On the Approximability of NP-complete Optimization Problems

More documents

Recommendations

Info

36 Chapter 4. Approximability classes Proof F ∈ Fptas PB ⇒∃algorithm A(x, ε), polynomial p(|x| , 1/ε) such that the time complexity of A is p(|x| , 1/ε), A approximates F within 1 + ε and there is a polynomial q(|x|) such that optF (x) ≤ q(|x|). 1 Compute A x, which approximates F within 1+1/(q(|x|)+1). q(|x|)+1 The time for this is p(|x| ,q(|x|) + 1), i.e. polynomial time. Let s = mF (x, A(x, 1/(q(|x|) + 1))). First suppose that optF =max. Then 0 optF (x) − 1 Thus opt F (x) − 1
Section 4.9. The existence of intermediate degrees 37 4.9 The existence of intermediate degrees Are there problems between Apx and the NPO-complete problems, that is problems which are not in Apx and are not NPO-complete? The following proposition answers this question affirmative. Proposition 4.12 (Crescenzi and Panconesi [24]) If P = NP there is an NPO problem F such that F /∈ Apx, F is not NPOcomplete and F is Apx-hard under P-reductions. Proposition 4.13 (Crescenzi and Panconesi [24]) If P = NP there is an NPO problem F such that F /∈ Apx, F is not NPOcomplete and F is not Apx-hard under P-reductions. Thus there are intuitively strange problems which are strong enough to be not approximable within a constant, but not strong enough to represent all problems which are approximable within a constant. 4.10 Classes defined by logical formulas An alternative, logical characterization of NP was made by Fagin in 1974 [29]. It is interesting since it is the only definition which does not involve computation. As an example we shall show that Sat ∈ NP using this characterization. Example 4.1 Given a number of disjunctive clauses the Sat problem is to determine if there is a truth assignment that satisfies all clauses. Let the input be given as two predicates P and N where P (x, c) =true iff variable x appears positive in clause c and N(x, c) =true iff variable x appears negated in clause c. The clauses can be satisfied if and only if ∃T ∀c∃x(P (x, c) ∧ x ∈ T ) ∨ (N(x, c) ∧ x/∈ T ). T shall be interpreted as the set of true variables. In general, a language L is in NP if and only if there is a quantifier free first-order formula ΦL such that I ∈ L ⇔∃S ∀x∃yΦL(I,S,x, y) where I is a finite structure describing the instance, S is a finite structure, x and y are finite vectors. This formalism was used by Papadimitriou and Yannakakis in 1988 to define two classes of NPO problems: Max NP (maximization NP) andMax SNP (maximization strict NP) [88]. Later Panconesi and Ranjan introduced the class Max Π1 [84] and recently Kolaitis and Thakur completed the picture with four class hierarchies Max Πi, Max Σi, Min Πi and Min Σi [61, 60]. All these classes will be presented later in this section. The definitions of Max NP and Max SNP have been interpreted differently by different authors. In this theses the notations Max NP and Max SNP
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 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: 4.7 NPO Section 4.7. NPO 35 The cla
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
Page 89 and 90: A B 1 2 3 C D Section 6.2. Maximum
Page 91 and 92: Section 6.3. Maximum common edge su
Page 93 and 94: −→ a ←−a ←− c −→ c
Page 95 and 96:
Chapter 7 The travellingsalesperson
Page 97 and 98:
Section 7.3. Tsp with triangle ineq
Page 99 and 100:
2−r1 Section 7.3. Tsp with triang
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Section 8.2. Problems in the plane
Page 111 and 112:
Section 8.3. Summary of how all pro
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
109 set of G of the same size. Once
Page 119 and 120:
111 x0 may take the values [1..m],
Page 121 and 122:
Section B.1. Graph theory 113 Algor
Page 123 and 124:
Section B.1. Graph theory 115 I = {
Page 125 and 126:
Section B.1. Graph theory 117 graph
Page 127 and 128:
Section B.1. Graph theory 119 S(〈
Page 129 and 130:
Section B.1. Graph theory 121 opt =
Page 131 and 132:
Section B.1. Graph theory 123 Ai is
Page 133 and 134:
Section B.1. Graph theory 125 n Al
Page 135 and 136:
Section B.2. Network design 127 CES
Page 137 and 138:
Section B.2. Network design 129 I =
Page 139 and 140:
Section B.3. Sets and partitions 13
Page 141 and 142:
Section B.3. Sets and partitions 13
Page 143 and 144:
Section B.5. Sequencing and schedul
Page 145 and 146:
Section B.6. Mathematical programmi
Page 147 and 148:
m(〈A, b, c〉 ,x)=cT x = ncixi i
Page 149 and 150:
Section B.7. Logic 141 I = {〈U, F
Page 151 and 152:
Section B.8. Automata and language
Page 153 and 154:
Section B.8. Automata and language
Page 155 and 156:
Max PB 0 − 1 Programming, 36, 107
Page 157 and 158:
feasible solution, 8 Fptas, 11, 30,
Page 159 and 160:
graph colouring, 47, 100, 114 heigh
Page 161 and 162:
Bibliography [1] A. Aho, J. E. Hopc
Page 163 and 164:
Bibliography 155 [26] D. Z. Du and
Page 165 and 166:
Bibliography 157 [55] V. Kann. On t
Page 167 and 168:
Bibliography 159 [83] P. Orponen an
show all

On the Approximability of NP-complete Optimization Problems

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?