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Introduction to Linear and Integer Programming Lecture 7: Feb 1
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Overview 1.Mathematical programming 2.Linear and integer programming 3.Examples 4.Geometric interpretation 5.Agenda
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Mathematical Programming Input: An objective function f: R n -> R A set of constraint functions: g i : R n -> R A set of constraint values: b i Goal: Find x in R n which: 1. maximizes f(x) 2. satisfies g i (x) <= b i
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Linear Programming Input: A linear objective function f: R n -> R A set of linear constraint functions: g i : R n -> R A set of constraint values: b i Goal: Find x in R n which: 1. maximizes f(x) 2. satisfies g i (x) <= b i
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Integer Linear Programming Input: A linear objective function f: Z n -> Z A set of linear constraint functions: g i : Z n -> Z A set of constraint values: b i Goal: Find x in Z n which: 1. maximizes f(x) 2. satisfies g i (x) <= b i
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Maximum Matchings (degree constraints) Every solution is a matching!
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Maximum Satisfiability Goal: Find a truth assignment to satisfy all clauses NP-complete!
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Integer Linear Programming Input: A linear objective function f: Z n -> Z A set of linear constraint functions: g i : Z n -> Z A set of constraint values: b i Goal: Find x in Z n which: 1. maximizes f(x) 2. satisfies g i (x) <= b i NP-complete!
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Mathematical Programming Input: An objective function f: R n -> R A set of constraint functions: g i : R n -> R A set of constraint values: b i Goal: Find x in R n which: 1. maximizes f(x) 2. satisfies g i (x) <= b i NP-complete!
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Algorithms for Linear Programming (Dantzig 1951) Simplex method Very efficient in practice Exponential time in worst case (Khachiyan 1979) Ellipsoid method Not efficient in practice Polynomial time in worst case
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Linear Programming Relaxation Replace By Surprisingly, this works for many problems!
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Bipartite matchings General matchings Maximum flows Stable matchings Shortest paths Minimum spanning trees Minimum Cost Flows Linear programming Submodular Flows Linear Programming Relaxation
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Maximum Matchings Every matching is a solution. But not every solution is a matching!!
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Geometric Interpretation Linear inequalities as hyperplanes Goal: Optimize over integers! Objective function is also a hyperplane Not a good relaxation!
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Good Relaxation Every vertex could be the unique optimal solution for some objective function. So, we need every vertex to be integral!
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Matching Polytope Goal: define a polytope which is the convex hull of matchings. x1 x3x2 x1 x2 x3 (0.5,0.5,0.5)
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x1 x3x2 x1 x2 x3 Matching Polytope x1 x2 x3
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Valid Inequalities Enough? Inequalities which are satisfied by integer solutions but kill unwanted fractional solution.
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Valid Inequalities Enough? Odd set inequalities Yes, that’s enough. [Edmonds 1965]
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Exponentially Many Inequalities Can take care by the ellipsoid method. Just need a separation oracle, which determines whether a solution is feasible. If not, find a violating inequality. How to construct a separation oracle for matchings?
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Questions Prove that bipartite matching polytope is defined by the degree constraints. Try to confirm what Edmonds said. Write a linear program for the stable matching problem.
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Plan How to prove integrality? Convex combination Totally unimodular matrix Iterative rounding Randomized rounding Totally dual integrality Uncrossing technique
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Plan How to prove min-max theorems? Duality theorem How to solve linear programs? Simplex method Ellipsoid method Primal-dual method Can see combinatorial algorithms!
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