Cutting Plane Technique for Solving Integer Programs

1
Cutting Plane Technique for Solving Integer
Programs

2
Motivating Example for Cutting Planes

• Recall the bad-case example for the LP-rounding
  algorithm
• Integer Program LP relaxation
• max x1 5x2 max x1 5x2
• s.t. x1 10x2 ? 20 s.t. x1 10x2 ? 20
• x1 ? 2 x1 ? 2
• x1 , x2 0 integer x1 , x2 0
• Solution to LP-relaxation (2, 1.8)
• Rounded IP solution
• (2, 1) with value 7
• IP optimal solution
• (0, 2) with value 10
• Conclusion Rounded solution too far
• from optimal solution

x1 2
x1 10x2 20
Z11
3

• How can we improve the performance of the
  LP-rounding?
• Add the following new constraint to the problem
  x1 2x2 ? 4 .
• New Integer Program New LP relaxation
• max x1 5x2 max x1 5x2
• s.t. x1 10x2 ? 20 s.t. x1 10x2 ? 20
• x1 ? 2 x1 ? 2
• x1 2x2 ? 4 x1 2x2 ? 4
• x1 , x2 0 integer x1 , x2 0
• The set of feasible integer points
• is the same for the old and new IPs
• But the feasible region of
• the new LP-relaxation is different
• some of the fractional points are cut off
• As a result, the optimal solution of
• the new LP-relaxation, (0,2)
• is also the optimal IP solution.

x1 2
(0, 2)
x1 10x2 20
Z10
x1 2x2 4
4
General Idea of Cutting Plane Technique

• Add new constraints (cutting planes) to the
  problem such that
• (i) the set of feasible integer solutions
  remains the same, i.e., we still have the same
  integer program.
• (ii) the new constraints cut off some of the
  fractional solutions making the feasible region
  of the LP-relaxation smaller.
• Smaller feasible region might result in a better
  LP value (i.e., closer to the IP value), thus
  making the search for the optimal IP solution
  more efficient.
• Each integer program might have many different
  formulations.
• Important modeling skill
• Give as tight formulation as possible.
• How? Find cutting planes that make the
  formulation of the original IP tighter.

5
Example of making a formulation tighter Bin
Packing Problem

• Given n items with sizes s1, s2, , sn
• bins with size W (where W si , any
  i1,,n).
• Goal Pack the items into the bins
• using as few bins as possible.
• Example
• n13 items with sizes
• 20, 20, 20, 20, 20, 81, 81, 81, 81, 82, 91, 49,
  51
• Bin size is W100.
• Minimum number of bins needed is 8.

6
Example of making a formulation tighter Bin
Packing Problem

• Want an IP formulation for this problem.
• Let M be an upper bound on the number of bins
  needed.
• (Mn is a safe upper bound
• but should try for smaller values)
• Define the following variables.
• For j1,,M, let
• For each i1,,n and j1,,M, let

7
Example of making a formulation tighter Bin
Packing Problem

• Our objective is to minimize the number of used
  bins
• Minimize sumj in 1..Mopenj
• We need the following functional constraints.
• Each item should be packed in exactly one bin
• (C1) sumj in 1..Massigni,j 1 , for each
  i1,,n
• Each bin can contain items of total size at most
  W
• (C2) sumi in 1..nsiassigni,j ? W , for
  each j1,,M
• Items can be packed only in open bins
• (C3) assigni,j ? openj , for each i1,,n
  and j1,,M
• Set constraints All variables are binary.

8
Example of making a formulation tighter Bin
Packing Problem

• The optimal solution to the LP relaxation
• openj 1/M , assigni,j 1/M ,
• for each i1,,n and j1,,M
• with optimal value M 1/M 1 .
• Lets check that it really satisfies the
  constraints
• (C1) sumj in 1..Massigni,j 1 , for each
  i1,,n
• For this solution, M 1/M 1 .
• (C2) sumi in 1..nsiassigni,j ? W , for
  each j1,,M
• For this solution, sumi in 1..n si / M ? W
  .
• (C3) assigni,j ? openj , for each i1,,n
  and j1,,M
• For this solution, 1/M ? 1/M .

9
Example of making a formulation tighter Bin
Packing Problem

• The optimal solution with value 1
• might be too far from the optimal IP solution.
• E.g., recall that we needed 8 bins for our
  example with 13 items.
• Thus, the bound given by the LP-relaxation is
  too loose.
• How to make the IP formulation tighter?
• Replace constraints (C2) with the following
  constraints
• (C2) sumi in 1..nsiassigni,j ?
  Wopenj ,
• for each j1,,M
• Note that these constraints are valid for the
  integer program
• (i.e., no feasible integer point is cut off).
• But it cuts off some of the fractional points,
• particularly the optimal solution of the old
  LP-relaxation.
• The optimal solution of the new LP-relaxation
  has value 6.97 for our example.
• This is a much tighter lower bound for the
  optimal IP value 8.

10
Methods of getting Cutting Planes

• Exploit the special structure of the problem
• to get cutting planes
• Often can be hard to get
• Topic of intensive research
• More general methods are also available
• Can be used automatically for many problems
• Often so-called branch-and-cut algorithms
• (some combination of branch-and-bound and
  cutting planes)
• are used to solve integer programs.
• More examples in the next handout