Chapter 6 Branch-and-Bound
Illustrating Branch-and-Bound
with the 0-1 Knapsack Problem
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Branch-and-Bound
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using state space tree like backtracking
not limit us to any way of traversing tree
use only for optimization problem
determine whether a node is promising
in the sate space tree by computing a
number (bound)
bound not better than the best solution
 nonpromising (e.g. 0-1 knapsack
problem)
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Branch-and-Bound
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we can visit the children of a promising
node which has the best bound for
rapidly reaching an optimal solution
breadth-first search (instead of
depth-first search)
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visit root first, followed by all nodes at level
1, followed by all nodes at level 2,....
See Fig. 6.1 pp. 250
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Branch-and-Bound
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Branch-and-Bound
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6.1 Illustrating Branch-andBound with the 0-1 Knapsack
Problem
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Breadth-First Search with Branchand-Bound Pruning
Best-First Search with Branch-andBound Pruning
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6.1.1 Breadth-First Search with Branch-andBound Pruning
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See Example 6.1 pp. 251
void breadth_first_branch_and_bound (tree T)
{ initialize (Q);
v=root of T;
enqueue(Q, v);
maxprofit = profit(v);
while (!empty(Q))
{ dequeue(Q, v);
for (each child u of v)
{ if (profit(v) is better than the maxprofit)
maxprofit = profit(v);
if (bound(v) is better than best)
enqueue(Q, u);
}
}
}
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6.1.1 Breadth-First Search with Branch-andBound Pruning
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6.1.2 Best-First Search with Branch-and-Bound
Pruning
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improve breadth-first search strategy
after visiting all children of a given node,
we look at all promising and
unexpanded nodes,
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expand the node with the best bound first
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See Example 6.2 pp. 257
it not guarantees that the node appears
to be best will actually lead to an
optimal solution
priority queue
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6.1.2 Best-First Search with Branch-and-Bound
Pruning
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The End
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