Implementation of 0/1 Knapsack using Branch and Bound

// C++ program to solve knapsack problem using
// branch and bound
#include <bits/stdc++.h>
using namespace std;

// Structure for Item which store weight and corresponding
// value of Item
struct Item
{
    float weight;
    int value;
};

// Node structure to store information of decision
// tree
struct Node
{
    // 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/
    int level, profit, bound;
    float weight;
};

// Comparison function to sort Item according to
// val/weight ratio
bool cmp(Item a, Item b)
{
    double r1 = (double)a.value / a.weight;
    double r2 = (double)b.value / b.weight;
    return r1 > r2;
}

// Returns bound of profit in subtree rooted with u.
// This function mainly uses Greedy solution to find
// an upper bound on maximum profit.
int bound(Node u, int n, int W, Item arr[])
{
    // if weight overcomes the knapsack capacity, return
    // 0 as expected bound
    if (u.weight >= W)
        return 0;

    // initialize bound on profit by current profit
    int profit_bound = u.profit;

    // start including items from index 1 more to current
    // item index
    int j = u.level + 1;
    int totweight = u.weight;

    // checking index condition and knapsack capacity
    // condition
    while ((j < n) && (totweight + arr[j].weight <= W))
    {
        totweight += arr[j].weight;
        profit_bound += arr[j].value;
        j++;
    }

    // If k is not n, include last item partially for
    // upper bound on profit
    if (j < n)
        profit_bound += (W - totweight) * arr[j].value /
                                        arr[j].weight;

    return profit_bound;
}

// Returns maximum profit we can get with capacity W
int knapsack(int W, Item arr[], int n)
{
    // sorting Item on basis of value per unit
    // weight.
    sort(arr, arr + n, cmp);

    // make a queue for traversing the node
    queue<Node> Q;
    Node u, v;

    // dummy node at starting
    u.level = -1;
    u.profit = u.weight = 0;
    Q.push(u);

    // One by one extract an item from decision tree
    // compute profit of all children of extracted item
    // and keep saving maxProfit
    int maxProfit = 0;
    while (!Q.empty())
    {
        // Dequeue a node
        u = Q.front();
        Q.pop();

        // If it is starting node, assign level 0
        if (u.level == -1)
            v.level = 0;

        // If there is nothing on next level
        if (u.level == n-1)
            continue;

        // Else if not last node, then increment level,
        // and compute profit of children nodes.
        v.level = u.level + 1;

        // Taking current level's item add current
        // level's weight and value to node u's
        // weight and value
        v.weight = u.weight + arr[v.level].weight;
        v.profit = u.profit + arr[v.level].value;

        // If cumulated weight is less than W and
        // profit is greater than previous profit,
        // update maxprofit
        if (v.weight <= W && v.profit > maxProfit)
            maxProfit = v.profit;

        // Get the upper bound on profit to decide
        // whether to add v to Q or not.
        v.bound = bound(v, n, W, arr);

        // If bound value is greater than profit,
        // then only push into queue for further
        // consideration
        if (v.bound > maxProfit)
            Q.push(v);

        // Do the same thing, but Without taking
        // the item in knapsack
        v.weight = u.weight;
        v.profit = u.profit;
        v.bound = bound(v, n, W, arr);
        if (v.bound > maxProfit)
            Q.push(v);
    }

    return maxProfit;
}

// driver program to test above function
int main()
{
    int W = 10; // Weight of knapsack
    Item arr[] = {{2, 40}, {3.14, 50}, {1.98, 100},
                {5, 95}, {3, 30}};
    int n = sizeof(arr) / sizeof(arr[0]);

    cout << "Maximum possible profit = "
        << knapsack(W, arr, n);

    return 0;
}