Combinatorial Optimization

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Combinatorial Optimization

• Chapter 13
• Integer Linear Programming
• Suggestion try to study this Chapter from the
  slides refer to the book for more details only
  if needed

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Integer Linear Programming

• min c x
• s.t. Ax b
• x 0
• x integer

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Polyhedral representation
Problem Vertices of the polytope are not
necessariliy integral points
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Polyhedral representation (2)
Solution choosing nearest integer point May be
infeasible
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Polyhedral representation (3)
In fact the whole problem might be infeasible
assessing feasibility is already a difficult task
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Other reasons against rounding

• Sometimes variables are tricks to make the
  model be correct
• Sometimes variables naturally express integral
  entities

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Example 13.1 Modelling TSP

• min Si,j i?j cij xij
• s.t. Sj xij 1 for all i
• Si xij 1 for all j
• 0 x 1
• Si e S je,V\S xij 1 for every S subset
  of V
• xij integer

Exponential number of constraints
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Example 13.2 (cont.)

• min Si,j i?j cij xij
• s.t. Sj cij xij 1 for all i
• Si cij xij 1 for all j
• 0 x 1
• ui uj nxij n-1
• xij integer
• ui 0 for all i

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Correctness

• Let p be a solution to TSP, that is a cyclical
  permutation of the cities, then we let ui t if
  p(t)i. Now, consider the t-th arc in the tour,
  then we set xij 1, uit-1 and ujt. It then
  holds that ui uj nxij -1 n n-1 as
  required. Moreover, if xij 0, the equation is
  always satisfied by the ui variables. Now we show
  that every feasible solution is a tour.
• Suppose not, left as an exercise...

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Example 13.2 Satisfiability

• Boolean variables x1, x2,...,xn.
• Literal ?i xi, or xi ( where xi true if
  xi is false and vice versa).
• Disjunctive Clause Cj (?i or ?k or ?l ... ?z)
  (or )
• Conjunction of Clauses (Cj and Cp and ... and
  Cm)
• (and )
• Satisfiability Given a boolean expression which
  is a conjunction of disjunctive clauses, is there
  a thruth assignment for the literatal which makes
  the boolean expression true?

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ILP for Satisfiability

• SAT (x1 x2 x3 ) (x1 x2 ) (x2 x3)
  (x3 x1) ( x1 x2 x3).
• ?
• ILP
• x1 x2 x3 1
• x1 (1- x2) 1
• x2 (1-x3) 1
• (1-x1) x3 1
• (1-x1) (1-x2) (1-x3) 1
• 0 x1 x2 x3 1
• x1 x2 x3 integer

SAT has truth assignment if ILP has a feasible
solution!
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Total Unimodularity

• Definition 13.1 A square integer Matrix B is
  called unimodular (UM) if its determinant det(B)
  1 or det(B) -1. An integer Matrix A is called
  totally unimodular (TUM) if every square
  nonsingular submatrix of A is UM.

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LP in standard form and TUM

• Let polytope R1(A) be defined as
• Ax b
• x 0
• Theorem 13.1 If A is TUM, then all vertices of
  R1(A) are integer for any integer vector b.
• Skip the proof!

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LP in canonical form and TUM

• Let polytope R2(A) be defined as
• Ax b
• x 0
• Theorem 13.2 If A is TUM, then all vertices of
  R2(A) are integer for any integer vector b.
• Skip the proof!

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Applications of TUM

• Theorem 13.3 An integer matrix A with
• aij -1,0,1 is TUM if no more than two nonzero
  entries appear in any column, and if the rows of
  A can be partitioned into two sets I1 and I2 such
  that
• If a column has two entries of the same sign,
  their rows are in different sets
• If a column has two entries of different signs,
  their rows are in the same set.

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Rest of chapter

• Skip 13.3

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Exercises

• Slide 9.
• Show that if there exists a polynomial algorithm
  for ILP, then there exists a polynomial algorithm
  for SAT.
• Is the LP for shortest path TUM? Motivate your
  answer
• Same as under 3 for Max Flow?