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

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IEOR269 notes, Prof. Hochbaum, 2010 ii 10 General Matching Problem 14 10.1 Maximum Matching Problem in Bipartite Graphs . . . . . . . . . . . . . . . . . . . . 14 10.2 Maximum Matching Problem in Non-Bipartite Graphs . . . . . . . . . . . . . . . . . 15 10.3 Constraint Matrix Analysis for Matching Problems . . . . . . . . . . . . . . . . . . . 16 11 Traveling Salesperson Problem (TSP) 17 11.1 IP Formulation for TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 12 Discussion of LP-Formulation for MST 18 13 Branch-and-Bound 20 13.1 The Branch-and-Bound technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 13.2 Other Branch-and-Bound techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 14 Basic graph definitions 23 15 Complexity analysis 24 15.1 Measuring quality of an algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 15.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 15.2 Growth of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 15.3 Definitions for asymptotic comparisons of functions . . . . . . . . . . . . . . . . . . . 26 15.4 Properties of asymptotic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 15.5 Caveats of complexity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 16 Complexity classes and NP-completeness 28 16.1 Search vs. Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 16.2 The class NP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 17 Addendum on Branch and Bound 30 18 Complexity classes and NP-completeness 31 18.1 Search vs. Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 18.2 The class NP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 18.2.1 Some Problems in NP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 18.3 The class co-NP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 18.3.1 Some Problems in co-NP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 18.4 NPand co-NP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 18.5 NP-completeness and reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 18.5.1 Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 18.5.2 NP-Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 19 The Chinese Checkerboard Problem and a First Look at Cutting Planes 40 19.1 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 19.2 The First Integer Programming Formulation . . . . . . . . . . . . . . . . . . . . . . . 40 19.3 An Improved ILP Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 20 Cutting Planes 43 20.1 Chinese checkers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 20.2 Branch and Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
IEOR269 notes, Prof. Hochbaum, 2010 iii 21 The geometry of Integer Programs and Linear Programs 45 21.1 Cutting planes for the Knapsack problem . . . . . . . . . . . . . . . . . . . . . . . . 45 21.2 Cutting plane approach for the TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 22 Gomory cuts 46 23 Generating a Feasible Solution for TSP 47 24 Diophantine Equations 48 24.1 Hermite Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 25 Optimization with linear set of equality constraints 51 26 Balas’s additive algorithm 51 27 Held and Karp’s algorithm for TSP 54 28 Lagrangian Relaxation 55 29 Lagrangian Relaxation 59 30 Complexity of Nonlinear Optimization 61 30.1 Input for a Polynomial Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 30.2 Table Look-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 30.3 Examples of Non-linear Optimization Problems . . . . . . . . . . . . . . . . . . . . . 62 30.4 Impossibility of strongly polynomial algorithms for nonlinear (non-quadratic) optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 31 The General Network Flow Problem Setup 65 32 Shortest Path Problem 68 33 Maximum Flow Problem 69 33.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 33.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 33.2.1 Ford-Fulkerson algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 33.2.2 Capacity scaling algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 33.3 Maximum-flow versus Minimum Cost Network Flow . . . . . . . . . . . . . . . . . . 75 33.4 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 34 Minimum Cut Problem 77 34.1 Minimum s-t Cut Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 34.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 35 Selection Problem 79 36 A Production/Distribution Network: MCNF Formulation 81 36.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 36.2 Formulation as a Minimum Cost Network Flow Problem . . . . . . . . . . . . . . . . 83 36.3 The Optimal Solution to the Chairs Problem . . . . . . . . . . . . . . . . . . . . . . 84
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