Introduction to Linear and Integer Programming

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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 -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

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
By
Surprisingly, this works for many problems!
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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 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
This provides an efficient way to check whether a
solution is a vertex.
A basic solution is formed by a set B of m
linearly independent columns, so that
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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.
What about general matching?
Write a linear program for the stable matching
problem.
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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
Moving along this direction improves the cost.
There is always one neighbour which improves the
cost.
We are here
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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.
Key show that the volume decreases fast
enough

• 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 ltb for
  which c is violated.
• Find a minimum ellipsoid which contains the
  intersection of
• the previous ellipsoid and ax lt b.
• Continue the process with the new (smaller)
  ellipsoid.

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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