graph. Reduction from multi-source, multi-sink flow to single-source, single-sink flow Form graph G = (V, E ), where V = {s, t} + V, and Solve for maximum s-t flow on G. E = {(s, a, ) a S} + E + {(b, t, ) b T } (b) Each vertex also has a capacity on the maximum flow that can enter it. Solution: Let C v denote the capacity of vertex v. (Just as C e or C (u,v) shall denote the capacity of an edge.) Reduction from node-capacitated max-flow to simple max-flow. From graph G = (V, E), where and, V = {a in a V } + {a out a V } E = {(a in, a out, C a ) a V } + {(a out, b in, C (a,b) ) (a, b) E} Solve for maximum s in -t out flow on G. (c) Each edge has not only a capacity, but also a lower bound on the flow it must carry. Solution: - Let i : N e be a bijective mapping between the natural numbers and edges in the graph. - Let C i be the capacity of edge i. - Let L i be the lower bound of i. - Let f i be the flow along edge i. Maximize: Subject to: i into v i (s,v) f i C i i E (Capacities) f i f i L i i E (Lower bounds) f i = v V {s, t} (Conservation) j out of v f j (d) The outgoing flow from each node u is not the same as the incoming flow, but is smaller by a factor of (1 ɛ u ), where ɛ u is a loss coefficient associated with node u.

10 Solution: - Let i : N e be a bijective mapping between the natural numbers and edges in the graph. - Let C i be the capacity of edge i. - Let e v be the loss coefficient along each node, so that outflow(v) = (1 e v )inflow(v) for all internal nodes. - Let f i be the flow along edge i. Maximize: Subject to: i into v i (s,v) f i f i C i i E (Capacities) (1 e v )f i = v V {s, t} (Lossy Conservation) j out of v f j Each of these can be solved efficiently. Show this by reducing (a) and (b) to the original max-flow problem, and reducing (c) and (d) to linear programming.

### 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

### Solutions to Exercises 8

Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

### 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

### 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

### Solutions to Homework 6

Solutions to Homework 6 Debasish Das EECS Department, Northwestern University ddas@northwestern.edu 1 Problem 5.24 We want to find light spanning trees with certain special properties. Given is one example

### 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

### 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

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

### CSE 20: Discrete Mathematics for Computer Science. Prof. Miles Jones. Today s Topics: Graphs. The Internet graph

Today s Topics: CSE 0: Discrete Mathematics for Computer Science Prof. Miles Jones. Graphs. Some theorems on graphs. Eulerian graphs Graphs! Model relations between pairs of objects The Internet graph!

### 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

### Homework 15 Solutions

PROBLEM ONE (Trees) Homework 15 Solutions 1. Recall the definition of a tree: a tree is a connected, undirected graph which has no cycles. Which of the following definitions are equivalent to this definition

### ! 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

### 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

### Two Ways of Thinking about Dynamic Programming Jeff Edmonds

Two Ways of Thinking about Dynamic Programming Jeff Edmonds Here are two set of steps that you could follow to design a new dynamic programming algorithm. The first are the steps that I teach in 3101 using

### GRAPH THEORY and APPLICATIONS. Trees

GRAPH THEORY and APPLICATIONS Trees Properties Tree: a connected graph with no cycle (acyclic) Forest: a graph with no cycle Paths are trees. Star: A tree consisting of one vertex adjacent to all the others.

### ! 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

### 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

### 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

### Lecture 6: Approximation via LP Rounding

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

### 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

### Homework 13 Solutions. X(n) = 3X(n 1) + 5X(n 2) : n 2 X(0) = 1 X(1) = 2. Solution: First we find the characteristic equation

Homework 13 Solutions PROBLEM ONE 1 Solve the recurrence relation X(n) = 3X(n 1) + 5X(n ) : n X(0) = 1 X(1) = Solution: First we find the characteristic equation which has roots r = 3r + 5 r 3r 5 = 0,

### Minimum Spanning Trees

Minimum Spanning Trees Algorithms and 18.304 Presentation Outline 1 Graph Terminology Minimum Spanning Trees 2 3 Outline Graph Terminology Minimum Spanning Trees 1 Graph Terminology Minimum Spanning Trees

### Network Flow I. Lecture 16. 16.1 Overview. 16.2 The Network Flow Problem

Lecture 6 Network Flow I 6. Overview In these next two lectures we are going to talk about an important algorithmic problem called the Network Flow Problem. Network flow is important because it can be

### 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;

### COT5405 Analysis of Algorithms Homework 3 Solutions

COT0 Analysis of Algorithms Homework 3 Solutions. Prove or give a counter example: (a) In the textbook, we have two routines for graph traversal - DFS(G) and BFS(G,s) - where G is a graph and s is any

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

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

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

### The Knapsack Problem. Decisions and State. Dynamic Programming Solution Introduction to Algorithms Recitation 19 November 23, 2011

The Knapsack Problem You find yourself in a vault chock full of valuable items. However, you only brought a knapsack of capacity S pounds, which means the knapsack will break down if you try to carry more

### Math 316 Solutions To Sample Exam 2 Problems

Solutions to Sample Eam 2 Problems Math 6 Math 6 Solutions To Sample Eam 2 Problems. (a) By substituting appropriate values into the binomial theorem, find a formula for the sum ( ) ( ) ( ) ( ) ( ) n n

### Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

### Sample Problems in Discrete Mathematics

Sample Problems in Discrete Mathematics This handout lists some sample problems that you should be able to solve as a pre-requisite to Computer Algorithms Try to solve all of them You should also read

### CSE 5311 Homework 3 Solution

CSE 5311 Homework 3 Solution Problem 15.1-1 Show that equation (15.4) follows from equation (15.3) and the initial condition T (0) = 1. We can verify that T (n) = n is a solution to the given recurrence

### 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

### The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson)

Chapter 11 Graph Theory The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson) (N. Let us start with a formal definition of what is a graph. Definition 72. A graph

### 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

### Linear Programming. March 14, 2014

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

### 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

### Basic Notions on Graphs. Planar Graphs and Vertex Colourings. Joe Ryan. Presented by

Basic Notions on Graphs Planar Graphs and Vertex Colourings Presented by Joe Ryan School of Electrical Engineering and Computer Science University of Newcastle, Australia Planar graphs Graphs may be drawn

### 10.1 Integer Programming and LP relaxation

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

### Part 2: Community Detection

Chapter 8: Graph Data Part 2: Community Detection Based on Leskovec, Rajaraman, Ullman 2014: Mining of Massive Datasets Big Data Management and Analytics Outline Community Detection - Social networks -

### Minimum cost maximum flow, Minimum cost circulation, Cost/Capacity scaling

6.854 Advanced Algorithms Lecture 16: 10/11/2006 Lecturer: David Karger Scribe: Kermin Fleming and Chris Crutchfield, based on notes by Wendy Chu and Tudor Leu Minimum cost maximum flow, Minimum cost circulation,

### 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

### Why graph clustering is useful?

Graph Clustering Why graph clustering is useful? Distance matrices are graphs as useful as any other clustering Identification of communities in social networks Webpage clustering for better data management

### Lecture Notes on Spanning Trees

Lecture Notes on Spanning Trees 15-122: Principles of Imperative Computation Frank Pfenning Lecture 26 April 26, 2011 1 Introduction In this lecture we introduce graphs. Graphs provide a uniform model

### Examples of decision problems:

Examples of decision problems: The Investment Problem: a. Problem Statement: i. Give n possible shares to invest in, provide a distribution of wealth on stocks. ii. The game is repetitive each time we

### 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,

### The Taxman Game. Robert K. Moniot September 5, 2003

The Taxman Game Robert K. Moniot September 5, 2003 1 Introduction Want to know how to beat the taxman? Legally, that is? Read on, and we will explore this cute little mathematical game. The taxman game

### 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)

### Approximation Algorithms. Scheduling. Approximation algorithms. Scheduling jobs on a single machine

Approximation algorithms Approximation Algorithms Fast. Cheap. Reliable. Choose two. NP-hard problems: choose 2 of optimal polynomial time all instances Approximation algorithms. Trade-off between time

### Social Media Mining. Graph Essentials

Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures

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

### CSE 373 Analysis of Algorithms Date: Nov. 1, Solution to HW3. Total Points: 100

CSE 373 Analysis of Algorithms Date: Nov. 1, 2016 Steven Skiena Problem 1 5-4 [5] Solution to HW3 Total Points: 100 Let T be the BFS-tree of the graph G. For any e in G and e T, we have to show that e

### Cost Model: Work, Span and Parallelism. 1 The RAM model for sequential computation:

CSE341T 08/31/2015 Lecture 3 Cost Model: Work, Span and Parallelism In this lecture, we will look at how one analyze a parallel program written using Cilk Plus. When we analyze the cost of an algorithm

### 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

### 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.

### 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

### Chapter 15 Introduction to Linear Programming

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

### TRANSPORTATION PROBLEMS

Chapter 5 TRANSPORTATION PROBLEMS 5.1. Classic transportation problem 5.1.1. The problem The standard form of the transportation problem is as follows: inf m n c ij x ij, where: i=1 j=1 n x ij = a i, 1

### 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

### Graph Theory Problems and Solutions

raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is

### Graph. Consider a graph, G in Fig Then the vertex V and edge E can be represented as:

Graph A graph G consist of 1. Set of vertices V (called nodes), (V = {v1, v2, v3, v4...}) and 2. Set of edges E (i.e., E {e1, e2, e3...cm} A graph can be represents as G = (V, E), where V is a finite and

### 15-750 Graduate Algorithms Spring 2010

15-750 Graduate Algorithms Spring 2010 1 Cheap Cut (25 pts.) Suppose we are given a directed network G = (V, E) with a root node r and a set of terminal nodes T V. We would like to remove the fewest number

### Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction.

Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction. February 21, 2006 1 Proof by Induction Definition 1.1. A subset S of the natural numbers is said to be inductive if n S we have

### 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

### CSPs: Arc Consistency

CSPs: Arc Consistency CPSC 322 CSPs 3 Textbook 4.5 CSPs: Arc Consistency CPSC 322 CSPs 3, Slide 1 Lecture Overview 1 Recap 2 Consistency 3 Arc Consistency CSPs: Arc Consistency CPSC 322 CSPs 3, Slide 2

### Spring 2007 Math 510 Hints for practice problems

Spring 2007 Math 510 Hints for practice problems Section 1 Imagine a prison consisting of 4 cells arranged like the squares of an -chessboard There are doors between all adjacent cells A prisoner in one

### 1. What s wrong with the following proofs by induction?

ArsDigita University Month : Discrete Mathematics - Professor Shai Simonson Problem Set 4 Induction and Recurrence Equations Thanks to Jeffrey Radcliffe and Joe Rizzo for many of the solutions. Pasted

### 6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium

6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline Pricing-Congestion Game Example Existence of a Mixed

### 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

### A Practical Scheme for Wireless Network Operation

A Practical Scheme for Wireless Network Operation Radhika Gowaikar, Amir F. Dana, Babak Hassibi, Michelle Effros June 21, 2004 Abstract In many problems in wireline networks, it is known that achieving

### Chapter 4. Trees. 4.1 Basics

Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.

### Linear Programming. April 12, 2005

Linear Programming April 1, 005 Parts of this were adapted from Chapter 9 of i Introduction to Algorithms (Second Edition) /i by Cormen, Leiserson, Rivest and Stein. 1 What is linear programming? The first

### AIMA Chapter 3: Solving problems by searching

AIMA Chapter 3: Solving problems by searching Ana Huaman January 13, 2011 1. Define in your own words the following terms: state, state space, search tree, search node, goal, action, transition model and

### 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

### Review: Induction and Strong Induction. CS311H: Discrete Mathematics. Mathematical Induction. Matchstick Example. Example. Matchstick Proof, cont.

Review: Induction and Strong Induction CS311H: Discrete Mathematics Mathematical Induction Işıl Dillig How does one prove something by induction? What is the inductive hypothesis? What is the difference

### Midterm Practice Problems

6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator

### Mechanism Design without Money I: House Allocation, Kidney Exchange, Stable Matching

Algorithmic Game Theory Summer 2015, Week 12 Mechanism Design without Money I: House Allocation, Kidney Exchange, Stable Matching ETH Zürich Peter Widmayer, Paul Dütting Looking at the past few lectures

### 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

### IE 680 Special Topics in Production Systems: Networks, Routing and Logistics*

IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* Rakesh Nagi Department of Industrial Engineering University at Buffalo (SUNY) *Lecture notes from Network Flows by Ahuja, Magnanti

### 1 Unrelated Parallel Machine Scheduling

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

### Monotone Partitioning. Polygon Partitioning. Monotone polygons. Monotone polygons. Monotone Partitioning. ! Define monotonicity

Monotone Partitioning! Define monotonicity Polygon Partitioning Monotone Partitioning! Triangulate monotone polygons in linear time! Partition a polygon into monotone pieces Monotone polygons! Definition

### CS188 Spring 2011 Section 3: Game Trees

CS188 Spring 2011 Section 3: Game Trees 1 Warm-Up: Column-Row You have a 3x3 matrix of values like the one below. In a somewhat boring game, player A first selects a row, and then player B selects a column.

### Acceptance of reservations for a rent-a-car company J.Alberto Conejero 1, Cristina Jordan 2, and Esther Sanabria-Codesal 1

Int. J. Complex Systems in Science vol. 2(1) (2012), pp. 27 32 Acceptance of reservations for a rent-a-car company J.Alberto Conejero 1, Cristina Jordan 2, and Esther Sanabria-Codesal 1 1 Instituto Universitario

### Mathematical induction & Recursion

CS 441 Discrete Mathematics for CS Lecture 15 Mathematical induction & Recursion Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Proofs Basic proof methods: Direct, Indirect, Contradiction, By Cases,

### Jade Yu Cheng ICS 311 Homework 10 Oct 7, 2008

Jade Yu Cheng ICS 311 Homework 10 Oct 7, 2008 Question for lecture 13 Problem 23-3 on p. 577 Bottleneck spanning tree A bottleneck spanning tree of an undirected graph is a spanning tree of whose largest

### 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

### Minimize subject to. x S R

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

### Graph Theory. Introduction. Distance in Graphs. Trees. Isabela Drămnesc UVT. Computer Science Department, West University of Timişoara, Romania

Graph Theory Introduction. Distance in Graphs. Trees Isabela Drămnesc UVT Computer Science Department, West University of Timişoara, Romania November 2016 Isabela Drămnesc UVT Graph Theory and Combinatorics

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

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

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

The multi-integer set cover and the facility teral cover problem Dorit S. Hochbaum Asaf Levin December 5, 2007 Abstract The facility teral cover problem is a generalization of the vertex cover problem.

### CS5314 Randomized Algorithms. Lecture 16: Balls, Bins, Random Graphs (Random Graphs, Hamiltonian Cycles)

CS5314 Randomized Algorithms Lecture 16: Balls, Bins, Random Graphs (Random Graphs, Hamiltonian Cycles) 1 Objectives Introduce Random Graph Model used to define a probability space for all graphs with

### CS420/520 Algorithm Analysis Spring 2010 Lecture 04

CS420/520 Algorithm Analysis Spring 2010 Lecture 04 I. Chapter 4 Recurrences A. Introduction 1. Three steps applied at each level of a recursion a) Divide into a number of subproblems that are smaller

### 6.042/18.062J Mathematics for Computer Science October 3, 2006 Tom Leighton and Ronitt Rubinfeld. Graph Theory III

6.04/8.06J Mathematics for Computer Science October 3, 006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Graph Theory III Draft: please check back in a couple of days for a modified version of these

### 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:

### Fast Algorithms for Connectivity Problems in Networks

Fast Algorithms for Connectivity Problems in Networks Steiner Cuts, Gomory-Hu Trees, and Edge Splitting Strand Life Sciences 29 May 2008/ MCDES The Setting A network G of n nodes and m edges. Many nodes,

### Network File Storage with Graceful Performance Degradation

Network File Storage with Graceful Performance Degradation ANXIAO (ANDREW) JIANG California Institute of Technology and JEHOSHUA BRUCK California Institute of Technology A file storage scheme is proposed

### 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

### 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