! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. #approximation algorithm.


 Abel Price
 6 years ago
 Views:
Transcription
1 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 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 Algorithm Design by Éva Tardos and Jon Kleinberg Copyright 005 Addison Wesley Slides by Kevin Wayne Load Balancing 111 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 makespan Algorithm Design by Éva Tardos and Jon Kleinberg Copyright 005 Addison Wesley Slides by Kevin Wayne 4
2 Load Balancing on Machines Load Balancing: List Scheduling Claim Load balancing is hard even if only machines Pf PARTITION $ P LOADBALANCE Listscheduling 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 ListScheduling(m, n, t 1,t,,t n ) { for i = 1 to m { L i % 0 load on machine i J(i) % & jobs assigned to machine i machine 1 machine length of job f a d Machine 1 f b c Machine e g yes for j = 1 to n { i = argmin k L k J(i) % J(i) ' {j L i % L i + t j machine i has smallest load assign job j to machine i update load of machine i 0 Time L Implementation O(n log n) using a priority queue 5 6 Load Balancing: List Scheduling Analysis Load Balancing: List Scheduling Analysis Theorem [Graham, 1966] Greedy algorithm is a approximation First worstcase analysis of an approximation algorithm Need to compare resulting solution with optimal makespan L* Lemma 1 The optimal makespan L* ) max j t j Pf Some machine must process the most timeconsuming job Theorem Greedy algorithm is a approximation Pf Consider load L i of bottleneck machine i Let j be 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 1 $ k $ m Lemma The optimal makespan Pf The total processing time is j t j L * " m 1 # j t j One of m machines must do at least a 1/m fraction of total work machine i blue jobs scheduled before j j 0 L i  t j L = L i 7 8
3 Load Balancing: List Scheduling Analysis Load Balancing: List Scheduling Analysis Theorem Greedy algorithm is a approximation Pf Consider load L i of bottleneck machine i Let j be 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 1 $ k $ m Sum inequalities over all k and divide by m: Q Is our analysis tight? A Essentially yes Ex: m machines, m(m1) jobs length 1 jobs, one job of length m L i " t j # 1 m $ k L k = 1 m $ k t k # L * Now L i = (L i " t j ) t j # L * { # L* # L* Lemma 1 Lemma Machine m = 10, list scheduling makespan = Machine idle Machine 3 idle Machine 4 idle Machine 5 idle Machine 6 idle Machine 7 idle Machine 8 idle Machine 9 idle Machine 10 idle 9 10 Load Balancing: List Scheduling Analysis Load Balancing: LPT Rule Q Is our analysis tight? A Essentially yes Longest processing time (LPT) Sort n jobs in descending order of processing time, and then run list scheduling algorithm Ex: m machines, m(m1) jobs length 1 jobs, one job of length m LPTListScheduling(m, n, t 1,t,,t n ) { Sort jobs so that t 1 t t n for i = 1 to m { L i % 0 J(i) % & load on machine i jobs assigned to machine i for j = 1 to n { i = argmin k L k J(i) % J(i) ' {j L i % L i + t j machine i has smallest load assign job j to machine i update load of machine i m = 10, optimal makespan =
4 Load Balancing: LPT Rule Load Balancing: LPT Rule Observation If at most m jobs, then listscheduling is optimal Pf Each job put on its own machine Q Is our 3/ analysis tight? A No Lemma 3 If there are more than m jobs, L* ) t m+1 Pf Consider first m+1 jobs t 1,, t m+1 Since the t i 's are in descending order, each takes at least t m+1 time There are m+1 jobs and m machines, so by pigeonhole principle, at least one machine gets two jobs Theorem LPT rule is a 3/ approximation algorithm Pf Same basic approach as for list scheduling L i = (L i " t j ) t j { # 3 L * # L* # 1 L* Theorem [Graham, 1969] LPT rule is a 4/3approximation Pf More sophisticated analysis of same algorithm Q Is Graham's 4/3 analysis tight? A Essentially yes Ex: m machines, n = m+1 jobs, jobs of length m+1, m+,, m1 and one job of length m Lemma 3 ( by observation, can assume number of jobs > m ) Center Selection Problem 11 Center Selection Input Set of n sites s 1,, s n Center selection problem Select k centers C so that maximum distance from a site to nearest center is minimized k = 4 r(c) center site Algorithm Design by Éva Tardos and Jon Kleinberg Copyright 005 Addison Wesley Slides by Kevin Wayne 16
5 Center Selection Problem Center Selection Example Input Set of n sites s 1,, s n Center selection problem Select k centers C so that maximum distance from a site to nearest center is minimized 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 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 r(c) Distance function properties dist(x, x) = 0 (identity) dist(x, y) = dist(y, x) (symmetry) dist(x, y) $ dist(x, z) + dist(z, y) (triangle inequality) center site Greedy Algorithm: A False Start Center Selection: Greedy Algorithm 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 Greedy algorithm Repeatedly choose the next center to be the site farthest from any existing center Remark: arbitrarily bad greedy center 1 GreedyCenterSelection(k, n, s 1,s,,s n ) { C = & 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 k = centers center site Observation Upon termination all centers in C are pairwise at least r(c) apart Pf By construction of algorithm 19 0
6 Center Selection: Analysis of Greedy Algorithm Center Selection Theorem Let C* be an optimal set of centers Then r(c) $ r(c*) Pf (by contradiction) Assume r(c*) < r(c) For each site c i in C, consider ball of radius 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*) *inequality $ r(c*) since c i * is closest center 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 eg, points in the plane r(c) r(c) Question Is there hope of a 3/approximation? 4/3? C* sites r(c) s c i c i * Theorem Unless P = NP, there no #approximation for centerselection problem for any # < 1 Weighted Vertex Cover 114 The Pricing Method: Vertex Cover Weighted vertex cover Given a graph G with vertex weights, find a vertex cover of minimum weight weight = weight = 9 Algorithm Design by Éva Tardos and Jon Kleinberg Copyright 005 Addison Wesley Slides by Kevin Wayne 4
7 Weighted Vertex Cover PrimalDual Algorithm Pricing method Each edge must be covered by some vertex Edge e pays price p e ) 0 to use edge Primaldual algorithm 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 w e= ( i, j) i 4 9 WeightedVertexCoverApprox(G, w) { foreach e in E p e = 0 p = w e e = ( i, j) while ( edge ij such that neither i nor j are tight) select such an edge e increase p e without violating fairness i Claim For any vertex cover S and any fair prices p e :, e p e $ w(s) Proof " pe " " pe " wi = w( S) e# E i# S e= ( i, j) i# S 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 5 6 Primal Dual Algorithm: Analysis Theorem Primaldual algorithm is a approximation Pf Algorithm terminates since at least one new node becomes tight after each iteration of while loop 116 LP Rounding: Vertex Cover Let S = set of all tight nodes upon termination of algorithm S is a vertex cover: if some edge ij is uncovered, then either i or j is not tight But then while loop would not terminate Let S* be optimal vertex cover We show w(s) $ w(s*) w( S) = " wi = " " pe " " pe = " pe i# S i# S e= ( i, j) i# V e= ( i, j) e# E w( S*) all nodes in S are tight S + V, prices ) 0 each edge counted twice fairness lemma 7 Algorithm Design by Éva Tardos and Jon Kleinberg Copyright 005 Addison Wesley Slides by Kevin Wayne
8 Weighted Vertex Cover Weighted Vertex Cover: IP Formulation Weighted vertex cover Given an undirected graph G = (V, E) with vertex weights w i ) 0, find a minimum weight subset of nodes S such that every edge is incident to at least one vertex in S 10 A 6F 9 Weighted vertex cover Integer programming formulation (ILP) min # w i x i i " V s t x i + x j $ 1 (i, j) " E x i " {0,1 i " V 16 B 7G C H 9 Observation If x* is optimal solution to (ILP), then S = {i " V : x* i = 1 is a min weight vertex cover 3 D I 33 7 E 10 J 3 total weight = Integer Programming Linear Programming INTEGERPROGRAMMING Given integers a ij and b i, find integers x j that satisfy: n " a ij x j # b i 1$ i $ m j=1 x j # 0 1$ j $ n x j integral 1$ j $ n Observation Vertex cover formulation proves that integer programming is NPhard search problem (even if all coefficients are 0/1 and at most two nonzeros per inequality) 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 " 0 Linear No x, xy, arccos(x), x(1x), etc (P) max " c j x j n j=1 n s t " a ij x j = b i 1# i # m j=1 x j $ 0 1# j # n Simplex algorithm [Dantzig 1947] Can solve LP in practice Ellipsoid algorithm [Khachian 1979] Can solve LP in polytime 31 3
9 Weighted Vertex Cover: LP Relaxation Weighted Vertex Cover Weighted vertex cover Linear programming formulation (LP) min # w i x i i " V s t x i + x j $ 1 (i, j) " E x i $ 0 i " V Observation Optimal value of (LP) is $ optimal value of (ILP) Theorem If x* is optimal solution to (LP), then S = {i " V : x* i ) 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 ) 1, either x* i ) or x* j ) ( (i, j) covered Pf [S has desired cost] Let S* be optimal vertex cover Then Note LP is not equivalent to vertex cover w i i " S* * # $ # w i x i $ 1 w # i i " S i " S LP is a relaxation x* i ) Q How can solving LP help us find a small vertex cover? A Solve LP and round fractional values Weighted Vertex Cover Good news approximation algorithm is basis for most practical heuristics can solve LP with min cut ( faster primaldual schema ( linear time (see book) PTAS for planar graphs Solvable in polytime on bipartite graphs using network flow 117 Load Balancing Reloaded Bad news [DinurSafra, 001] If P NP, then no #approximation for # < 13607, even with unit weights 10 / Algorithm Design by Éva Tardos and Jon Kleinberg Copyright 005 Addison Wesley Slides by Kevin Wayne
10 Generalized Load Balancing Generalized Load Balancing: Integer Linear Program and Relaxation Input Set of m machines M; set of n jobs J Job j must run contiguously on an authorized machine in M j + M Job j has processing time t j Each 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 = max i L i Generalized load balancing Assign each job to an authorized machine to minimize makespan ILP formulation x ij = time machine i spends processing job j LP relaxation (IP) min (LP) min L s t " x i j = t j for all j # J i " $ L for all i # M x i j j x i j # {0, t j for all j # J and i # M j x i j = 0 for all j # J and i % M j L s t " x i j = t j for all j # J i " $ L for all i # M x i j j x i j % 0 for all j # J and i # M j x i j = 0 for all j # J and i & M j Generalized Load Balancing: Lower Bounds Generalized Load Balancing: Structure of LP Solution Lemma 1 Let L be the optimal value to the LP Then, the optimal makespan L* ) L Pf LP has fewer constraints than IP formulation Lemma The optimal makespan L* ) max j t j Pf Some machine must process the most timeconsuming job Lemma 3 Let x be an extreme point solution to LP Let G(x) be the graph with an edge from machine i to job j if x ij > 0 Then, G(x) is acyclic Pf (we prove contrapositive) Let x be a feasible solution to the LP such that G(x) has a cycle Define % x ij ±" y ij = & ' x ij (i, j) # C (i, j) $ C % x ij m " z ij = & ' x ij (i, j) # C (i, j) $ C The variables y and z are feasible solutions to the LP Observe x = y + z Thus, x is not an extreme point j +0 i
11 Generalized Load Balancing: Rounding Generalized Load Balancing: Analysis Rounded solution Find extreme point LP solution x Root forest G(x) at some arbitrary machine node r If job j is a leaf node, assign j to its parent machine i If job j is not a leaf node, assign j to one of its children Lemma 4 Rounded solution only assigns jobs to authorized machines Pf If job j is assigned to machine i, then x ij > 0 LP solution can only assign positive value to authorized machines Lemma 5 If job j is a leaf node and machine i = parent(j), then x ij = t j Pf Since i is a leaf, x ij = 0 for all j parent(i) LP constraint guarantees i x ij = t j Lemma 6 At most one nonleaf job is assigned to a machine Pf The only possible nonleaf job assigned to machine i is parent(i) job machine 41 4 Generalized Load Balancing: Analysis Conclusions Theorem Rounded solution is a approximation Pf Let J(i) be the jobs assigned to machine i By Lemma 6, the load L i on machine i has two components: Running time The bottleneck operation in our approximation is solving one LP with mn + 1 variables Remark Possible to solve LP using max flow techniques (see text) leaf nodes parent(i) t j j " J(i) j is a leaf Lemma 5 # = # x ij $ # x ij $ L $ L * j " J(i) j is a leaf Lemma t parent(i) " L * j " J LP Lemma 1 (LP is a relaxation) optimal value of LP Extensions: unrelated parallel machines [LenstraShmoysTardos 1990] Job j takes t ij time if processed on machine i approximation algorithm via LP rounding No 3/approximation algorithm unless P = NP Thus, the overall load L i $ L* 43 44
12 Polynomial Time Approximation Scheme 118 Knapsack Problem PTAS (1 + 0)approximation algorithm for any constant 0 > 0 Load balancing [HochbaumShmoys 1987] Euclidean TSP [Arora 1996] FPTAS PTAS that is polynomial in input size and 1/0 Consequence PTAS produces arbitrarily high quality solution, but trades off accuracy for time This section FPTAS for knapsack problem via rounding and scaling Algorithm Design by Éva Tardos and Jon Kleinberg Copyright 005 Addison Wesley Slides by Kevin Wayne 46 Knapsack Problem Knapsack is NPComplete Knapsack problem Given n objects and a "knapsack" Item i has value v i > 0 and weighs w i > 0 Knapsack can carry weight up to W we'll assume w i $ W Goal: fill knapsack so as to maximize total value Ex: { 3, 4 has value 40 Item 1 Value 1 6 Weight 1 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 w i i"s v i i"s # % V 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 1,, u n, U) of SUBSETSUM, create KNAPSACK instance: v i = w i = u i # u i $ U i"s V = W = U # u i % U i"s 47 48
13 Knapsack Problem: Dynamic Programming 1 Knapsack Problem: Dynamic Programming II Def OPT(i, w) = max value subset of items 1,, i with weight limit w Case 1: OPT does not select item i OPT selects best of 1,, i 1 using up to weight limit w Case : OPT selects item i new weight limit = w w i OPT selects best of 1,, i 1 using up to weight limit w w i # 0 if i = 0 % OPT(i, w) = $ OPT(i "1, w) if w i > w % & max{ OPT(i "1, w), v i + OPT(i "1, 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 1,, i that yields value exactly v Case 1: OPT does not select item i OPT selects best of 1,, i1 that achieves exactly value v Case : OPT selects item i consumes weight w i, new value needed = v v i OPT selects best of 1,, i1 that achieves exactly value v $ 0 if v = 0 & " if i = 0, v > 0 OPT(i, v) = % & OPT(i #1, v) if v i > v '& min{ OPT(i #1, v), w i + OPT(i #1, v # v i ) otherwise 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 Knapsack: FPTAS 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 Knapsack: FPTAS Knapsack FPTAS Round up all values: v max = largest value in original instance 0 = precision parameter 1 = scaling factor = 0 v max / n # v i = v i % $ " & ", v ˆ # i = v i % $ " & Item Value Weight 1 134, ,34 3 1,810, ,17, ,343,199 7 W = 11 original instance Item Value Weight W = 11 rounded instance Observation Optimal solution to problems with or are equivalent Intuition close to v so optimal solution using is nearly optimal; small and integral so dynamic programming algorithm is fast v ˆ v Running time O(n 3 / 0) Dynamic program II running time is O(n v ˆ max ), where v ˆ max = # v max $ " % & = # n % $ ' & v v v ˆ 51 5
14 Knapsack: FPTAS Knapsack FPTAS Round up all values: # v i = v i % $ " & " Extra Slides Theorem If S is solution found by our algorithm and S* is any other feasible solution then (1+") % v i # % v i i $ S i $ S* Pf Let S* be any feasible solution satisfying weight constraint v i i " S* # $ # v i i " S* $ # v i i " S $ # (v i + %) i " S always round up solve rounded instance optimally never round up by more than 1 $ # v i + n% i" S S $ n DP alg can take v max $ (1+&) # v i i" S n 1 = 0 v max, v max $ i"s v i 53 Algorithm Design by Éva Tardos and Jon Kleinberg Copyright 005 Addison Wesley Slides by Kevin Wayne Center Selection: Hardness of Approximation Knapsack: State of the Art Theorem Unless P = NP, there is no (  0) approximation algorithm for kcenter problem for any 0 > 0 Pf We show how we could use a (  0) approximation algorithm for k center to solve DOMINATINGSET in polytime Let G = (V, E), k be an instance of DOMINATINGSET Construct instance G' of kcenter with sites V and distances d(u, v) = if (u, v) " E d(u, v) = 1 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*) = 1 Thus, if G has a dominating set of size k, a (  0)approximation algorithm on G' must find a solution C* with r(c*) = 1 since it cannot use any edge of distance This lecture "Rounding and scaling" method finds a solution within a (1 + 0) factor of optimum for any 0 > 0 Takes O(n 3 / 0) time and space IbarraKim (1975), Lawler (1979) Faster FPTAS: O(n log (1 / 0) + 1 / 0 4 ) time Idea: group items by value into "large" and "small" classes run dynamic programming algorithm only on large items insert small items according to ratio v i / w i clever analysis 55 56
! 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 informationChapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
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
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 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 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 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 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 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 information! X is a set of strings. ! Instance: string s. ! Algorithm A solves problem X: A(s) = yes iff s! X.
Decision Problems 8.2 Definition of NP Decision problem. X is a set of strings. Instance: string s. Algorithm A solves problem X: A(s) = yes iff s X. Polynomial time. Algorithm A runs in polytime if for
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationHow To Find An Optimal Search Protocol For An Oblivious Cell
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 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 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 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 informationClassification  Examples
Lecture 2 Scheduling 1 Classification  Examples 1 r j C max given: n jobs with processing times p 1,...,p n and release dates r 1,...,r n jobs have to be scheduled without preemption on one machine taking
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 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 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 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 information4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal
More informationEfficient and Robust Allocation Algorithms in Clouds under Memory Constraints
Efficient and Robust Allocation Algorithms in Clouds under Memory Constraints Olivier Beaumont,, Paul RenaudGoud Inria & University of Bordeaux Bordeaux, France 9th Scheduling for Large Scale Systems
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 informationThe Generalized Assignment Problem with Minimum Quantities
The Generalized Assignment Problem with Minimum Quantities Sven O. Krumke a, Clemens Thielen a, a University of Kaiserslautern, Department of Mathematics PaulEhrlichStr. 14, D67663 Kaiserslautern, Germany
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 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 informationImproved Algorithms for Data Migration
Improved Algorithms for Data Migration Samir Khuller 1, YooAh Kim, and Azarakhsh Malekian 1 Department of Computer Science, University of Maryland, College Park, MD 20742. Research supported by NSF Award
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 information9th MaxPlanck Advanced Course on the Foundations of Computer Science (ADFOCS) PrimalDual Algorithms for Online Optimization: Lecture 1
9th MaxPlanck Advanced Course on the Foundations of Computer Science (ADFOCS) PrimalDual Algorithms for Online Optimization: Lecture 1 Seffi Naor Computer Science Dept. Technion Haifa, Israel Introduction
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 informationSchool Timetabling in Theory and Practice
School Timetabling in Theory and Practice Irving van Heuven van Staereling VU University, Amsterdam Faculty of Sciences December 24, 2012 Preface At almost every secondary school and university, some
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 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 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 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 informationDistributed 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.
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 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 informationScheduling Parallel Jobs with Monotone Speedup 1
Scheduling Parallel Jobs with Monotone Speedup 1 Alexander Grigoriev, Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands, {a.grigoriev@ke.unimaas.nl,
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 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 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 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 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 informationOnline and Offline Selling in Limit Order Markets
Online and Offline Selling in Limit Order Markets Kevin L. Chang 1 and Aaron Johnson 2 1 Yahoo Inc. klchang@yahooinc.com 2 Yale University ajohnson@cs.yale.edu Abstract. Completely automated electronic
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 informationLoad balancing of temporary tasks in the l p norm
Load balancing of temporary tasks in the l p norm Yossi Azar a,1, Amir Epstein a,2, Leah Epstein b,3 a School of Computer Science, Tel Aviv University, Tel Aviv, Israel. b School of Computer Science, The
More informationPractical Guide to the Simplex Method of Linear Programming
Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear
More informationSingle 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
More informationLecture 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
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 informationCIS 700: algorithms for Big Data
CIS 700: algorithms for Big Data Lecture 6: Graph Sketching Slides at http://grigory.us/bigdataclass.html Grigory Yaroslavtsev http://grigory.us Sketching Graphs? We know how to sketch vectors: v Mv
More informationA simpler and better derandomization of an approximation algorithm for Single Source RentorBuy
A simpler and better derandomization of an approximation algorithm for Single Source RentorBuy David P. Williamson Anke van Zuylen School of Operations Research and Industrial Engineering, Cornell University,
More informationApproximating Minimum Bounded Degree Spanning Trees to within One of Optimal
Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal ABSTACT Mohit Singh Tepper School of Business Carnegie Mellon University Pittsburgh, PA USA mohits@andrew.cmu.edu In the MINIMUM
More informationAn 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
More informationLoad Balancing. Load Balancing 1 / 24
Load Balancing Backtracking, branch & bound and alphabeta pruning: how to assign work to idle processes without much communication? Additionally for alphabeta pruning: implementing the youngbrotherswait
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 informationEmbedded Systems 20 BF  ES
Embedded Systems 201  Multiprocessor Scheduling REVIEW Given n equivalent processors, a finite set M of aperiodic/periodic tasks find a schedule such that each task always meets its deadline. Assumptions:
More informationCombinatorial Algorithms for Data Migration to Minimize Average Completion Time
Combinatorial Algorithms for Data Migration to Minimize Average Completion Time Rajiv Gandhi 1 and Julián Mestre 1 Department of Computer Science, Rutgers UniversityCamden, Camden, NJ 0810. Research partially
More informationLoad Balancing and Rebalancing on Web Based Environment. Yu Zhang
Load Balancing and Rebalancing on Web Based Environment Yu Zhang This report is submitted as partial fulfilment of the requirements for the Honours Programme of the School of Computer Science and Software
More information3. 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
More informationThe Container Selection Problem
The Container Selection Problem Viswanath Nagarajan 1, Kanthi K. Sarpatwar 2, Baruch Schieber 2, Hadas Shachnai 3, and Joel L. Wolf 2 1 University of Michigan, Ann Arbor, MI, USA viswa@umich.edu 2 IBM
More informationEmbedded Systems 20 REVIEW. Multiprocessor Scheduling
Embedded Systems 0   Multiprocessor Scheduling REVIEW Given n equivalent processors, a finite set M of aperiodic/periodic tasks find a schedule such that each task always meets its deadline. Assumptions:
More informationThe LoadDistance Balancing Problem
The LoadDistance Balancing Problem Edward Bortnikov Samir Khuller Yishay Mansour Joseph (Seffi) Naor Yahoo! Research, Matam Park, Haifa 31905 (Israel) Department of Computer Science, University of Maryland,
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 informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 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
More informationEnergy Efficient Monitoring in Sensor Networks
Energy Efficient Monitoring in Sensor Networks Amol Deshpande, Samir Khuller, Azarakhsh Malekian, Mohammed Toossi Computer Science Department, University of Maryland, A.V. Williams Building, College Park,
More informationOne last point: we started off this book by introducing another famously hard search problem:
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani 261 Factoring One last point: we started off this book by introducing another famously hard search problem: FACTORING, the task of finding all prime factors
More informationONLINE DEGREEBOUNDED STEINER NETWORK DESIGN. Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015
ONLINE DEGREEBOUNDED STEINER NETWORK DESIGN Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015 ONLINE STEINER FOREST PROBLEM An initially given graph G. s 1 s 2 A sequence of demands (s i, t i ) arriving
More informationDynamic programming. Doctoral course Optimization on graphs  Lecture 4.1. Giovanni Righini. January 17 th, 2013
Dynamic programming Doctoral course Optimization on graphs  Lecture.1 Giovanni Righini January 1 th, 201 Implicit enumeration Combinatorial optimization problems are in general NPhard and we usually
More informationThe Design of Approximation Algorithms
The Design of Approximation Algorithms David P. Williamson David B. Shmoys Copyright c 2010 by David P. Williamson and David B. Shmoys. All rights reserved. To be published by Cambridge University Press.
More informationScheduling to Minimize Power Consumption using Submodular Functions
Scheduling to Minimize Power Consumption using Submodular Functions Erik D. Demaine MIT edemaine@mit.edu Morteza Zadimoghaddam MIT morteza@mit.edu ABSTRACT We develop logarithmic approximation algorithms
More informationTutorial 8. NPComplete Problems
Tutorial 8 NPComplete Problems Decision Problem Statement of a decision problem Part 1: instance description defining the input Part 2: question stating the actual yesorno question A decision problem
More informationLecture 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
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 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 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 informationIE 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
More informationAnalysis of Approximation Algorithms for kset Cover using FactorRevealing Linear Programs
Analysis of Approximation Algorithms for kset Cover using FactorRevealing Linear Programs Stavros Athanassopoulos, Ioannis Caragiannis, and Christos Kaklamanis Research Academic Computer Technology Institute
More information