On the Approximability of NP-complete Optimization Problems

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Akademisk avhandling för teknisk doktorsexamen vid Kungl. Tekniska Högskolan, Stockholm Maj 1992 c○ Viggo Kann 1992 ISBN 91-7170-082-X TRITA-NA-9206 • ISSN 1101-2250 • ISRN KTH/NA–9206 Repro Print AB, Stockholm mcmxcii
Abstract This thesis deals with the polynomial approximability of combinatorial optimization problems which are NP-complete in their decision problem version. Different measures of approximation quality are possible and give rise to different approximation preserving reductions and different approximation classes of problems. We present definitions and characteristics of existing measures, reductions and classes and make comparisons. The main contributions comprise the following four subjects. • We show that the definition of the approximability class Max SNP due to Panconesi and Ranjan differs from the original definition due to Papadimitriou and Yannakakis and that the same problem may or may not be included in the class depending on how it is encoded. A new definition of Max SNP without this limitation is proposed. • We prove that maximum three dimensional matching is Max SNP-hard using a reduction constructing a structure of rings of trees. Using similar techniques we can show that several other matching and packing problems are Max SNP-hard, for example the maximum H-matching problem (the problem of determining the maximum number of node-disjoint copies of a fixed graph H contained in a variable graph). Most of the problems are Max SNP-complete when bounding the degree of the input structure. • Some versions of the maximum common subgraph problem are studied and approximation algorithms are given. The maximum bounded common induced subgraph problem is shown to be Max SNP-hard and the maximum unbounded common induced subgraph problem is shown to be as hard to approximate as the maximum independent set problem. The maximum common induced connected subgraph problem is still harder to approximate and is shown to be NPO PB-complete, i.e. complete in the class of optimization problems with optimum value bounded by a polynomial. • An algorithm which solves the Euclidean travelling salesperson problem in two dimensions optimally is given. The worst case time bound for the algorithm is 2 O(√ n log n) . As an appendix there is a list of NP optimization problems including their definitions, approximation properties and references. Keywords: computational complexity, approximability, NP optimization problems, combinatorial problems, graph problems, optimization, approximation, approximation algorithms, approximation preserving reduction, approximability classes, relative error, performance ratio, Max SNP-completeness, maximum three dimensional matching, maximum H-matching, maximum common subgraph, travelling salesperson problem. i
Page 1: On the Approximability of NP-comple
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 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 4.11. 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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