Probabilistic Combinatorial Optimization on Graphs

, by Murat, Cé; cile; Paschos, Vangelis Th.

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ISBN: 9781905209330 | 1905209339
Cover: Hardcover
Copyright: 3/23/2006

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This title provides a comprehensive survey over the subject of probabilistic combinatorial optimization, discussing probabilistic versions of some of the most paradigmatic combinatorial problems on graphs, such as the maximum independent set, the minimum vertex covering, the longest path and the minimum coloring.Those who possess a sound knowledge of the subject mater will find the title of great interest, but those who have only some mathematical familiarity and knowledge about complexity and approximation theory will also find it an accessible and informative read.

Preface

(4)

Chapter 1. A Short Insight into Probabilistic Combinatorial Optimization

(20)

1.1. Motivations and applications

(4)

1.2. A formalism for probabilistic combinatorial optimization

(5)

1.3. The main methodological issues dealing with probabilistic combinatorial optimization

(7)

1.3.1. Complexity issues

(2)

1.3.1.1. Membership in NPO is not always obvious

(1)

1.3.1.2. Complexity of deterministic vs. complexity of probabilistic optimization problems

(2)

1.3.2. Solution issues

(11)

1.3.2.1. Characterization of optimal a priori solutions

(2)

1.3.2.2. Polynomial subcases

(1)

1.3.2.3. Exact solutions and polynomial approximation issues

(2)

1.4. Miscellaneous and bibliographic notes

(4)

FIRST PART. PROBABILISTIC GRAPH-PROBLEMS

(150)

Chapter 2. The Probabilistic Maximum Independent Set

(44)

2.1. The modification strategies and a preliminary result

(5)

2.1.1. Strategy M1

(1)

2.1.2. Strategies M2 and M3

(2)

2.1.3. Strategy M4

(1)

2.1.4. Strategy M5

(1)

2.1.5. A general mathematical formulation for the five functionals

(2)

2.2. PROBABILISTIC MAX INDEPENDENT SET1

(3)

2.2.1. Computing optimal a priori solutions

(1)

2.2.2. Approximating optimal solutions

(1)

2.2.3. Dealing with bipartite graphs

(1)

2.3. PROBABILISTIC MAX INDEPENDENT SET2 and 3

(8)

2.3.1. Expressions for E(G, S, M2) and E(G, S, M3)

(1)

2.3.2. An upper bound for the complexity of E(G, S, M2)

(1)

2.3.3. Bounds for E(G, S, M2)

(2)

2.3.4. Approximating optimal solutions

(2)

2.3.4.1. Using argmax{Σviepsilons pi} as an a priori solution

(2)

2.3.4.2. Using approximations of MAX INDEPENDENT SET

(1)

2.3.5. Dealing with bipartite graphs

(2)

2.4. PROBABILISTIC MAX INDEPENDENT SET4

(3)

2.4.1. An expression for E(G, S, M4)

(1)

2.4.2. Using S* or argmax{Σviepsilons pi} as an a priori solution

(1)

2.4.3. Dealing with bipartite graphs

(1)

2.5. PROBABILISTIC MAX INDEPENDENT SETS

(3)

2.5.1. In general graphs

(2)

2.5.2. In bipartite graphs

(1)

2.6. Summary of the results

(2)

2.7. Methodological questions

(8)

2.7.1. Maximizing a criterion associated with gain

(3)

2.7.1.1. The minimum gain criterion

(1)

2.7.1.2. The maximum gain criterion

(2)

2.7.2. Minimizing a criterion associated with regret

(2)

2.7.2.1. The maximum regret criterion

(2)

2.7.3. Optimizing expectation

(1)

2.8. Proofs of the results

(10)

2.8.1. Proof of Proposition 2.1

(3)

2.8.2. Proof of Theorem 2.6

(3)

2.8.3. Proof of Proposition 2.3

(1)

2.8.4. Proof of Theorem 2.13

(3)

Chapter 3. The Probabilistic Minimum Vertex Cover

(18)

3.1. The strategies M1, M2 and M3 and a general preliminary result

(2)

3.1.1. Specification of M1, M2 and M3

(2)

3.1.1.1. Strategy M1

(1)

3.1.1.2. Strategy M2

(1)

3.1.1.3. Strategy M3

(1)

3.1.2. A first expression for the functionals

(1)

3.2. PROBABILISTIC MIN VERTEX COVER1

(2)

3.3. PROBABILISTIC MIN VERTEX COVER2

(1)

3.4. PROBABILISTIC MIN VERTEX COVER3

(2)

3.4.1. Building E(G, C, M3)

(1)

3.4.2. Bounds for E(G, C, M3)

(1)

3.5. Some methodological questions

(2)

3.6. Proofs of the results

(8)

3.6.1. Proof of Theorem 3.3

(2)

3.6.2. On the the bounds obtained in Theorem 3.3

(6)

Chapter 4. The Probabilistic Longest Path

(26)

4.1. Probabilistic longest path in terms of vertices

100

(2)

4.2. Probabilistic longest path in terms of arcs

102

(7)

4.2.1. An interesting algebraic expression

104

(1)

4.2.2. Metric PROBABILISTIC ARC WEIGHTED LONGEST PATH

105

(4)

4.3. Why the strategies used are pertinent

109

(1)

4.4. Proofs of the results

110

(15)

4.4.1. Proof of Theorem 4.1

110

(2)

4.4.2. Proof of Theorem 4.2

112

(2)

4.4.3. An algebraic proof for Theorem 4.3

114

(2)

4.4.4. Proof of Lemma 4.1

116

(1)

4.4.5. Proof of Lemma 4.2

117

(1)

4.4.6. Proof of Theorem 4.4

117

(8)

Chapter 5. Probabilistic Minimum Coloring

125

(60)

5.1. The functional E(G,C)

127

(4)

5.2. Basic properties of probabilistic coloring

131

(1)

5.2.1. Properties under non-identical vertex-probabilities

131

(1)

5.2.2. Properties under identical vertex-probabilities

131

(1)

5.3. PROBABILISTIC MIN COLORING in general graphs

132

(7)

5.3.1. The complexity of probabilistic coloring

132

(1)

5.3.2. Approximation

132

(7)

5.3.2.1. The main result

132

(5)

5.3.2.2. Further approximation results

137

(2)

5.4. PROBABILISTIC MIN COLORING in bipartite graphs

139

(16)

5.4.1. A basic property

139

(2)

5.4.2. General bipartite graphs

141

(6)

5.4.3. Bipartite complements of bipartite matchings

147

(4)

5.4.4. Trees

151

(3)

5.4.5. Cycles

154

(1)

5.5. Complements of bipartite graphs

155

(1)

5.6. Split graphs

156

(8)

5.6.1. The complexity of PROBABILISTIC MIN COLORING

156

(3)

5.6.2. Approximation results

159

(5)

5.7. Determining the best k-coloring in k-colorable graphs

164

(11)

5.7.1. Bipartite graphs

164

(7)

5.7.1.1. PROBABILISTIC MIN 3-COLORING

164

(4)

5.7.1.2. PROBABILISTIC MIN k-COLORING for k > 3

168

(3)

5.7.1.3. Bipartite complements of bipartite matchings

171

(1)

5.7.2. The complements of bipartite graphs

171

(3)

5.7.3. Approximation in particular classes of graphs

174

(1)

5.8. Comments and open problems

175

(3)

5.9. Proofs of the different results

178

(9)

5.9.1. Proof of [5.5]

178

(1)

5.9.2. Proof of [5.4]

179

(1)

5.9.3. Proof of Property 5.1

180

(1)

5.9.4. Proof of Proposition 5.2

181

(2)

5.9.5. Proof of Lemma 5.11

183

(2)

SECOND PART. STRUCTURAL RESULTS

185

(54)

Chapter 6. Classification of Probabilistic Graph-problems

187

(24)

6.1. When MS is feasible

187

(11)

6.1.1. The a priori solution is a subset of the initial vertex-set

188

(3)

6.1.2. The a priori solution is a collection of subsets of the initial vertex-set

191

(2)

6.1.3. The a priori solution is a subset of the initial edge-set

193

(5)

6.2. When application of MS itself does not lead to feasible solutions

198

(7)

6.2.1. The functional associated with MSC

198

(1)

6.2.2. Applications

199

(15)

6.2.2.1. The a priori solution is a cycle

200

(1)

6.2.2.2. The a priori solution is a tree

201

(4)

6.3. Some comments

205

(1)

6.4. Proof of Theorem 6.4

206

(5)

Chapter 7. A Compendium of Probabilistic NPO Problems on Graphs

211

(28)

7.1. Covering and partitioning

214

(10)

7.1.1. MIN VERTEX COVER

214

(1)

7.1.2. MIN COLORING

214

(1)

7.1.3. MAX ACHROMATIC NUMBER

215

(1)

7.1.4. MIN DOMINATING SET

215

(1)

7.1.5. MAX DOMATIC PARTITION

216

(1)

7.1.6. MIN EDGE-DOMINATING SET

216

(1)

7.1.7. MIN INDEPENDENT DOMINATING SET

217

(1)

7.1.8. MIN CHROMATIC SUM

217

(1)

7.1.9. MIN EDGE COLORING

218

(1)

7.1.10. MIN FEEDBACK VERTEX-SET

219

(1)

7.1.11. MIN FEEDBACK ARC-SET

220

(1)

7.1.12. MAX MATCHING

220

(1)

7.1.13. MIN MAXIMAL MATCHING

220

(1)

7.1.14. MAX TRIANGLE PACKING

220

(1)

7.1.15. MAX H-MATCHING

221

(1)

7.1.16. MIN PARTITION INTO CLIQUES

222

(1)

7.1.17. MIN CLIQUE COVER

222

(1)

7.1.18. MIN k-CAPACITED TREE PARTITION

222

(1)

7.1.19. MAX BALANCED CONNECTED PARTITION

223

(1)

7.1.20. MIN COMPLETE BIPARTITE SUBGRAPH COVER

223

(1)

7.1.21. MIN VERTEX-DISJOINT CYCLE COVER

223

(1)

7.1.22. MIN CUT COVER

224

(1)

7.2. Subgraphs and supergraphs

224

(5)

7.2.1. MAX INDEPENDENT SET

224

(1)

7.2.2. MAX CLIQUE

224

(1)

7.2.3. MAX INDEPENDENT SEQUENCE

225

(1)

7.2.4. MAX INDUCED SUBGRAPH WITH PROPERTY π

225

(1)

7.2.5. MIN VERTEX DELETION TO OBTAIN SUBGRAPH WITH PROPERTY π

225

(1)

7.2.6. MIN EDGE DELETION TO OBTAIN SUBGRAPH WITH PROPERTY π

226

(1)

7.2.7. MAX CONNECTED SUBGRAPH WITH PROPERTY π

226

(1)

7.2.8. MIN VERTEX DELETION TO OBTAIN CONNECTED SUBGRAPH WITH PROPERTY π

226

(1)

7.2.9. MAX DEGREE-BOUNDED CONNECTED SUBGRAPH

226

(1)

7.2.10. MAX PLANAR SUBGRAPH

227

(1)

7.2.11. MIN EDGE DELETION k-PARTITION

227

(1)

7.2.12. MAX k-COLORABLE SUBGRAPH

227

(1)

7.2.13. MAX SUBFOREST

228

(1)

7.2.14. MAX EDGE SUBGRAPH or DENSE k-SUBGRAPH

228

(1)

7.2.15. MIN EDGE K-SPANNER

228

(1)

7.2.16. MAX k-COLORABLE INDUCED SUBGRAPH

229

(1)

7.2.17. MIN EQUIVALENT DIGRAPH

229

(1)

7.2.18. MIN CHORDAL GRAPH COMPLETION

229

(1)

7.3. Iso- and other morphisms

229

(2)

7.3.1. MAX COMMON SUBGRAPH

229

(1)

7.3.2. MAX COMMON INDUCED SUBGRAPH

230

(1)

7.3.3. MAX COMMON EMBEDDED SUBTREE

230

(1)

7.3.4. MIN GRAPH TRANSFORMATION

230

(1)

7.4. Cuts and connectivity

231

(8)

7.4.1. MAX CUT

231

(1)

7.4.2. MAX DIRECTED CUT

231

(1)

7.4.3. MIN CROSSING NUMBER

231

(1)

7.4.4. MAX k-CUT

232

(1)

7.4.5. MIN k-CUT

233

(1)

7.4.6. MIN NETWORK INHIBITION ON PLANAR GRAPHS

233

(1)

7.4.7. MIN VERTEX k-CUT

234

(1)

7.4.8. MIN MULTI-WAY CUT

234

(1)

7.4.9. MIN MULTI-CUT

234

(1)

7.4.10. MIN RATIO-CUT

235

(1)

7.4.11. MIN b-BALANCED CUT

236

(1)

7.4.12. MIN b-VERTEX SEPARATOR

236

(1)

7.4.13. MIN QUOTIENT CUT

236

(1)

7.4.14. MIN k-VERTEX CONNECTED SUBGRAPH

236

(1)

7.4.15. MIN k-EDGE CONNECTED SUBGRAPH

237

(1)

7.4.16. MIN BICONNECTIVITY AUGMENTATION

237

(1)

7.4.17. MIN STRONG CONNECTIVITY AUGMENTATION

237

(1)

7.4.18. MIN BOUNDED DIAMETER AUGMENTATION

237

(2)

Appendix A. Mathematical Preliminaries

239

(10)

A.1. Sets, relations and functions

239

(3)

A.2. Basic concepts from graph-theory

242

(4)

A.3. Elements from discrete probabilities

246

(3)

Appendix B. Elements of the Complexity and the Approximation Theory

249

(6)

B.1. Problem, algorithm, complexity

249

(1)

B.2. Some notorious complexity classes

250

(1)

B.3. Reductions and NP-completeness

251

(1)

B.4. Approximation of NP-hard problems

252

(3)

Bibliography

255

(6)

Index

261

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Probabilistic Combinatorial Optimization on Graphs

Summary

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Table of Contents

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Probabilistic Combinatorial Optimization on Graphs

Summary

Author Biography

Table of Contents

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