Algorithm Design

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654 Chapter 1! Approximation <strong>Algorithm</strong>s Exercises 655 6. To try understanding why they don’t seem to be encountering this factor-of-two behavior, you ask a bit about the kind of jobs and loads they see. You find out that the sizes of jobs range between 1 and.50, that is, 1
656 Chapter 11 Approximation Mgorithms reorder the input in such a way that when Greedy-Balance encounters the input in this order, it produces an optimal solution. So their question to you is simply the following: Is this always possible? Their conjecm_re is, For every instance of the load balancing problem from Section 11.1, there exists an order of the jobs so that ~vhen Greedy-Balance processes the jobs in this order, it produces an assignment of jobs to machines ~vith the minimum possible makespan. Decide whether you think this conjecture is true or false, and give either a proof or a counterexample. 9. Consider the following maximizationversion of the 3-DimensionalMatching Problem. Given disjoint sets X, Y, and Z, and given a set T _ X x Y x Z of ordered triples, a subset M c T is a 3-dimensional matching if each element of X u Y u Z is contained in at most one of these triples. The Maximum 3-Dimensional Matching Problem is to find a 3-dimensional matching M of maximum size. (The size of the matching, as usual, is the number of triples it contains. You may assume IXl = IYI = IZI ff you want.) Give a polvnomial-time algorithm that finds a 3-dimension.~ matching of size at least ½ times the ma~xirnum possible size. 10. Suppose you are given an n x n grid graph G, as in Figure 11.13. Associated with each node v is a weight ra(v), which is a nonnegative integer. You may assume that the weights of all nodes are distinct. Your Figure 11.13 A grid graph. 11. Exercises goal is to choose an independent set S of nodes of the grid, so that the sum of the weights of the nodes in Sis as large as possible. (The sum of the weights of the nodes in S will be called its total weight.) Consider the following greedy algorithm for this problem. The "heaviest-first" greedy algorithm: Start with S equal to the empty set While some node remiins in G Pick a node u~ of maximum weight Add u~ to S Delete uf and its neighbors from G Endwhi!e Ret ulql S (a) Let S be the independent set returned by the "heaviest-first" greedy algorithm, and let T be any other independent set in G. Show that, for each node v ~ T, either u ~ S, or there is a node u’ S so that w(u) _< and (v, v’) is an edge of G. (b) Show that the "heaviest-first" greedy algorithm returns an independent set of total weight at least ¼ times the maximum total:Weight of any independent set in the grid graph G. Recall that in the Knapsack Problem, we have n items, each with a weight tv~ and a value ui. We also have a weight bound W, and the problem is to select a set of items S of highest possible value subject to the condition that the total weight does not exceed W--that is, ~i~s w~ _< W. Here’s one way to look at the approximation algorithm that we designed in this chapter. If we are told there exists a subset (9 whose total weight is ~.,~ w~ _< W and whose total value is ~,~ ~ = V for some V, then our approximation algorithm can find a set A with total weight ~i~ w~ _< W and total value at least ~.~ u~ >_ V/(1 + ~). Thus the algorithm approximates the best value, while keeping the weights strictly under W. (Of course, returning the set (9 is always a valid solution, but since the problem is NP-hard, we don’t expect to always be able to find (9 itself; the approximation bound of 1 + ~ means that other sets .4, with slightly less value, can be valid answers as well.) Now, as is well known, you can always pack a Little bit more for a trip just by "sitting on your suitcase"--in other words, by slightly overflowing the allowed weight limit. This too suggests a way of formalizing the approximation question for the Knapsack Problem, but it’s the following, different, formalization. 657
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Cornell University Boston San Franc
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Contents About the Authors Preface
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X Contents 9 10 Extending the Limit
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xiv Preface problems out in the wor
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Preface Chapters 2 and 3 also prese
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xxii Preface Preface xxiii Sue Whit
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2 Chapter 1 Introduction: Some Repr
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6 Chapter 1 Introduction: Some Repr
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10 Chapter 1 Introduction: Some Rep
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14 Chapter ! Introduction: Some Rep
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30 Chapter 2 Basics of Algorithm An
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34 n= I0 n=30 n=50 = 100 = 1,000 10
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74 Chapter 3 Graphs 3.1 Basic Defin
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78 Chapter 3 Graphs Rooted trees ar
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82 Chapter 3 Graphs Current compone
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86 Chapter 3 Graphs Proof. Suppose
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90 Chapter 3 Graphs Two of the simp
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94 Chapter 3 Graphs most ~u nv = 2m
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98 Chapter 3 Graphs It is important
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102 Chapter 3 Graphs ordering, that
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106 Chapter 3 Graphs We can already
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110 Chapter 3 Graphs Exercises 111
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Greedy A~gorithms In Wall Street, t
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118 Chapter 4 Greedy Mgofithms o In
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122 Chapter 4 Greedy Algorithms Our
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126 Chapter 4 Greedy Algorithms Len
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Before swapping: After swapping: d
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134 Chapter 4 Greedy Algorithms wit
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138 Chapter 4 Greedy Algorithms 4.4
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142 Chapter 4 Greedy Algorithms 4.5
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146 Chapter 4 Greedy Algorithms (e
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150 Chapter 4 Greedy Algorithms her
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154 Chapter 4 Greedy Algorithms we
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158 Chapter 4 Greedy Algorithms spa
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162 Chapter 4 Greedy Algorithms ~ T
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166 Chapter 4 Greedy Algorithms g2(
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170 Chapter 4 Greedy Algorithms Sha
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174 Chapter 4 Greedy Algorithms ...
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178 Chapter 4 Greedy Algorithms Fig
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182 Chapter 4 Greedy Algorithms The
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186 Chapter 4 Greedy Algorithms Sol
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192 Chapter 4 Greedy Algorithms 8.
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204 Chapter 4 Greedy Algorithms Not
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Chapter Divide artd Cortquer Divide
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212 Chapter 5 Divide and Conquer Th
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216 cn time plus recursive calls Ch
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220 Chapter 5 Divide and Conquer mo
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224 Chapter 5 Divide and Conquer El
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228 Chapter 5 Divide and Conquer 5.
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232 Chapter 5 Divide and Conquer (a
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we’ll just give a simple example
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240 Chapter 5 Divide and Conquer po
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244 Chapter 5 Divide and Conquer Th
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248 Chapter 5 Divide and Conquer It
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252 Chapter 6 Dynamic Programming b
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256 Chapter 6 Dynamic Programming 6
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260 Index 1 2 3 4 5 6 L~ I = 2 Chap
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264 Chapter 6 Dynamic Programming w
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268 Chapter 6 Dynamic Programming t
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272 Chapter 6 Dynamic Programming s
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280 Chapter 6 Dynamic Programming t
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284 Figure 6.18 The OPT values for
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288 Chapter 6 Dynamic Programming U
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292 (a) Figure 6.21 (a) With negati
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296 Chapter 6 Dynamic Programming i
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300 Chapter 6 Dynamic Programming p
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304 Chapter 6 Dynamic Programming e
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308 Chapter 6 Dynamic Programming o
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312 Chapter 6 Dynamic Programming E
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316 Chapter 6 Dynamic Programming T
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320 Chapter 6 Dynamic Programming (
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328 Chapter 6 Dynamic Programming T
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336 Chapter 6 Dynamic Programming h
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54O Chapter 7 Network Flow define f
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344 Chapter 7 Network Flow This aug
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348 Chapter 7 Network Flow Proof. ~
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352 Chapter 7 Network Flow In the n
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356 Chapter 7 Network Flow (7.19) T
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360 Chapter 7 Network Flow preflow
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364 Chapter 7 Network Flow Heights
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368 Chapter 7 Network Flow ~ The Pr
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372 Chapter 7 Network Flow that are
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376 Chapter 7 Network Flow I~low ar
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380 Chapter 7 Network Flow Figure 7
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384 Chapter 7 Network Flow Lower bo
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388 BOS 6 DCA 7 DCA 8 Chapter 7 Net
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392 Chapter 7 Network Flow natural
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396 Chapter 7 Network Flow 7.11 Pro
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400 Chapter 7 Network Flow 7.12 Bas
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404 Chapter 7 Network Flow to cross
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4O8 Chapter 7 Network Flow Why are
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412 Chapter 7 Network Flow ProoL Th
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416 Chapter 7 Network Flow Figure 7
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420 Chapter 7 Network Flow 11. Now
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424 Figure 7.29 An instance of Cove
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428 Chapter 7 Network Flow 22. This
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432 Chapter 7 Network Flow generall
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436 Chapter 7 Network Flow (a) (b)
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440 Chapter 7 Network Flow going to
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Chapter 7 Network Flow 44. 45. Give
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448 Chapter 7 Network Flow 51. The
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452 Chapter 8 NP and Computational
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472 Chapter 8 NP and Computation!!
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5OO Chapter 8 NP and Computational
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532 Chapter 9 PSPACE: A Class of Pr
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756 Chapter 13 Randomized Algorithm
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792 Chapter !3 Randomized Algorithm
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796 Epilogue: Algorithms That Run F
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800 Epilogue: Algorithms That Run F
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8O4 Epilogue: Algorithms That Run F
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808 References L. R. Ford. Network
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812 References M. Mitzenmacher and
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816 Index Approximation algorithms
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820 Cushions in packet switching, 8
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824 Index Graphs (cont.) queues and
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828 Index Mapping genomes, 279, 521
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832 Index Index 833 Preflow-Push Ma
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836 Index Stale items in randomized
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Algorithm Design

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