Introduction to Linear and Integer Programming

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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 Rn -gt R
• A set of constraint functions gi Rn -gt R
• A set of constraint values bi

• Goal
• Find x in Rn which
• maximizes f(x)
• satisfies gi(x) lt bi

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Linear Programming

• Input
• A linear objective function f Rn -gt R
• A set of linear constraint functions gi Rn -gt
  R
• A set of constraint values bi

• Goal
• Find x in Rn which
• maximizes f(x)
• satisfies gi(x) lt bi

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Integer Linear Programming

• Input
• A linear objective function f Zn -gt Z
• A set of linear constraint functions gi Zn -gt
  Z
• A set of constraint values bi

• Goal
• Find x in Zn which
• maximizes f(x)
• satisfies gi(x) lt bi

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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 Zn -gt Z
• A set of linear constraint functions gi Zn -gt
  Z
• A set of constraint values bi

• Goal
• Find x in Zn which
• maximizes f(x)
• satisfies gi(x) lt bi

NP-complete!
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Mathematical Programming

• Input
• An objective function f Rn -gt R
• A set of constraint functions gi Rn -gt R
• A set of constraint values bi

• Goal
• Find x in Rn which
• maximizes f(x)
• satisfies gi(x) lt bi

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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Linear Programming Relaxation
Stable matchings
Bipartite matchings
Minimum spanning trees
General matchings
Maximum flows
Shortest paths
Minimum Cost Flows
Submodular Flows
Linear programming
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Maximum Matchings
But not every solution is a matching!!
Every matching is a solution.
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Geometric Interpretation
Goal Optimize over integers!
Linear inequalities as hyperplanes
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
x1
Goal define a polytope which is the
convex hull of matchings.
x1
(0.5,0.5,0.5)
x3
x2
x3
x2
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x1
Matching Polytope
x1
x3
x2
x1
x3
x2
x3
x2
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Valid Inequalities
Inequalities which are satisfied by integer
solutions but kill unwanted fractional solution.
Enough?
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Valid Inequalities
Odd set inequalities
Enough?
Yes, thats 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!