Chapter Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling


 Cleopatra Sims
 3 years ago
 Views:
Transcription
1 Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NPhard problem. What should I do? A. Theory says you're unlikely to find a polytime algorithm. Must sacrifice one of three desired features. Solve problem to optimality. Solve problem in polytime. Solve arbitrary instances of the problem. approximation algorithm. Guaranteed to run in polytime. Guaranteed to solve arbitrary instance of the problem Guaranteed to find solution within ratio of true optimum. Challenge. Need to prove a solution's value is close to optimum, without even knowing what optimum value is! Load Balancing. Load Balancing Input. m identical machines; n jobs; job j has processing time t j. Job j must run contiguously on one machine. A machine can process at most one job at a time. Def. Let J(i) be the subset of jobs assigned to machine i. The load of machine i is L i = j J(i) t j. Def. The makespan is the maximum load on any machine L = max i L i. Load balancing. Assign each job to a machine to minimize the makespan. Decision Version. Is the makespan bound by a number K? Load Balancing on Machines Load Balancing: Greedy Scheduling Claim. Load balancing is hard even if only machines. Pf. NUMBERPARTITION P LOADBALANCE. NPcomplete Greedyscheduling algorithm. Consider n jobs in some fixed order. Assign job j to machine whose load is smallest so far. e a b f c g d GreedyScheduling(m, n, t,t,,t n ) { for i = to m { L i load on machine i J(i) jobs assigned to machine i machine machine length of job f a d Machine f b c Machine e g yes for j = to n { i = argmin k { L k J(i) J(i) {j L i L i + t j return J(),, J(m) machine i has smallest load assign job j to machine i update load of machine i Time L Implementation: O(n log m) using a priority queue. 6
2 approximation Load Balancing: Greedy Scheduling Analysis An algorithm for an optimization problem is a approximation if the solution found by the algorithm is always within a factor of the optimal solution. Minimization Problem: = approximatesolution/optimalsolution Maximization Problem: = optimalsolution/approximatesolution In general,. If =, then the solution is optimal. Theorem. [Graham, 66] Greedy algorithm is a approximation. First worstcase analysis of an approximation algorithm. Need to compare resulting solution with optimal makespan L*. Lemma. The optimal makespan L* max j t j. Pf. Some machine must process the most timeconsuming job. Lemma. The optimal makespan Pf. The total processing time is j t j. L * m j t j. One of m machines must do at least a /m fraction of total work. 7 8 Load Balancing: Greedy Scheduling Analysis Load Balancing: Greedy Scheduling Analysis Theorem. Greedy algorithm is a approximation. Pf. Consider max load L i of bottleneck machine i. Let j be the last job scheduled on machine i. When job j assigned to machine i, i had smallest load. Its load before assignment is L i t j L i t j L k for all k m. Theorem. Greedy algorithm is a approximation. Pf. Consider max load L i of bottleneck machine i. Let j be the last job scheduled on machine i. When job j assigned to machine i, i had smallest load. Its load before assignment is L i t j L i t j L k for all k m. Sum inequalities over all k and divide by m: blue jobs scheduled before j L i t j m k L k m k t k Lemma L * machine i j Now L i (L i t j ) t j L *. L* L* Lemma L i t j L = L i Load Balancing: Greedy Scheduling Analysis Load Balancing: List Scheduling Analysis Q. Is our analysis tight? A. Essentially yes. Ex: m machines, m(m) jobs length jobs, one job of length m Greedy solution = m; Q. Is our analysis tight? A. Essentially yes. Ex: m machines, m(m) jobs length jobs, one job of length m Greedy solution = m; Optimal makespan = m; = /m. m = machine idle machine idle machine idle machine idle machine 6 idle machine 7 idle machine 8 idle machine idle machine idle m = list scheduling makespan = optimal makespan =
3 Load Balancing: LPT Rule Load Balancing: LPT Rule Longest processing time (LPT). Sort n jobs in descending order of processing time, and then run Greedy scheduling algorithm. Observation. If at most m jobs, then greedyscheduling is optimal. Pf. Each job put on its own machine. LPTGreedyScheduling(m, n, t,t,,t n ) { Sort jobs so that t t t n for i = to m { L i J(i) for j = to n { i = argmin k L k J(i) J(i) {j L i L i + t j return J(),, J(m) load on machine i jobs assigned to machine i machine i has smallest load assign job j to machine i update load of machine i Lemma. If there are more than m jobs, L* t m+. Pf. Consider first m+ jobs t,, t m+. Since the t i 's are in descending order, each takes at least t m+ time. There are m+ jobs and m machines, so by pigeonhole principle, at least one machine gets two jobs. Theorem. LPT rule is a / approximation algorithm. Pf. Same basic approach as for the first greedy scheduling. L i (L i t j ) t j L *. L* L* Lemma ( by observation, can assume number of jobs > m ) Load Balancing: LPT Rule Q. Is our / analysis tight? A. No.. Center Selection Theorem. [Graham, 6] LPT rule is a /approximation. Pf. More sophisticated analysis of the same algorithm. Q. Is Graham's / analysis tight? A. Essentially yes. Center Selection Problem Center Selection Problem Input. Set of n sites s,, s n and integer k >. Center selection problem. Select k centers C so that maximum distance from a site to nearest center is minimized. Input. Set of n sites s,, s n and integer k >. Center selection problem. Select k centers C so that maximum distance from a site to nearest center is minimized. r(c) k = Notation. dist(x, y) = distance between x and y. dist(s i, C) = min c C dist(s i, c) = distance from s i to closest center. r(c) = max i dist(s i, C) = smallest covering radius. Goal. Find set of centers C that minimizes r(c), subject to C = k. center site Distance function properties. dist(x, x) = (identity) dist(x, y) = dist(y, x) (symmetry) dist(x, y) dist(x, z) + dist(z, y) (triangle inequality) 7 8
4 Center Selection Example Greedy Algorithm: A False Start Ex: each site is a point in the plane, a center can be any point in the plane, dist(x, y) = Euclidean distance. Remark: search can be infinite! Greedy algorithm. Put the first center at the best possible location for a single center, and then keep adding centers so as to reduce the covering radius each time by as much as possible. Remark: arbitrarily bad! r(c) greedy center center k = centers site center site Center Selection: Greedy Algorithm Center Selection: Analysis of Greedy Algorithm Greedy algorithm. Repeatedly choose the next center to be the site farthest from any existing center. GreedyCenterSelection(k, n, s,s,,s n ) { C = { s repeat k times { Select a site s i with maximum dist(s i, C) Add s i to C site farthest from any center return C Theorem. Let C* = {c i * be an optimal set of centers. Then r(c) r(c*). Pf. (by contradiction) Assume r(c*) < ½ r(c). For each site c i in C, consider ball of radius ½ r(c) around it. Exactly one c i * in each ball; let c i be the site paired with c i *. Consider any site s and its closest center c i * in C*. dist(s, C) = dist(s, c i ) dist(s, c i *) + dist(c i *, c i ) r(c*). Thus r(c) r(c*). ½r(C) r(c*) since c i* is closest center ½r(C) Observation. Upon termination all centers in C are pairwise at least r(c) apart. Pf. By construction of algorithm. C* sites ½r(C) s c i c i * Center Selection Center Selection: Hardness of Approximation Theorem. Let C* be an optimal set of centers. Then r(c) r(c*). Theorem. Greedy algorithm is a approximation for center selection problem. Remark. Greedy algorithm always places centers at sites, but is still within a factor of of best solution that is allowed to place centers anywhere. e.g., points in the plane Question. Is there hope of a /approximation? /? Theorem. Unless P = NP, there no approximation for centerselection problem for any <. Theorem. Unless P = NP, there is no approximation algorithm for metric kcenter problem for any <. Pf. We show how we could use a (  ) approximation algorithm for k center to solve DOMINATINGSET in polytime. Let G = (V, E), k be an instance of DOMINATINGSET. see Exercise 8. Construct instance G' of kcenter with sites V and distances d(u, v) = if (u, v) E d(u, v) = if (u, v) E Note that G' satisfies the triangle inequality. Claim: G has dominating set of size k iff there exists k centers C* with r(c*) =. Thus, if G has a dominating set of size k, a (  )approximation algorithm on G' must find a solution C* with r(c*) = since it cannot use any edge of distance.
5 . The Pricing Method: ed Vertex Cover ed Vertex Cover ed vertex cover. Given a graph G with vertex weights, find a vertex cover of minimum weight. weight = + + weight = 6 Pricing Method Pricing Method Pricing method. Each edge must be covered by some vertex. Edge e = (i, j) pays price p e to use vertex i and j. Pricing method. Set prices and find vertex cover simultaneously. Fairness. Edges incident to vertex i should pay w i in total. for each vertex i : p e wi e ( i, j) Lemma. For any vertex cover S and any fair prices p e : e p e w(s). Pf. pe ee pe wi w( S). is e( i, j) is edvertexcoverapprox(g, w) { foreach e in E p e = pe wi e ( i, j) while ( edge ij such that neither i nor j are tight) select such an edge e increase p e as much as possible until i or j tight S set of all tight nodes return S each edge e covered by at least one node in S sum fairness inequalities for each node in S 7 8 Pricing Method Pricing Method: Analysis Theorem. Pricing method is a approximation. Pf. Algorithm terminates since at least one new node becomes tight after each iteration of while loop. Let S = set of all tight nodes upon termination of algorithm. S is a vertex cover: if some edge ij is uncovered, then neither i nor j is tight. But then while loop would not terminate. price of edge ab Let S* be optimal vertex cover. We show w(s) w(s*). vertex weight w(s) w i i S i S p e e(i,j) p e iv e(i, j) p e w(s*). e E Figure.8 all nodes in S are tight S V, prices each edge counted twice fairness lemma
6 ed Vertex Cover.6 LP Rounding: ed Vertex Cover ed vertex cover. Given an undirected graph G = (V, E) with vertex weights w i, find a minimum weight subset of nodes S such that every edge is incident to at least one vertex in S. A 6F 6 B 7G 6 C H D I 7 E J total weight = ed Vertex Cover: IP Formulation ed Vertex Cover: IP Formulation ed vertex cover. Given an undirected graph G = (V, E) with vertex weights w i, find a minimum weight subset of nodes S such that every edge is incident to at least one vertex in S. Integer programming formulation. Model inclusion of each vertex i using a / variable x i. if vertex i is not in vertex cover x i if vertex i is in vertex cover Vertex covers in  correspondence with / assignments: S = {i V: x i = ed vertex cover. Integer programming formulation. (ILP) min w i x i i V s. t. x i x j (i, j) E x i {, i V Observation. If x* is optimal solution to (ILP), then S = {i V: x* i = is a minimum weight vertex cover. Objective function: minimize i w i x i. For each edge (i, j), must take either i or j: x i + x j. Integer Programming Linear Programming INTEGERPROGRAMMING. Given integers a ij and b i, find integers x j that satisfy: max c t x s. t. Ax b x integral n a ij x j b i i m j x j j n x j integral j n Observation. Vertex cover formulation proves that integer programming is NPhard. Linear programming. Max/min linear objective function subject to linear inequalities. Input: integers c j, b i, a ij. Output: real numbers x j. (P) max c t x s. t. Ax b x Linear. No x, xy, arccos(x), x(x), etc. (P) max n c j x j j n s. t. a ij x j b i i m j x j j n even if all coefficients are / and at most two variables per inequality Simplex algorithm. [Dantzig 7] Can solve LP in practice. Ellipsoid algorithm. [Khachian 7] Can solve LP in polytime. 6 6
7 LP Feasible Region ed Vertex Cover: LP Relaxation LP geometry in D. ed vertex cover. Linear programming formulation. x = (LP) min w i x i i V s. t. x i x j (i, j) E x i i V Observation. Optimal value of (LP) is optimal value of (ILP). Pf. LP has fewer constraints. Note. LP is not equivalent to vertex cover. ½ ½ x + x = 6 x = x + x = 6 Q. How can solving LP help us find a small vertex cover? A. Solve LP and round fractional values. ½ 7 8 ed Vertex Cover ed Vertex Cover Theorem. If x* is optimal solution to (LP), then S = {i V : x* i ½ is a vertex cover whose weight is at most twice the min possible weight. Pf. [S is a vertex cover] Consider an edge (i, j) E. Since x* i + x* j, either x* i ½or x* j ½ (i, j) covered. Pf. [S has desired cost] Let S* be optimal vertex cover. Then Theorem. approximation algorithm for weighted vertex cover. Theorem. [DinurSafra ] If P NP, then no approximation for <.67, even with unit weights.  Open research problem. Close the gap. w i i S* * w i x i w i i S i S LP is a relaxation x* i ½ Polynomial Time Approximation Scheme.8 Knapsack Problem PTAS. ( + )approximation algorithm for any constant >. Load balancing. [HochbaumShmoys 87] Euclidean TSP. [Arora 6] Consequence. PTAS produces arbitrarily high quality solution, but trades off time for accuracy. This section. PTAS for knapsack problem via rounding and scaling. 7
8 Knapsack Problem Knapsack is NPComplete Knapsack problem. Given n objects and a "knapsack." i has value v i > and weighs w i >. we'll assume w i W Knapsack can carry weight up to W. Goal: fill knapsack so as to maximize total value. Ex: {, has value. 6 W = KNAPSACK: Given a finite set X, nonnegative weights w i, nonnegative values v i, a weight limit W, and a target value V, is there a subset S X such that: w i W is v i V is SUBSETSUM: Given a finite set X, nonnegative values u i, and an integer U, is there a subset S X whose elements sum to exactly U? Claim. SUBSETSUM P KNAPSACK. Pf. Given instance (u,, u n, U) of SUBSETSUM, create KNAPSACK instance: v i w i u i V W U u i U is u i U is Knapsack Problem: Dynamic Programming Knapsack Problem: Dynamic Programming II Def. OPT(i, w) = max value subset of items,..., i with weight limit w. Case : OPT does not select item i. OPT selects best of,, i using up to weight limit w Case : OPT selects item i. new weight limit = w w i OPT selects best of,, i using up to weight limit w w i if i OPT(i, w) OPT(i, w) if w i w maxopt(i, w), v i OPT(i, w w i ) otherwise Running time. O(n W). W = weight limit. Not polynomial in input size! Def. OPT(i, v) = min weight subset of items,, i that yields value exactly v. Case : OPT does not select item i. OPT selects best of,, i that achieves exactly value v Case : OPT selects item i. consumes weight w i, new value needed = v v i OPT selects best of,, i that achieves exactly value v if v if i, v > OPT(i, v) OPT(i, v) if v i v minopt(i, v), w i OPT(i, v v i ) otherwise 6 if v if i, v > OPT(i, v) OPT(i, v) if v i v minopt(i, v), w i OPT(i, v v i ) otherwise if v if i, v > OPT(i, v) OPT(i, v) if v i v minopt(i, v), w i OPT(i, v v i ) otherwise i = or v = i =, v = 8 x x x x x x x x x x x x x x x 6 W = 8 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 6 W = 7 8 8
9 if v if i, v > OPT(i, v) OPT(i, v) if v i v minopt(i, v), w i OPT(i, v v i ) otherwise if v if i, v > OPT(i, v) OPT(i, v) if v i v minopt(i, v), w i OPT(i, v v i ) otherwise i =, v = i =, v = 8 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 6 W = 8 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 6 8 x x x x x x x x x x 6 W = if v if i, v > OPT(i, v) OPT(i, v) if v i v minopt(i, v), w i OPT(i, v v i ) otherwise if v if i, v > OPT(i, v) OPT(i, v) if v i v minopt(i, v), w i OPT(i, v v i ) otherwise i =, v = i =, v = 8 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 6 8 x x x x x x x x x x x x x x x x 6 W = 8 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 6 8 x x x x x x x x x x x x x x x x W = Knapsack Problem: Dynamic Programming II Def. OPT(i, v) = min weight subset of items,, i that yields value exactly v. Case : OPT does not select item i. OPT selects best of,, i that achieves exactly value v Case : OPT selects item i. Intuition for approximation algorithm. Round all values up to lie in smaller range. Run dynamic programming algorithm on rounded instance. Return optimal items in rounded instance. consumes weight w i, new value needed = v v i OPT selects best of,, i that achieves exactly value v if v if i, v > OPT(i, v) OPT(i, v) if v i v minopt(i, v), w i OPT(i, v v i ) otherwise,,6, 7,8,,7,8 6 7,, V* n v max Running time. O(n V*) = O(n v max ). V* = optimal value = maximum v such that OPT(n, v) W. Not polynomial in input size! original instance W = rounded instance W =
10 Knapsack PTAS. Round up all values: v max = largest value in original instance = precision parameter = scaling factor = v max / n Observation. Optimal solution to problems with or are equivalent. v Intuition. close to v so optimal solution using is nearly optimal; small and integral so dynamic programming algorithm is fast. v ˆ Running time. O(n / ). vi vi vi, vˆ i v ˆ Dynamic program II running time is O(n v ˆ max ), where v ˆ max vmax n v v v Knapsack PTAS. Round up all values: i vi Theorem. If S is solution found by our algorithm and S* is an optimal solution of the original problem, then () v i v i Pf. Let S* be an optimal solution satisfying weight constraint. v i i S* v i i S* v i i S i S (v i ) i S v i n i S () v i i S i S* always round up solve rounded instance optimally never round up by more than S n DP alg can take v max n = v max, v max is v i 6
! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. !approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. #approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of three
More information11. 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
More informationAlgorithm 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;
More informationApplied 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
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NPCompleteness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More information2.3 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
More informationAnswers 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 kcenter algorithm first to D and then for each center in D
More informationApproximation Algorithms. Scheduling. Approximation algorithms. Scheduling jobs on a single machine
Approximation algorithms Approximation Algorithms Fast. Cheap. Reliable. Choose two. NPhard problems: choose 2 of optimal polynomial time all instances Approximation algorithms. Tradeoff between time
More information5 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
More informationTopic: 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
More informationGuessing Game: NPComplete?
Guessing Game: NPComplete? 1. LONGESTPATH: Given a graph G = (V, E), does there exists a simple path of length at least k edges? YES 2. SHORTESTPATH: Given a graph G = (V, E), does there exists a simple
More informationApproximation 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
More informationMinimum 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,
More informationCMPSCI611: Approximating MAXCUT Lecture 20
CMPSCI611: Approximating MAXCUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NPhard problems. Today we consider MAXCUT, which we proved to
More informationScheduling 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
More informationLecture 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
More informationNear Optimal Solutions
Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NPComplete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.
More informationFairness 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
More information5.1 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 FordFulkerson
More informationDiscrete (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
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationWhy? 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
More informationComputer Algorithms. NPComplete Problems. CISC 4080 Yanjun Li
Computer Algorithms NPComplete Problems NPcompleteness The quest for efficient algorithms is about finding clever ways to bypass the process of exhaustive search, using clues from the input in order
More informationPrivate Approximation of Clustering and Vertex Cover
Private Approximation of Clustering and Vertex Cover Amos Beimel, Renen Hallak, and Kobbi Nissim Department of Computer Science, BenGurion University of the Negev Abstract. Private approximation of search
More informationLinear 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.
More informationComplexity 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
More informationprinceton 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
More informationLecture 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
More informationBicolored Shortest Paths in Graphs with Applications to Network Overlay Design
Bicolored Shortest Paths in Graphs with Applications to Network Overlay Design Hongsik Choi and HyeongAh Choi Department of Electrical Engineering and Computer Science George Washington University Washington,
More informationScheduling and (Integer) Linear Programming
Scheduling and (Integer) Linear Programming Christian Artigues LAAS  CNRS & Université de Toulouse, France artigues@laas.fr Master Class CPAIOR 2012  Nantes Christian Artigues Scheduling and (Integer)
More informationScheduling Parallel Machine Scheduling. Tim Nieberg
Scheduling Parallel Machine Scheduling Tim Nieberg Problem P C max : m machines n jobs with processing times p 1,..., p n Problem P C max : m machines n jobs with processing times p 1,..., p { n 1 if job
More informationThe multiinteger set cover and the facility terminal cover problem
The multiinteger 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.
More information8.1 Makespan Scheduling
600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Dynamic Programing: MinMakespan and Bin Packing Date: 2/19/15 Scribe: Gabriel Kaptchuk 8.1 Makespan Scheduling Consider an instance
More informationGood luck, veel succes!
Final exam Advanced Linear Programming, May 7, 13.0016.00 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
More informationDiversity Coloring for Distributed Data Storage in Networks 1
Diversity Coloring for Distributed Data Storage in Networks 1 Anxiao (Andrew) Jiang and Jehoshua Bruck California Institute of Technology Pasadena, CA 9115, U.S.A. {jax, bruck}@paradise.caltech.edu Abstract
More informationCSC2420 Spring 2015: Lecture 3
CSC2420 Spring 2015: Lecture 3 Allan Borodin January 22, 2015 1 / 1 Announcements and todays agenda Assignment 1 due next Thursday. I may add one or two additional questions today or tomorrow. Todays agenda
More informationChapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS 3.2 TERMINOLOGY
Chapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS Once the problem is formulated by setting appropriate objective function and constraints, the next step is to solve it. Solving LPP
More informationDefinition 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
More informationUsing the Simplex Method in Mixed Integer Linear Programming
Integer Using the Simplex Method in Mixed Integer UTFSM Nancy, 17 december 2015 Using the Simplex Method in Mixed Integer Outline Mathematical Programming Integer 1 Mathematical Programming Optimisation
More informationLinear Programming: Introduction
Linear Programming: Introduction Frédéric Giroire F. Giroire LP  Introduction 1/28 Course Schedule Session 1: Introduction to optimization. Modelling and Solving simple problems. Modelling combinatorial
More informationmax 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
More informationMapReduce 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
More informationPh.D. Thesis. Judit NagyGyörgy. Supervisor: Péter Hajnal Associate Professor
Online algorithms for combinatorial problems Ph.D. Thesis by Judit NagyGyörgy Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai
More informationDiscuss the size of the instance for the minimum spanning tree problem.
3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can
More informationLecture 6 Online and streaming algorithms for clustering
CSE 291: Unsupervised learning Spring 2008 Lecture 6 Online and streaming algorithms for clustering 6.1 Online kclustering To the extent that clustering takes place in the brain, it happens in an online
More information1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.
Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S
More informationPage 1. CSCE 310J Data Structures & Algorithms. CSCE 310J Data Structures & Algorithms. P, NP, and NPComplete. PolynomialTime Algorithms
CSCE 310J Data Structures & Algorithms P, NP, and NPComplete Dr. Steve Goddard goddard@cse.unl.edu CSCE 310J Data Structures & Algorithms Giving credit where credit is due:» Most of the lecture notes
More informationLecture 4 Online and streaming algorithms for clustering
CSE 291: Geometric algorithms Spring 2013 Lecture 4 Online and streaming algorithms for clustering 4.1 Online kclustering To the extent that clustering takes place in the brain, it happens in an online
More informationPermutation 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
More informationLecture 7: NPComplete Problems
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NPComplete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit
More informationA constantfactor approximation algorithm for the kmedian problem
A constantfactor approximation algorithm for the kmedian problem Moses Charikar Sudipto Guha Éva Tardos David B. Shmoys July 23, 2002 Abstract We present the first constantfactor approximation algorithm
More informationJUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004
Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February
More informationNotes 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
More information1 Approximating Set Cover
CS 05: Algorithms (Grad) Feb 224, 2005 Approximating Set Cover. Definition An Instance (X, F ) of the setcovering problem consists of a finite set X and a family F of subset of X, such that every elemennt
More informationGENERATING LOWDEGREE 2SPANNERS
SIAM J. COMPUT. c 1998 Society for Industrial and Applied Mathematics Vol. 27, No. 5, pp. 1438 1456, October 1998 013 GENERATING LOWDEGREE 2SPANNERS GUY KORTSARZ AND DAVID PELEG Abstract. A kspanner
More informationOutline. NPcompleteness. When is a problem easy? When is a problem hard? Today. Euler Circuits
Outline NPcompleteness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2pairs sum vs. general Subset Sum Reducing one problem to another Clique
More informationInverse 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
More information6. Mixed Integer Linear Programming
6. Mixed Integer Linear Programming Javier Larrosa Albert Oliveras Enric RodríguezCarbonell Problem Solving and Constraint Programming (RPAR) Session 6 p.1/40 Mixed Integer Linear Programming A mixed
More informationMinimal Cost Reconfiguration of Data Placement in a Storage Area Network
Minimal Cost Reconfiguration of Data Placement in a Storage Area Network Hadas Shachnai Gal Tamir Tami Tamir Abstract VideoonDemand (VoD) services require frequent updates in file configuration on the
More informationCSC 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
More informationSIMS 255 Foundations of Software Design. Complexity and NPcompleteness
SIMS 255 Foundations of Software Design Complexity and NPcompleteness Matt Welsh November 29, 2001 mdw@cs.berkeley.edu 1 Outline Complexity of algorithms Space and time complexity ``Big O'' notation Complexity
More informationSmall 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
More informationThe 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,
More informationMath Real Analysis I
Math 431  Real Analysis I Solutions to Homework due October 1 In class, we learned of the concept of an open cover of a set S R n as a collection F of open sets such that S A. A F We used this concept
More informationCSC2420 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
More informationLoadBalanced Virtual Backbone Construction for Wireless Sensor Networks
LoadBalanced Virtual Backbone Construction for Wireless Sensor Networks Jing (Selena) He, Shouling Ji, Yi Pan, Zhipeng Cai Department of Computer Science, Georgia State University, Atlanta, GA, USA, {jhe9,
More informationCan linear programs solve NPhard problems?
Can linear programs solve NPhard problems? p. 1/9 Can linear programs solve NPhard problems? Ronald de Wolf Linear programs Can linear programs solve NPhard problems? p. 2/9 Can linear programs solve
More informationIntroduction to Scheduling Theory
Introduction to Scheduling Theory Arnaud Legrand Laboratoire Informatique et Distribution IMAG CNRS, France arnaud.legrand@imag.fr November 8, 2004 1/ 26 Outline 1 Task graphs from outer space 2 Scheduling
More informationDuplicating and its Applications in Batch Scheduling
Duplicating and its Applications in Batch Scheduling Yuzhong Zhang 1 Chunsong Bai 1 Shouyang Wang 2 1 College of Operations Research and Management Sciences Qufu Normal University, Shandong 276826, China
More informationDefinition 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
More informationConstant Factor Approximation Algorithm for the Knapsack Median Problem
Constant Factor Approximation Algorithm for the Knapsack Median Problem Amit Kumar Abstract We give a constant factor approximation algorithm for the following generalization of the kmedian problem. We
More informationARTICLE IN PRESS. European Journal of Operational Research xxx (2004) xxx xxx. Discrete Optimization. Nan Kong, Andrew J.
A factor 1 European Journal of Operational Research xxx (00) xxx xxx Discrete Optimization approximation algorithm for twostage stochastic matching problems Nan Kong, Andrew J. Schaefer * Department of
More informationApproximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs
Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and HingFung Ting 2 1 College of Mathematics and Computer Science, Hebei University,
More informationTHE 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
More informationA 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
More informationCHAPTER 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
More informationOn 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
More informationPermutation 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
More informationPlanar Tree Transformation: Results and Counterexample
Planar Tree Transformation: Results and Counterexample Selim G Akl, Kamrul Islam, and Henk Meijer School of Computing, Queen s University Kingston, Ontario, Canada K7L 3N6 Abstract We consider the problem
More informationScheduling Home Health Care with Separating Benders Cuts in Decision Diagrams
Scheduling Home Health Care with Separating Benders Cuts in Decision Diagrams André Ciré University of Toronto John Hooker Carnegie Mellon University INFORMS 2014 Home Health Care Home health care delivery
More informationNan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA
A Factor 1 2 Approximation Algorithm for TwoStage Stochastic Matching Problems Nan Kong, Andrew J. Schaefer Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA Abstract We introduce
More information8.1 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
More informationCSE 101: Design and Analysis of Algorithms Homework 4 Winter 2016 Due: February 25, 2016
CSE 101: Design and Analysis of Algorithms Homework 4 Winter 2016 Due: February 25, 2016 Exercises (strongly recommended, do not turn in) 1. (DPV textbook, Problem 7.11.) Find the optimal solution to the
More informationDynamic Programming. Applies when the following Principle of Optimality
Dynamic Programming Applies when the following Principle of Optimality holds: In an optimal sequence of decisions or choices, each subsequence must be optimal. Translation: There s a recursive solution.
More informationOnline Adwords Allocation
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
More informationIntroduction to computer science
Introduction to computer science Michael A. Nielsen University of Queensland Goals: 1. Introduce the notion of the computational complexity of a problem, and define the major computational complexity classes.
More informationProximal mapping via network optimization
L. Vandenberghe EE236C (Spring 234) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:
More informationA counterexample to the Hirsch conjecture
A counterexample to the Hirsch conjecture Francisco Santos Universidad de Cantabria, Spain http://personales.unican.es/santosf/hirsch Bernoulli Conference on Discrete and Computational Geometry Lausanne,
More informationKey words. multiobjective optimization, approximate Pareto set, biobjective shortest path
SMALL APPROXIMATE PARETO SETS FOR BI OBJECTIVE SHORTEST PATHS AND OTHER PROBLEMS ILIAS DIAKONIKOLAS AND MIHALIS YANNAKAKIS Abstract. We investigate the problem of computing a minimum set of solutions that
More informationPartitioning edgecoloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edgecoloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political
More informationResource 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
More informationAnalysis of Server Provisioning for Distributed Interactive Applications
Analysis of Server Provisioning for Distributed Interactive Applications Hanying Zheng and Xueyan Tang Abstract Increasing geographical spreads of modern distributed interactive applications DIAs make
More informationOffline sorting buffers on Line
Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com
More informationEcient approximation algorithm for minimizing makespan. on uniformly related machines. Chandra Chekuri. November 25, 1997.
Ecient approximation algorithm for minimizing makespan on uniformly related machines Chandra Chekuri November 25, 1997 Abstract We obtain a new ecient approximation algorithm for scheduling precedence
More informationTriangle 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,
More informationIntroduction to Algorithms. Part 3: P, NP Hard Problems
Introduction to Algorithms Part 3: P, NP Hard Problems 1) Polynomial Time: P and NP 2) NPCompleteness 3) Dealing with Hard Problems 4) Lower Bounds 5) Books c Wayne Goddard, Clemson University, 2004 Chapter
More informationTransportation 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,
More informationRANDOMIZATION 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,
More informationWelcome to the course Algorithm Design
HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Welcome to the course Algorithm Design Summer Term 2011 Friedhelm Meyer auf der Heide Lecture 6, 20.5.2011 Friedhelm Meyer auf
More information