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Introduction to Linear and Integer

Programming

Lecture 9: Feb 14

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Mathematical Programming

Input:

• An objective function f: Rn -> R

• A set of constraint functions: gi: Rn -> R

• A set of constraint values: bi

Goal:

Find x in Rn which:

1. maximizes f(x)

2. satisfies gi(x) <= bi

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Linear Programming

Input:

• A linear objective function f: Rn -> R

• A set of linear constraint functions: gi: Rn -> R

• A set of constraint values: bi

Goal:

Find x in Rn which:

1. maximizes f(x)

2. satisfies gi(x) <= bi

Integer linear program

requires the solution

to be in Zn

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Perfect Matching

(degree constraints)

Every solution is a perfect matching!

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Maximum Satisfiability

Goal: Find a truth assignment to satisfy all clauses

NP-complete!

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Different Forms

canonical form standard form

The general form (with equalities, unconstrained variables)

can be reduced to these forms.

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Linear Programming Relaxation

Replace

ByBy

Surprisingly, this works for many problems!

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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 “corner” could be the unique optimal

solution for some objective function.

So, we need every “corner” to be integral!

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Vertex Solutions

This says we can restrict our attention to vertex solutions.

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Basic Solutions

A basic solution is formed by a set B of m linearlyindependent columns, so that

This provides an efficient way to check whether a solution is a vertex.

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Basic Solutions

Tight inequalities: inequalities achieved as equalities

Basic solution:unique solution of n linearly independent tight inequalities

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Questions

Prove that the LP for perfect matching

is integral for bipartite graphs.

Write a linear program for the stable matching problem.

What about general matching?

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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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Simplex Method

Simplex method:

A simple and effective approach to

solve linear programs in practice.

It has a nice geometric interpretation.

Idea: Focus only on vertex solutions,

since no matter what is the objective function,

there is always a vertex which attains optimality.

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Simplex Method

Simplex Algorithm:

• Start from an arbitrary vertex.

• Move to one of its neighbours

which improves the cost. Iterate.

Key: local minimum = global minimum

Global minimum

We are here

Moving along this direction improves the cost. There is always one neighbour

which improves the cost.

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Simplex Method

Simplex Algorithm:

• Start from an arbitrary vertex.

• Move to one of its neighbours which improves the cost. Iterate.

Which one?

There are many different rules to choose a neighbour,

but so far every rule has a counterexample so that

it takes exponential time to reach an optimum vertex.

MAJOR OPEN PROBLEM: Is there a polynomial time simplex algorithm?

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Ellipsoid Method

Goal: Given a bounded convex set P, find a point x in P.

Ellipsoid Algorithm:

Start with a big ellipsoid which contains P.

Test if the center c is inside P.

If not, there is a linear inequality ax <=b for which c is violated.

Find a minimum ellipsoid which contains the intersection of

the previous ellipsoid and ax <= b.

Continue the process with the new (smaller) ellipsoid.

Key: show that the volume decreases fast enough

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Ellipsoid Method

Goal: Given a bounded convex set P, find a point x in P.

Why it is enough to test if P contains a point?

Because optimization problem can be reduced to this testing problem.

Do binary search until we find an “almost” optimal solution.

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Ellipsoid Method

Important property:

We just need to know the previous

ellipsoid and a violated inequality.

This can help to solve some exponential size LP if we have a separation oracle.

Separation orcale: given a point x, decide in polynomial time

whether x is in P or output a violating inequality.

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Looking Forward

• Prove that many combinatorial problems have an integral LP.

• Study LP duality and its applications

• Study primal dual algorithms