On the Approximability of NP-complete Optimization Problems

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30 Chapter 4. Approximability classes Definition 4.2 Given an approximability class C and an approximation preserving reduction ≤. The closure of C under the reduction ≤ is C = {F ∈ NPO : ∃G ∈ C : F ≤ G}. In the following sections the most common approximability classes are defined and their characteristics are presented. 4.2 Fptas An Fptas or a fully polynomial time approximation scheme was defined in Definition 2.15 as a polynomial approximation scheme with a time complexity bounded by a polynomial in 1/ε. Let us define the class Fptas of optimization problems which can be approximated by a fully polynomial time approximation scheme. For simplicity the class Fptas has the same name as the approximation scheme Fptas. This should not create any confusion. Definition 4.3 Fptas = {F ∈ NPO : F has an Fptas} The property of having an Fptas is closely related to the property, introduced by Garey and Johnson, of having a pseudo-polynomial time algorithm. A pseudo-polynomial time algorithm is an algorithm whose time complexity is bounded by a polynomial in both the size of the input and the magnitude of the largest number in the input. Definition 4.4 [35] Given an NPO problem F . Let maxintF (x) denotethe magnitude of the largest integer occurring in x where x ∈IF , or 0 if no integers occur in x. Definition 4.5 [35] Given an NPO problem F . An algorithm that solves F is called a pseudo-polynomial time algorithm if its time complexity is bounded from above by a polynomial function of the two variables maxintF (x) and|x| where x ∈IF . Proposition 4.1 (Garey and Johnson [34]) Given an NPO problem F such that for each x ∈IF , opt F (x) is bounded from above by a polynomial in both maxintF (x) and |x|. If F ∈ Fptas then it can be solved by a pseudo-polynomial time algorithm. This is also related to the concept of strong NP-completeness. Definition 4.6 [35] A decision problem F ∈ NP is NP-complete in the strong sense if there exists a polynomial p : N → N for which the problem, when restricting to the instances x that satisfy is NP-complete. maxintF (x) ≤ p(|x|),
Section 4.2. Fptas 31 Many of the problems mentioned in Computers and Intractability: a guide to the theory of NP-completeness by Garey and Johnson are NP-complete in the strong sense [34]. Proposition 4.2 (Garey and Johnson [34]) Given an NPO problem F . If the decision problem version of F is NP-complete in the strong sense,then F cannot be solved by a pseudo-polynomial time algorithm unless P = NP. Thus, given an NPO problem F such that for each x ∈IF , optF (x) is bounded from above by a polynomial in both maxintF (x)and|x|; if the decision problem version of F is NP-complete in the strong sense, then F /∈ Fptas unless P = NP. Paz and Moran have completely characterized the problems in Fptas using a property called p-simpleness. The connections between p-simpleness and NP-completeness in the strong sense are discussed in [5]. Definition 4.7 [90] An NPO problem F is p-simple if there is some polynomial q : N × N → N such that for each k ∈ N the set {x ∈IF : optF (x) ≤ k} is recognizable in time q(|x| ,k). Proposition 4.3 (Paz and Moran [90]) Given an NPO problem F. 1. If F is p-simple and for all x ∈IF , optF (x) is bounded from above by a polynomial in |x|,then F is polynomially solvable,i.e. F ∈ PO. 2. If F is p-simple and for all x ∈IF , optF (x) is bounded from above by a polynomial in maxintF (x) and |x|,then F can be solved by a pseudopolynomial time algorithm. 3. If F can be solved by a pseudo-polynomial time algorithm and for all instances x ∈IF , maxintF (x) is bounded from above by a polynomial in |x| and optF (x),then F is p-simple. Proposition 4.4 (Paz and Moran [90]) Given an NPO problem F. F ∈ Fptas if and only if the following conditions hold. 1. F is p-simple. 2. There is a function b : IF × Z + → N and a polynomial q : N → N such that for each instance x ∈IF and each integer h>0 0 ≤ optF (x) − b(x, h) ≤ q(|x|) if optF =max, h 0 ≤ b(x, h) − optF (x) ≤ q(|x|) if optF =min h where b has time complexity bounded by a polynomial in |x| and optF (x) . h
Page 1 and 2: On the Approximability of NP-comple
Page 3 and 4: Abstract This thesis deals with the
Page 5 and 6: Contents 1 Introduction 1 1.1 Backg
Page 7 and 8: Contents v 7 The travelling salespe
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: Chapter 4 Approximability classes 4
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
Page 61 and 62: Section 4.11. The Max Ind Set class
Page 63 and 64: Section 5.2. Maximum three dimensio
Page 65 and 66: w T x y T T y w w x Section 5.2. Ma
Page 67 and 68: Section 5.2. Maximum three dimensio
Page 69 and 70: Section 5.3. Maximum three-set pack
Page 71 and 72: Section 5.5. Maximum k-dimensional
Page 73 and 74: Section 5.7. Maximum H-matching 65
Page 75 and 76: a b c b Section 5.7. Maximum H-matc
Page 79 and 80: a c b A B Section 5.7. Maximum H-ma
Page 83 and 84: Section 6.2. Maximum common induced
Page 85 and 86: x 1 − x 1 x 2 − x 2 x 3 − x 3
Page 87 and 88: 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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