0 1 knapsack using branch and bound

0/1 Knapsack Problem
(using BRANCH & BOUND)
Presented by
41.ABHISHEK KUMAR SINGH

0/1 Knapsack Problem
Given two integer arrays val[0..n-1] and wt[0..n-1] that represent values and weights
associated with n items respectively.
Find out the maximum value subset of val[] such that sum of the weights of this
subset is smaller than or equal to Knapsack capacity W.
We have ‘n’ items with value v1 , v2 . . . vn and weight of the corresponding items
is w1 , w2 . . . Wn .
Max capacity is W .
We can either choose or not choose an item. We have x1 , x2 . . . xn.
Here xi = { 1 , 0 }.
xi = 1 , item chosen
xi = 0 , item not chosen

Different approaches of this problem :
 Dynamic programming
 Brute force
 Backtracking
 Branch and bound

Why branch and bound ?
 Greedy approach works only for fractional knapsack problem.
 If weights are not integers , dynamic programming will not work.
 There are 2n possible combinations of item , complexity for brute force
goes exponentially.

What is branch and bound ?
 Branch and bound is an algorithm design paradigm which is
generally used for solving combinatorial optimization
problems.
 These problems typically exponential in terms of time
complexity and may require exploring all possible
permutations in worst case.
 Branch and Bound solve these problems relatively quickly.

 Combinatorial optimization problem is an
optimization problem, where an optimal solution
has to be identified from a finite set of solutions.

 Sort all items in decreasing order of ratio of value per unit
weight so that an upper bound can be computed using Greedy
Approach.
 Initialize maximum profit, maxProfit = 0
 Create an empty queue, Q.
 Create a dummy node of decision tree and enqueue it to Q.
Profit and weight of dummy node are 0.

ALGORITHM

 Do following while Q is not empty.

 Extract an item from Q. Let the extracted item be u.
 Compute profit of next level node. If the profit is more than maxProfit, then update
maxProfit.
 Compute bound of next level node. If bound is more than maxProfit, then add next
level node to Q.
 Consider the case when next level node is not considered as part of solution and
add a node to queue with level as next, but weight and profit without considering
next level nodes.

EXAMPLE :

$90
12
$98
Max weight = 16
maxProfit = 0
$ 0
0
$115

$90
12
$98
$115
2
$40
Max weight = 16
maxProfit = 40

$90
12
$98
Max weight = 16
maxProfit = 40
$ 0
0
$82

$90
12
$98
Max weight = 16
maxProfit = 70
$70
7
$115

$90
12
$98
2
$40
$98
Max weight = 16
maxProfit = 70

$90
12
$98
Max weight = 16
maxProfit = 70
$30
5
$82

$90
12
$98
Max weight = 16
maxProfit = 70
$ 0
0
$60

$90
12
$98
Max weight = 16
maxProfit = 70
$120
17
$0

$90
12
$98
Max weight = 16
maxProfit = 70
$70
7
$80

Max weight = 16
maxProfit = 90

struct Item
{
float weight;
int value;
}
Node structure to store information of decision tree
struct Node
{
int level, profit, bound;
float weight;
// level ---> Level of node in decision tree (or index ) in arr[]
// profit ---> Profit of nodes on path from root to this node (including this node)
// bound ---> Upper bound of maximum profit in subtree of this node
}
Data items used in the Algorithm :

knapsack(int W, Item arr[], int n)
queue<Node> Q
Node u, v //u.level = -1
Q.push(u) //u.profit = u.weight = 0
while ( !Q.empty() ) //int maxProfit = 0
u = Q.front() & Q.pop()
v.level = u.level + 1 // selecting the item
v.weight = u.weight + arr[v.level].weight
v.profit = u.profit + arr[v.level].value
if (v.weight <= W && v.profit > maxProfit)
maxProfit = v.profit
v.bound = bound(v, n, W, arr)
if (v.bound > maxProfit)
Q.push(v)
v.weight = u.weight // not selecting the item
v.profit = u.profit
v.bound = bound(v, n, W, arr)
If (v.bound > maxProfit)
Q.push(v)
return (maxProfit)
Algorithm for maxProfit :

bound(Node u, int n, int W, Item a[])
if (u.weight >= W)
return (0)
int u_bound <- u.profit
int j <- u.level + 1
int totweight <- u.weight
while ((j < n) && (totweight + a[j].weight <= W))
totweight <- totweight + a[j].weight
u_bound <- u_bound + a[j].value
j++
if (j < n)
u_bound <- u_bound + ( W - totweight ) * a[j].value /a[j].weight
return (u_bound)
Procedure to calculate upper bound :

0 1 knapsack using branch and bound

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