Integer Programming

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Integer Programming

• ECE 665
• Professor Maciej Ciesielski
• By DFG

2
Outline

• Mathematical programming paradigm
• Linear Programming
• Integer Programming
• Integer Programming
• Example
• Unimodularity LP -gt IP
• Theorem
• Conclusion
• Special Linear Programming with Integer Solutions
• Assignment Problem
• Network Flow Problem
• Review
• Conclusions

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Mathematical Programming Paradigm

• A mathematical program is an optimization
  problem of the form  
• Maximize (or Minimize) f(x)
•   subject to
• g(x) 0
• h(x) 0
•  
• where x x1,...xn is a subset of Rn, the
  functions g and h are called constraints, and f
  is called the objective function.

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Linear Programming (LP)

• The goal of Linear programming is to
• (Max)Minimize CTx
• Subject to Ax b
• x 0
• Where CT is a coefficient vector for f, A is a
  constraint matrix, and b es a constraint vector.
• The constraint set x Ax b is a convex
  polyhedron, and f is linear so it is convex.
• Therefore, LP has convexity, and the local
  min/max is the global min/max

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Integer Programming (IP)

• The goal of Integer programming is to
• (Max)Minimize CTx
• Subject to Ax ? b
• xi integer
• Typically, xi 0,1
• (0,1 Integer Programming)

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IP Example (Matching Problem)

• Given a graph G , find maximal set of edges in
  G, such that no two edges are adjacent to the
  same vertex.
• Maximum matching matching of maximum
  cardinality.
• Weighted matching matching with

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IP Example (Matching Problem)
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IP Example (Matching Problem)

• In matrix form
• Where b 1,1T , A incidence matrix of
  G

1 2 3
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IP Example (Matching Problem)
max 1x1 1x2 1x3
x1 , x2 , x3 0, 1
One solution to this IP problem
x1 1 , x2 0 , x3 0
Other possible solutions
x1 0 , x2 1 , x3 0, or x1 0 , x2 0 ,
x3 1
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Question

• Can the solution to this IP problem be obtained
  by dropping the integrality constraint
• xi 0, 1
• And solving the LP problem instead?
• In our example, solution to the IP is not
  obtainable from LP.
• Reason
• matrix A does not have certain property (total
  unimodularity) needed to guarantee integer
  solutions.

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Question
If xi 0, 1 is relaxed, such that xi ? 0 ,
then the solution to the associated LP problem is
non integer
Reason the A matrix
Is not totally unimodular A - 2
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Unimodularity LP -gt IP
Given a constraint set in standard form where
A, b are integer Partition A B/N x xB,
xN B is nonsingular m x m basis, N is non-basic
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Unimodularity LP -gt IP
Basic solution is
In particular, when B I and B -1 I
then
XB b
a solution can be obtained by inspection (as in
the initial step of Simplex method).
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Unimodularity LP -gt IP
Since xB B 1 b with xN 0, b integer
A sufficient condition for a basic solution xB to
be integer is that
B 1 be an integer matrix
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Unimodularity
A square matrix B is called unimodular if D
det B 1 An integer matrix A is totally
unimodular if every square, nonsingular submatrix
of A is unimodular. Equivalently A is totally
unimodular if every subdeterminant of A is 0, 1,
or 1.
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Unimodularity
Recall that for B nonsingular
Where B , adjoint matrix
i j cofactor of element a i j in det A T

B and det B are integer if B 1 is
integer
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Unimodularity
Cofactor of ai j Determinant obtained by
omitting the ith row and the jth column of A and
then multiplying by (-1) i j .
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Unimodularity
For B unimodular, B 1 integer If A is
totally unimodular, every basis matrix B is
unimodular and every basic solution ( xB , yN)
( B 1 b, 0 ) Is integer. In particular, the
optimal solution is integer
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Theorem 1
If A is totally unimodular then every basic
solution of Ax b is integer.
For LPs with equality constraints total
unimodularity is sufficient but not
necessary. For LP with inequality constraints,
Ax ? b, total unimodularity of A is both
necessary and sufficient for all extreme
points of s x Ax ? b , x ? 0 to be
integer for every integer vector b.
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Conclusion
Any IP with totally unimodular constraint matrix
can be solved as an LP. Totally unimodular matrix
a i j 0 , 1, -1 Also, every determinant
of A must be 0, 1 or -1
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Special Linear Programs with Integer Solutions
Network flow problems

• max flow
• min cost flow
• assignment problem
• shortest path
• transportation problem

are LP with the property that they possess
optimal solutions in integers.
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Some Totally Unimodular Linear Programs
Assignment problem (special case of min cost
capacitated flow problems) m jobs x m men
c i j cost of assigning man i to job j
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Some Totally Unimodular Linear Programs
s. to.
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Some Totally Unimodular Linear Programs
X11 X12 X13 X21 X22 X23 X31
X32 X33
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Some Totally Unimodular Linear Programs
In matrix notation
Ax 1 where
m2
m
2m
...
Exactly m 1s in each row Exactly 2 1s in each
column
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Network Flow Problems
Given

• a set of origins V1 each origin i ? V1
  supplies a1 of commodity.
• a set of destinations V2 each destination j ?
  V2 has a demand bi of commodity.
• cost per unit commodity cij associated with
  sending commodity through (i, j ).

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Network Flow Problems
Constraint Set (Totally unimodular)
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Network Flow Problems
Special case for single source, single
destination, max flow problem
in any case Ax ? b
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Review
No matter which problem it is
the general format is Ax ? b It can be shown
that

• If A is totally unimodular then A/I is also
  totally unimodular
• The transpose of totally unimodular matrix is
  also totally unimodular
• .

where A incidence matrix of the corresponding
digraph, totally unimodular. I identity
matrix, A is totally unimodular.
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Review
No efficient algorithms exist for general Integer
Programming problems

• Exploit a special structure of the problem (total
  unimodularity, etc.) to obtain integer solutions
  by solving simpler problems.
• Transform the problem to another problem for
  which an approximate solution is easier to find.
• Once the structure of the problem is well
  understood, use heuristic, but stay away from
  brute force approach.

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Conclusions
A large number of CAD problems can be cast in
analytical form

• Graph Theory
• Mathematical Optimization

For some problems efficient algorithms
exist. For others need to resort to heuristic,
suboptimal solutions. Some known successful
heuristic approaches

• Simulated annealing
• LP rounding