On the Approximability of NP-complete Optimization Problems

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iv Contents 4 Approximability classes 29 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 9 4.2 Fptas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Ptas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.4 Fptas ∞ and Ptas ∞ . . . . . . . . . . . . . . . . . . . . . . . . 33 4.5 Rptas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.6 Apx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.7 NPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.8 NPO problems with polynomially bounded optimum . . . . . . 35 4.9 The existence of intermediate degrees . . . . . . . . . . . . . . 37 4.10 Classes defined by logical formulas . . . . . . . . . . . . . . . . 37 4.10.1 Syntactic Max NP . . . . . . . . . . . . . . . . . . . 38 4.10.2 Syntactic Max SNP .................. 40 4.10.3 Closures of Max NP and Max SNP . . . . . . . . . . 42 4.10.4 Relations between the classes Syntactic Max SNP and Max SNP ......................... 44 4.10.5 New definitions of Max SNP and Max NP ....... 45 4.10.6 The hierarchies Max Πi, Max Σi, Min Πi and Min Σi . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.10.7 Closures of these classes . . . . . . . . . . . . . . . . . . 48 4.10.8 Other syntactically defined classes . . . . . . . . . . . . 49 4.11 The Max Ind Set class . . . . . . . . . . . . . . . . . . . . . . 52 5 Matching and packing problems 54 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2Maximum three dimensional matching . . . . . . . . . . . . . . 54 5.3 Maximum three-set packing . . . . . . . . . . . . . . . . . . . . 61 5.4 Maximum k-set packing . . . . . . . . . . . . . . . . . . . . . . 62 5.5 Maximum k-dimensional matching . . . . . . . . . . . . . . . . 63 5.6 Maximum triangle packing. . . . . . . . . . . . . . . . . . . . . 63 5.7 Maximum H-matching . . . . . . . . . . . . . . . . . . . . . . . 64 6 Maximum common subgraph problems 74 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.2 Maximum common induced subgraph . . . . . . . . . . . . . . 74 6.2.1 Approximation of maximum bounded common induced subgraph . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.2.2 Approximation of maximum unbounded common induced subgraph . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.3 Maximum common edge subgraph . . . . . . . . . . . . . . . . 82 6.3.1 Approximation of maximum bounded common edge subgraph ............................ 82 6.3.2Approximation of maximum unbounded common edge subgraph . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.4 Maximum common induced connected subgraph . . . . . . . . 86
Contents v 7 The travelling salesperson problem 87 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.2General Tsp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.3 Tsp with triangle inequality . . . . . . . . . . . . . . . . . . . . 88 7.3.1 Approximation algorithm by Christofides . . . . . . . . 88 7.3.2 Tsp with distances one and two . . . . . . . . . . . . . 89 7.3.3 2-dimensional Euclidean Tsp ............... 90 8 How problems are related approximability-wise 100 8.1 Master and slave problems . . . . . . . . . . . . . . . . . . . . . 100 8.2Problemsintheplane ....................... 100 8.3 Summary of how all problems are related . . . . . . . . . . . . 102 8.3.1 Problems with constant absolute error guarantee . . . . 102 8.3.2Problems with approximation schemes . . . . . . . . . . 103 8.3.3 Problems which can be approximated within a constant 104 8.3.4 Problems not known to be approximable within a constant106 Appendices 108 A Proofs of several reductions 108 B A list of optimization problems 112 B.1 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 B.2 Network design . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 B.3 Setsandpartitions......................... 130 B.4 Storage and retrieval . . . . . . . . . . . . . . . . . . . . . . . . 134 B.5 Sequencing and scheduling . . . . . . . . . . . . . . . . . . . . . 135 B.6 Mathematical programming . . . . . . . . . . . . . . . . . . . . 137 B.7 Logic................................. 139 B.8 Automataandlanguagetheory .................. 143 Index 146 Bibliography 153
Page 1 and 2: On the Approximability of NP-comple
Page 3 and 4: Abstract This thesis deals with the
Page 5: Contents 1 Introduction 1 1.1 Backg
Page 9 and 10: Chapter 1 Introduction Is there any
Page 11 and 12: Section 1.3. Degrees of approximabi
Page 13 and 14: Section 1.5. Approximability of oth
Page 15 and 16: Chapter 2 Definitions 2.1 Basic not
Page 17 and 18: Section 2.3. How approximation is m
Page 19 and 20: Section 2.4. Approximation schemes
Page 21 and 22: Section 2.6. Why do NP-complete pro
Page 23 and 24: IF A ′ Section 3.2. Reencoding of
Page 25 and 26: Section 3.4. Relative error preserv
Page 27 and 28: iii) opt G(t1(x)) ≤ α · opt F (
Page 29 and 30: Section 3.5. Ratio preserving reduc
Page 31 and 32: Section 3.5. Ratio preserving reduc
Page 33 and 34: Section 3.7. Non-constructive reduc
Page 35 and 36: Section 3.8. Structure preserving r
Page 37 and 38: Chapter 4 Approximability classes 4
Page 39 and 40: Section 4.2. Fptas 31 Many of the p
Page 41 and 42: Section 4.4. Fptas ∞ and Ptas ∞
Page 43 and 44: 4.7 NPO Section 4.7. NPO 35 The cla
Page 45 and 46: Section 4.9. The existence of inter
Page 47 and 48: Section 4.10. Classes defined by lo
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Section 4.10. Classes defined by lo
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Section 4.10. Classes defined by lo
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Section 4.11. The Max Ind Set class
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Section 5.2. Maximum three dimensio
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w T x y T T y w w x Section 5.2. Ma
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Section 5.2. Maximum three dimensio
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Section 5.3. Maximum three-set pack
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Section 5.5. Maximum k-dimensional
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Section 5.7. Maximum H-matching 65
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a b c b Section 5.7. Maximum H-matc
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a c b A B Section 5.7. Maximum H-ma
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Section 6.2. Maximum common induced
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x 1 − x 1 x 2 − x 2 x 3 − x 3
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x - x b Section 6.2. Maximum common
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A B 1 2 3 C D Section 6.2. Maximum
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Section 6.3. Maximum common edge su
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−→ a ←−a ←− c −→ c
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Chapter 7 The travellingsalesperson
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Section 7.3. Tsp with triangle ineq
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2−r1 Section 7.3. Tsp with triang
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Section 8.2. Problems in the plane
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Section 8.3. Summary of how all pro
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109 set of G of the same size. Once
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111 x0 may take the values [1..m],
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Section B.1. Graph theory 113 Algor
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Section B.1. Graph theory 115 I = {
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Section B.1. Graph theory 117 graph
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Section B.1. Graph theory 119 S(〈
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Section B.1. Graph theory 121 opt =
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Section B.1. Graph theory 123 Ai is
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Section B.1. Graph theory 125 n Al
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Section B.2. Network design 127 CES
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Section B.2. Network design 129 I =
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Section B.3. Sets and partitions 13
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Section B.3. Sets and partitions 13
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Section B.5. Sequencing and schedul
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Section B.6. Mathematical programmi
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m(〈A, b, c〉 ,x)=cT x = ncixi i
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Section B.7. Logic 141 I = {〈U, F
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Section B.8. Automata and language
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Section B.8. Automata and language
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Max PB 0 − 1 Programming, 36, 107
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feasible solution, 8 Fptas, 11, 30,
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graph colouring, 47, 100, 114 heigh
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Bibliography [1] A. Aho, J. E. Hopc
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Bibliography 155 [26] D. Z. Du and
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Bibliography 157 [55] V. Kann. On t
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Bibliography 159 [83] P. Orponen an
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On the Approximability of NP-complete Optimization Problems

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