Branch and Bound

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Branch and Bound

• Similar to backtracking in generating a search
  tree and looking for one or more solutions
• Different in that the objective is constrained
  to maximization or minimization of a given
  function
• We will consider minimization problems in this
  section.
• A bound (for the maximum or the minimum) is
  calculated for each node x in the tree
• If the bound is worse than a previous bound, the
  subtree rooted at x is blocked

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Branch and Bound

• In order for Branch and Bound to work, the cost
  function Cost must satisfy the following
  condition on all partial solutions (x1, x2, ...,
  xk1) and their extensions (x1, x2, ..., xk)
• Cost (x1, x2, ..., xk1) ? Cost (x1, x2, ..., xk)
• Therefore, a partial solution can be discarded if
  its cost is greater than or equal to a previously
  computed solution

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Branch and Bound Example

• Traveling Salesman Problem (TSP) Given a set of
  cities and a cost function that is defined on
  each pair of cities, find a tour of minimum cost
• The cost function may be .........................
  ..........
• An instance of TSP is given by a matrix of
  non-negative values
• A lower bound y is associated such that the cost
  of any complete tour that visits cities x1, x2,
  ..., xk , in this order, must be at least y.

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Branch and Bound Example

• Important Notes
• Each complete tour must contain exactly one edge
  and its associated cost from each row and each
  column of the cost matrix.
• If a constant r is subtracted from every entry in
  a row or a column of the cost matrix, the cost of
  any tour under the new matrix is exactly r less
  than the cost of the same tour under the original
  matrix.

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Branch and Bound Example

• Idea of the Solution Reduce the cost matrix so
  that each row or column contains at least one
  entry that is equal to 0. Such a matrix is called
  a reduction of the original matrix.
• In general, let (r1, r2, ..., rn) and (c1, c2,
  ..., cn) be the amounts subtracted from rows 1 to
  n and columns 1 to n, respectively, in an n ? n
  cost matrix A. Then
• is a lower bound on the cost of any complete tour

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Branch and Bound Example

• After creating the reduction matrix
• The lower bound is computed
• Initially, it is the value y of the reduction
  matrix
• An edge (x, y) is chosen from the matrix
• Two children of the current node are constructed
  where
• The right node represents all solutions that
  exclude edge (x, y)
• That is why the cost of (x, y) is set to ?
• The left node represents all solutions that
  include edge (x, y)
• Row x and column y are removed from the cost
  matrix. Why?
• Since all solutions use the edge (x, y), they
  will not use edge (y, x) and therefore the cost
  of (y, x) can be set to ?
• The above is also applied for the current paths
• The lower bound in each child is computed by
  adding the current value of y into the current
  lower bound when computing the reduction matrix
  of each child.
• Choose the subtree with minimum bound in order to
  apply branch and bound

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