Stability and Robustness: Reliability in the World of Uncertainty

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The Central Tree ProblemThe Central TreeProblemHeuristics for theCTPFinal RemarksRecall that the cospanning tree ¯T of a spanning tree T is theedge complement of T in G, i.e. ¯T = (V, E ¯T), E ¯T = E\E T .Also, if a graph G has n vertices and c connected components,then the rank ρ(G) of graph G is n − c.Definition. [Deo, 1966]The Central Tree Problem (shortly CTP) consists in finding centraltree T ∗ such that the rank ρ( ¯T ∗ ) of cospanning tree ¯T ∗ isminimal, i.e.ρ( ¯T ∗ ) ≤ ρ( ¯T) ∀ T ∈ T G .Thus, a central tree is a spanning tree whose edge removalleads to splitting graph G into the maximum number ofcomponents.Heuristics for the Central Tree Problem Yury Nikulin - p. 4/33
The CTP vs. MDTPThe Central TreeProblemThe Maximally Distant Tree Problem (MDTP) consists infinding a pair of extremal trees with the largest distance.Heuristics for theCTPFinal RemarksHeuristics for the Central Tree Problem Yury Nikulin - p. 5/33
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OutlineThe World of UncertaintyRobu
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Interval UncertaintyOutlineThe Worl
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ExampleOutlineThe World of Uncertai
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Page 57 and 58: OutlineThe World of UncertaintyRobu
Page 59 and 60: Introduction: Sensitivity analysis
Page 61 and 62: Dealing with uncertainty16/04/2009
Page 63 and 64: Classical approachKouvelis P. and Y
Page 65 and 66: Telecommunication and MST16/04/2009
Page 67 and 68: Multicommodity reformulation formin
Page 69 and 70: Dual problem16/04/2009 Yury Nikulin
Page 71 and 72: Interval data and scenariosG = ( E,
Page 73 and 74: RST problem reformulation as mixedi
Page 75 and 76: Robust Spanning Tree Problem within
Page 77 and 78: Example: weak edgesweaknon-weak12-5
Page 79 and 80: Example: Strong edgesstrong12-553-4
Page 81 and 82: CPLEX results on complete graphs16/
Page 83 and 84: Branching strategyNode d with the s
Page 85 and 86: Lower bound16/04/2009 Yury Nikulin,
Page 87 and 88: SA: preprocessing, initial solution
Page 89 and 90: SA: neighborhood search moveCase2.1
Page 91 and 92: Running time16/04/2009 Yury Nikulin
Page 93 and 94: SA summary• In spite of SA is gen
Page 95 and 96: 16/04/2009 Yury Nikulin, Sensitivit
Page 97 and 98: Heuristics for Central Tree Problem
Page 99 and 100: Distance MeasureThe Central TreePro
Page 101 and 102: Distance MeasureThe Central TreePro
Page 103: The Central Tree ProblemThe Central
Page 107 and 108: The CTP vs. MDTPThe Central TreePro
Page 109 and 110: Short ExampleThe Central TreeProble
Page 111 and 112: Complexity of the CTPThe Central Tr
Page 113 and 114: Complexity of the CTPThe Central Tr
Page 115 and 116: Equivalence between 0 − 1 RSTP an
Page 117 and 118: One more evidence of NP-hardnessThe
Page 119 and 120: MILP FormulationCorollary from the
Page 121 and 122: Approximation for the CTPThe Centra
Page 123 and 124: The Central TreeProblemHeuristics f
Page 125 and 126: Greedy Construction HeuristicThe Ce
Page 127 and 128: ExampleThe Central TreeProblemHeuri
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Page 145 and 146: ConclusionThe Central TreeProblemHe
Page 147 and 148: Sensitivity Analysis in Game Theory
Page 149 and 150: Game descriptionProblemFormulationS
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Page 153 and 154: EquilibriaProblemFormulationStabili
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Original problemProblemFormulationS
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ExampleProblemFormulationStability
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Stability FunctionProblemFormulatio
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SufficiencyProblemFormulationStabil
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Formula for Stability FunctionProbl
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Sketch of the proofProblemFormulati
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Formula for Stability RadiusProblem
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Independent players, no bansProblem
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Accuracy Function and RadiusProblem
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ExampleProblemFormulationStability
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Concluding remarksProblemFormulatio
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Fuzziness: one more way of dealingw
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Crisp and Fuzzy Numbers28/04/2009 Y
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Geometrical interpretation: Union28
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Geometrical interpretation:Intersec
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Fuzzy sets28/04/2009 Yury Nikulin,
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Comparison of fuzzy numbers:strong
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Maximal and Minimalfuzzy numbersμ(
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Fuzzy maximum and minimum28/04/2009
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Statistical data about flight delay
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Possibility distribution28/04/2009
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Fuzzy Number representing delays28/
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Stability and Robustness: Reliability in the World of Uncertainty

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