Integer Programming

1
Integer Programming

• Difference from linear programming
• Variables xi must take on integral values, not
  real values
• Lots of interesting problems can be formulated as
  integer programs
• Unfortunately, integer programming is NP-hard
• Even when we restrict to choices 0 and 1 for vars
• Despite this difficulty, formulating a problem as
  an integer program can still be very useful in
  developing a good approximation algorithm for a
  problem

2
Algorithmic Techniques

• Formulate problem as an integer program (IP)
• Relax it to create a linear program (LP)
• Solvable in polynomial time
• The LP solution is a lower (upper) bound on the
  IP solution
• The gap between these is often referred to as
  the integrality gap
• Get an integral solution
• Round solution to create an integer solution with
  some form of performance guarantee
• Argue problem has structure that optimal solution
  is guaranteed to be integral
• Use LP solution as a guide for your algorithm
• Primal-dual methods to gain good approximation
  bounds

3
Formulating Integer Programs

• Indicator variables
• A variable xi often takes values 0 or 1 to encode
  a choice in the final solution structure
• Value 0 means that some element is NOT part of
  the final solution
• Value 1 means that some element IS part of the
  final solution
• See associated worksheet for exercise on
  formulating problems as integer programs

4
Set Cover

• Universe of elements U u1, , un
• Subsets S1, , Sm of U
• Weights (costs) wj 0 for each subset Sj
• Goal Find a collection I from 1, , m that
  minimizes ?j in I wj such that the union of Sj T

5
Approximation Algorithm

• Formulate an integer program IP
• Relax IP to create an LP
• Solve LP to create optimal solution x
• Round solution to create integral solution as
  follows
• If xi gt 1/f then choose the corresponding set
• Define f maximum number of sets that contain
  any item
• Prove
• Rounding algorithm creates set cover
• Rounding algorithm guarantees approx ratio of ?

6
Prove IP is NP-hard

• Reduction from SAT
• Input instance for SAT
• Set of variables xj
• Set of clauses Cj disjunction of literals
• Goal Find a truth assignment T to the variables
  that satisfies at least one literal from each
  clause

7
Details
Any SAT instance has boolean variables and
clauses. Our Integer programming problem will
have twice as many variables, one for each
variable and its complement, as well as the
following inequalities 0 ? vi ? 1 and 0 ?
?vi ? 1 1 ? vi ?vi ? 1 for each clause C
v1, ?v2, ... vi v1 ?v2 vi ? 1