On the Approximability of NP-complete Optimization Problems

More documents

Recommendations

Info

vi Contents
Chapter 1 Introduction Is there any way to find efficient algorithms which always give good approximations to optimization problems to which it is hard to find the optimal solution? This is the main question motivating the work in this thesis. The answer to the question is: it depends on the problem. Of problems which are equally hard to solve optimally, some can be very well approximated with efficient algorithms, while others, under the well-known and commonly believed assumption that P = NP, cannot be approximated with efficient algorithms at all. 1.1 Background Identifying which combinatorial problems are easy to solve and which are hard is an important and challenging task, which has occupied theoretical computer scientists for many years. In order to translate the everyday expression “easy to solve” to mathematical theory the concept of polynomial time algorithms has been introduced. An algorithm is said to run in polynomial time if there is a polynomial p such that the algorithm applied to an input of size n always finds a solution in time p(n), that is after performing p(n) simple instructions. Note that we measure the worst case complexity, that is the time in which we are sure that the algorithm ends regardless of which input of size n we have fed it. The execution time of a polynomial time algorithm grows slowly enough with increasing input size to be able to be run on a computer, but if the execution time grows exponentially the algorithm is useless for all but the smallest inputs. One of the most accepted ways to prove that a problem is hard is to prove it NP-complete. If an optimization problem is NP-complete we are almost certain that it cannot be solved optimally in polynomial time. If we are given an algorithm solving a problem, we can often use it to solve other, similar problems. We can even, given two problems A and B, specify in advance how to use any algorithm for problem B to solve problem A. Such a specification is called a reduction from A to B –wereduce A to B. If the reduction itself is easy to execute, then we have shown that problem B is at least as hard to solve as A, in the sense that if we can solve B we can also solve 1
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: Contents v 7 The travelling salespe
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
Page 59 and 60:
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 77 and 78:
Page 79 and 80:
a c b A B Section 5.7. Maximum H-ma
Page 81 and 82:
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

Create successful ePaper yourself

Delete template?

Save as template?