IEOR 269, Spring 2010 Integer Programming and Combinatorial ...

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IEOR269 notes, Prof. Hochbaum, 2010 iv 37 Transhipment Problem 87 38 Transportation Problem 87 38.1 Production/Inventory Problem as Transportation Problem . . . . . . . . . . . . . . . 88 39 Assignment Problem 90 40 Maximum Flow Problem 90 40.1 A Package Delivery Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 41 Shortest Path Problem 91 41.1 An Equipment Replacement Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 42 Maximum Weight Matching 93 42.1 An Agent Scheduling Problem with Reassignment . . . . . . . . . . . . . . . . . . . 93 43 MCNF Hierarchy 96 44 The maximum/minimum closure problem 96 44.1 A practical example: open-pit mining . . . . . . . . . . . . . . . . . . . . . . . . . . 96 44.2 The maximum closure problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 45 Integer programs with two variables per inequality 101 45.1 Monotone IP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 45.2 Non–monotone IP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 46 Vertex cover problem 104 46.1 Vertex cover on bipartite graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 46.2 Vertex cover on general graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 47 The convex cost closure problem 107 47.1 The threshold theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 47.2 Naive algorithm for solving (ccc) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 47.3 Solving (ccc) in polynomial time using binary search . . . . . . . . . . . . . . . . . . 111 47.4 Solving (ccc) using parametric minimum cut . . . . . . . . . . . . . . . . . . . . . . . 111 48 The s-excess problem 113 48.1 The convex s-excess problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 48.2 Threshold theorem for linear edge weights . . . . . . . . . . . . . . . . . . . . . . . . 114 48.3 Variants / special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 49 Forest Clearing 116 50 Producing memory chips (VLSI layout) 117 51 Independent set problem 117 51.1 Independent Set v.s. Vertex Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 51.2 Independent set on bipartite graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
IEOR269 notes, Prof. Hochbaum, 2010 v 52 Maximum Density Subgraph 119 52.1 Linearizing ratio problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 52.2 Solving the maximum density subgraph problem . . . . . . . . . . . . . . . . . . . . 119 53 Parametric cut/flow problem 120 53.1 The parametric cut/flow problem with convex function (tentative title) . . . . . . . 121 54 Average k-cut problem 122 55 Image segmentation problem 122 55.1 Solving the normalized cut variant problem . . . . . . . . . . . . . . . . . . . . . . . 123 55.2 Solving the λ-question with a minimum cut procedure . . . . . . . . . . . . . . . . . 125 56 Duality of Max-Flow and MCNF Problems 127 56.1 Duality of Max-Flow Problem: Minimum Cut . . . . . . . . . . . . . . . . . . . . . . 127 56.2 Duality of MCNF problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 57 Variant of Normalized Cut Problem 128 57.1 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 57.2 Review of Maximum Closure Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 130 57.3 Review of Maximum s-Excess Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 131 57.4 Relationship between s-excess and maximum closure problems . . . . . . . . . . . . 132 57.5 Solution Approach for Variant of the Normalized Cut problem . . . . . . . . . . . . 132 58 Markov Random Fields 133 59 Examples of 2 vs. 3 in Combinatorial Optimization 134 59.1 Edge Packing vs. Vertex Packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 59.2 Chinese Postman Problem vs. Traveling Salesman Problem . . . . . . . . . . . . . . 135 59.3 2SAT vs. 3SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 59.4 Even Multivertex Cover vs. Vertex Cover . . . . . . . . . . . . . . . . . . . . . . . . 136 59.5 Edge Cover vs. Vertex Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 59.6 IP Feasibility: 2 vs. 3 Variables per Inequality . . . . . . . . . . . . . . . . . . . . . 138 These notes are based on “scribe” notes taken by students attending Professor Hochbaum’s course IEOR 269 in the spring semester of 2010. The current version has been updated and edited by Professor Hochbaum 2010
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