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# 0/1 Knapsack using 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.

Let us consider below 0/1 Knapsack problem to understand Branch and Bound. 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. Let us explore all approaches for this problem.

1. A Greedy approach is to pick the items in decreasing order of value per unit weight. The Greedy approach works only for fractional knapsack problem and may not produce correct result for 0/1 knapsack.
2. We can use Dynamic Programming (DP) for 0/1 Knapsack problem. In DP, we use a 2D table of size n x W. The DP Solution doesn’t work if item weights are not integers.
3. Since DP solution doesn’t always work, a solution is to use Brute Force. With n items, there are 2n solutions to be generated, check each to see if they satisfy the constraint, save maximum solution that satisfies constraint. This solution can be expressed as tree
4. We can use Backtracking to optimize the Brute Force solution. In the tree representation, we can do DFS of tree. If we reach a point where a solution no longer is feasible, there is no need to continue exploring. In the given example, backtracking would be much more effective if we had even more items or a smaller knapsack capacity.

Branch and BoundThe backtracking based solution works better than brute force by ignoring infeasible solutions. We can do better (than backtracking) if we know a bound on best possible solution subtree rooted with every node. If the best in subtree is worse than current best, we can simply ignore this node and its subtrees. So we compute bound (best solution) for every node and compare the bound with current best solution before exploring the node. Example bounds used in below diagram are, A down can give \$315, B down can \$275, C down can \$225, D down can \$125 and E down can \$30. In the next article, we have discussed the process to get these bounds.

Branch and bound is very useful technique for searching a solution but in worst case, we need to fully calculate the entire tree. At best, we only need to fully calculate one path through the tree and prune the rest of it.

### Pseudo code

function knapsack(items, max_weight):
best_value = 0
queue = [{items: [], value: 0, weight: 0}]
while queue is not empty:
node = queue.pop()
if node is a leaf node:
update best_value if necessary
else:
for each remaining item:
child = create child node for item
if child is promising:
queue.append(child)
return best_value

function is_promising(node, max_weight, best_value):
if node.weight > max_weight:
return False
if node.value + bound(node.items) < best_value:
return False
return True

function bound(items):
# Calculate an upper bound on the value of the remaining items
# using some heuristic (e.g., the fractional knapsack algorithm)
…

Example:

## Java

 `import` `java.util.*;`   `class` `KnapsackNode {` `  ``// The items that have been included in the knapsack so far` `  ``List items;` `  ``// The total value of the items in the knapsack so far` `  ``int` `value;` `  ``// The total weight of the items in the knapsack so far` `  ``int` `weight;`   `  ``public` `KnapsackNode(List items, ``int` `value, ``int` `weight) {` `    ``this``.items = items;` `    ``this``.value = value;` `    ``this``.weight = weight;` `  ``}` `}`   `class` `Item {` `  ``// The value of the item` `  ``int` `value;` `  ``// The weight of the item` `  ``int` `weight;` `  ``// The value-to-weight ratio of the item` `  ``double` `ratio;`   `  ``public` `Item(``int` `value, ``int` `weight) {` `    ``this``.value = value;` `    ``this``.weight = weight;` `    ``this``.ratio = (``double``) value / weight;` `  ``}` `}`   `public` `class` `Knapsack {` `  ``// The maximum weight capacity of the knapsack` `  ``int` `maxWeight;` `  ``// The list of items` `  ``Item[] items;`   `  ``public` `Knapsack(``int` `maxWeight, Item[] items) {` `    ``this``.maxWeight = maxWeight;` `    ``this``.items = items;` `  ``}`   `  ``// Solves the 0/1 knapsack problem using branch and bound` `  ``public` `int` `solve() {` `    ``// Sort the items in decreasing order of value per unit weight` `    ``Arrays.sort(items, (a, b) -> -Double.compare(a.ratio, b.ratio));`   `    ``// The best value found so far` `    ``int` `bestValue = ``0``;`   `    ``// The queue of nodes to be explored` `    ``Queue queue = ``new` `LinkedList<>();` `    ``queue.add(``new` `KnapsackNode(``new` `ArrayList<>(), ``0``, ``0``));`   `    ``while` `(!queue.isEmpty()) {` `      ``KnapsackNode node = queue.poll();` `      ``int` `i = node.items.size();` `      ``if` `(i == items.length) {` `        ``// This is a leaf node, so update the best value if necessary` `        ``bestValue = Math.max(bestValue, node.value);` `      ``} ``else` `{` `        ``// Add the child nodes for the remaining items` `        ``Item item = items[i];` `        ``KnapsackNode withItem = ``new` `KnapsackNode(` `          ``new` `ArrayList<>(node.items), node.value + item.value, node.weight + item.weight);` `        ``if` `(isPromising(withItem, maxWeight, bestValue)) {` `          ``queue.add(withItem);` `        ``}` `        ``KnapsackNode withoutItem = ``new` `KnapsackNode(` `          ``new` `ArrayList<>(node.items), node.value, node.weight);` `        ``if` `(isPromising(withoutItem, maxWeight, bestValue)) {` `          ``queue.add(withoutItem);` `        ``}` `      ``}` `    ``}`   `    ``return` `bestValue;` `  ``}`   `  ``// Returns true if the given node is promising (i.e., it may lead to a better solution than the current best)` `  ``private` `boolean` `isPromising(KnapsackNode node, ``int` `maxWeight, ``int` `best`

## Javascript

 `class KnapsackNode {` `  ``constructor(items, value, weight) {` `    ``this``.items = items;` `    ``this``.value = value;` `    ``this``.weight = weight;` `  ``}` `}`   `class Item {` `  ``constructor(value, weight) {` `    ``this``.value = value;` `    ``this``.weight = weight;` `    ``this``.ratio = value / weight;` `  ``}` `}`   `class Knapsack {` `  ``constructor(maxWeight, items) {` `    ``this``.maxWeight = maxWeight;` `    ``this``.items = items;` `  ``}`   `  ``solve() {` `    ``this``.items.sort((a, b) => b.ratio - a.ratio);` `    ``let bestValue = 0;` `    ``const queue = [``new` `KnapsackNode([], 0, 0)];`   `    ``while` `(queue.length > 0) {` `      ``const node = queue.shift();` `      ``const i = node.items.length;`   `      ``if` `(i === ``this``.items.length) {` `        ``bestValue = Math.max(bestValue, node.value);` `      ``} ``else` `{` `        ``const item = ``this``.items[i];` `        ``const withItem = ``new` `KnapsackNode(` `          ``[...node.items, i],` `          ``node.value + item.value,` `          ``node.weight + item.weight` `        ``);` `        ``if` `(``this``.isPromising(withItem, ``this``.maxWeight, bestValue)) {` `          ``queue.push(withItem);` `        ``}` `        ``const withoutItem = ``new` `KnapsackNode(` `          ``[...node.items],` `          ``node.value,` `          ``node.weight` `        ``);` `        ``if` `(``this``.isPromising(withoutItem, ``this``.maxWeight, bestValue)) {` `          ``queue.push(withoutItem);` `        ``}` `      ``}` `    ``}`   `    ``return` `bestValue;` `  ``}`   `  ``isPromising(node, maxWeight, bestValue) {` `    ``return` `(` `      ``node.weight <= maxWeight &&` `      ``node.value + ``this``.getBound(node) > bestValue` `    ``);` `  ``}`   `  ``getBound(node) {` `    ``let remainingWeight = ``this``.maxWeight - node.weight;` `    ``let bound = node.value;`   `    ``for` `(let i = node.items.length; i < ``this``.items.length; i++) {` `      ``const item = ``this``.items[i];`   `      ``if` `(remainingWeight >= item.weight) {` `        ``bound += item.value;` `        ``remainingWeight -= item.weight;` `      ``} ``else` `{` `        ``bound += remainingWeight * item.ratio;` `        ``break``;` `      ``}` `    ``}`   `    ``return` `bound;` `  ``}` `}`

## Python3

 `from` `queue ``import` `Queue` `from` `typing ``import` `List`     `class` `KnapsackNode:` `    ``def` `__init__(``self``, items: ``List``[``int``], value: ``int``, weight: ``int``):` `        ``self``.items ``=` `items` `        ``self``.value ``=` `value` `        ``self``.weight ``=` `weight`     `class` `Item:` `    ``def` `__init__(``self``, value: ``int``, weight: ``int``):` `        ``self``.value ``=` `value` `        ``self``.weight ``=` `weight` `        ``self``.ratio ``=` `value ``/` `weight`     `class` `Knapsack:` `    ``def` `__init__(``self``, maxWeight: ``int``, items: ``List``[Item]):` `        ``self``.maxWeight ``=` `maxWeight` `        ``self``.items ``=` `items`   `    ``def` `solve(``self``) ``-``> ``int``:` `        ``self``.items.sort(key``=``lambda` `x: x.ratio, reverse``=``True``)` `        ``bestValue ``=` `0` `        ``queue ``=` `[KnapsackNode([], ``0``, ``0``)]`   `        ``while` `queue:` `            ``node ``=` `queue.pop(``0``)` `            ``i ``=` `len``(node.items)`   `            ``if` `i ``=``=` `len``(``self``.items):` `                ``bestValue ``=` `max``(bestValue, node.value)` `            ``else``:` `                ``item ``=` `self``.items[i]` `                ``withItem ``=` `KnapsackNode(` `                    ``node.items ``+` `[i],` `                    ``node.value ``+` `item.value,` `                    ``node.weight ``+` `item.weight` `                ``)` `                ``if` `self``.isPromising(withItem, ``self``.maxWeight, bestValue):` `                    ``queue.append(withItem)` `                ``withoutItem ``=` `KnapsackNode(` `                    ``node.items,` `                    ``node.value,` `                    ``node.weight` `                ``)` `                ``if` `self``.isPromising(withoutItem, ``self``.maxWeight, bestValue):` `                    ``queue.append(withoutItem)`   `        ``return` `bestValue`   `    ``def` `isPromising(``self``, node: KnapsackNode, maxWeight: ``int``, bestValue: ``int``) ``-``> ``bool``:` `        ``return` `node.weight <``=` `maxWeight ``and` `node.value ``+` `self``.getBound(node) > bestValue`   `    ``def` `getBound(``self``, node: KnapsackNode) ``-``> ``float``:` `        ``remainingWeight ``=` `self``.maxWeight ``-` `node.weight` `        ``bound ``=` `node.value`   `        ``for` `i ``in` `range``(``len``(node.items), ``len``(``self``.items)):` `            ``item ``=` `self``.items[i]`   `            ``if` `remainingWeight >``=` `item.weight:` `                ``bound ``+``=` `item.value` `                ``remainingWeight ``-``=` `item.weight` `            ``else``:` `                ``bound ``+``=` `remainingWeight ``*` `item.ratio` `                ``break`   `        ``return` `bound`

## C++

 `#include ` `#include ` `#include ` `#include `   `using` `namespace` `std;`   `class` `Item {` `public``:` `    ``int` `value;` `    ``int` `weight;` `    ``double` `ratio;`   `    ``Item(``int` `value, ``int` `weight) {` `        ``this``->value = value;` `        ``this``->weight = weight;` `        ``this``->ratio = (``double``)value / weight;` `    ``}` `};`   `class` `KnapsackNode {` `public``:` `    ``vector<``int``> items;` `    ``int` `value;` `    ``int` `weight;`   `    ``KnapsackNode(vector<``int``> items, ``int` `value, ``int` `weight) {` `        ``this``->items = items;` `        ``this``->value = value;` `        ``this``->weight = weight;` `    ``}` `};`   `class` `Knapsack {` `public``:` `    ``int` `maxWeight;` `    ``vector items;`   `    ``Knapsack(``int` `maxWeight, vector items) {` `        ``this``->maxWeight = maxWeight;` `        ``this``->items = items;` `    ``}`   `    ``int` `solve() {` `        ``sort(``this``->items.begin(), ``this``->items.end(), [](``const` `Item& a, ``const` `Item& b) {` `            ``return` `a.ratio > b.ratio;` `        ``});`   `        ``int` `bestValue = 0;` `        ``queue q;` `        ``q.push(KnapsackNode({}, 0, 0));`   `        ``while` `(!q.empty()) {` `            ``KnapsackNode node = q.front();` `            ``q.pop();` `            ``int` `i = node.items.size();`   `            ``if` `(i == ``this``->items.size()) {` `                ``bestValue = max(bestValue, node.value);` `            ``} ``else` `{` `                ``Item item = ``this``->items[i];` `                ``KnapsackNode withItem(node.items, node.value + item.value, node.weight + item.weight);` `                ``if` `(isPromising(withItem, ``this``->maxWeight, bestValue)) {` `                    ``q.push(withItem);` `                ``}` `                ``KnapsackNode withoutItem(node.items, node.value, node.weight);` `                ``if` `(isPromising(withoutItem, ``this``->maxWeight, bestValue)) {` `                    ``q.push(withoutItem);` `                ``}` `            ``}` `        ``}`   `        ``return` `bestValue;` `    ``}`   `    ``bool` `isPromising(KnapsackNode node, ``int` `maxWeight, ``int` `bestValue) {` `        ``return` `node.weight <= maxWeight && node.value + getBound(node) > bestValue;` `    ``}`   `    ``int` `getBound(KnapsackNode node) {` `        ``int` `remainingWeight = ``this``->maxWeight - node.weight;` `        ``int` `bound = node.value;`   `        ``for` `(``int` `i = node.items.size(); i < ``this``->items.size(); i++) {` `            ``Item item = ``this``->items[i];`   `            ``if` `(remainingWeight >= item.weight) {` `                ``bound += item.value;` `                ``remainingWeight -= item.weight;` `            ``} ``else` `{` `                ``bound += remainingWeight * item.ratio;` `                ``break``;` `            ``}` `        ``}`   `        ``return` `bound;` `    ``}` `};`

## C#

 `using` `System;` `using` `System.Collections.Generic;` `using` `System.Linq;`   `class` `Item {` `    ``public` `int` `value;` `    ``public` `int` `weight;` `    ``public` `double` `ratio;`   `    ``public` `Item(``int` `value, ``int` `weight)` `    ``{` `        ``this``.value = value;` `        ``this``.weight = weight;` `        ``this``.ratio = (``double``)value / weight;` `    ``}` `}`   `class` `KnapsackNode {` `    ``public` `List<``int``> items;` `    ``public` `int` `value;` `    ``public` `int` `weight;` `    ``public` `KnapsackNode(List<``int``> items, ``int` `value,` `                        ``int` `weight)` `    ``{` `        ``this``.items = items;` `        ``this``.value = value;` `        ``this``.weight = weight;` `    ``}` `}`   `class` `Knapsack {` `    ``public` `int` `maxWeight;` `    ``public` `List items;` `    ``public` `Knapsack(``int` `maxWeight, List items)` `    ``{` `        ``this``.maxWeight = maxWeight;` `        ``this``.items = items;` `    ``}`   `    ``public` `int` `Solve()` `    ``{` `        ``items = items.OrderByDescending(i = > i.ratio)` `                    ``.ToList();`   `        ``int` `bestValue = 0;` `        ``Queue q = ``new` `Queue();` `        ``q.Enqueue(``new` `KnapsackNode(``new` `List<``int``>(), 0, 0));`   `        ``while` `(q.Count > 0) {` `            ``KnapsackNode node = q.Dequeue();` `            ``int` `i = node.items.Count;`   `            ``if` `(i == items.Count) {` `                ``bestValue = Math.Max(bestValue, node.value);` `            ``}` `            ``else` `{` `                ``Item item = items[i];` `                ``KnapsackNode withItem = ``new` `KnapsackNode(` `                    ``new` `List<``int``>(node.items),` `                    ``node.value + item.value,` `                    ``node.weight + item.weight);` `                ``if` `(IsPromising(withItem, maxWeight,` `                                ``bestValue)) {` `                    ``q.Enqueue(withItem);` `                ``}` `                ``KnapsackNode withoutItem = ``new` `KnapsackNode(` `                    ``new` `List<``int``>(node.items), node.value,` `                    ``node.weight);` `                ``if` `(IsPromising(withoutItem, maxWeight,` `                                ``bestValue)) {` `                    ``q.Enqueue(withoutItem);` `                ``}` `            ``}` `        ``}`   `        ``return` `bestValue;` `    ``}`   `    ``private` `bool` `IsPromising(KnapsackNode node,` `                             ``int` `maxWeight, ``int` `bestValue)` `    ``{` `        ``return` `node.weight <= maxWeight` `            ``&& node.value + GetBound(node) > bestValue;` `    ``}`   `    ``private` `int` `GetBound(KnapsackNode node)` `    ``{` `        ``int` `remainingWeight = maxWeight - node.weight;` `        ``int` `bound = node.value;`   `        ``for` `(``int` `i = node.items.Count; i < items.Count;` `             ``i++) {` `            ``Item item = items[i];`   `            ``if` `(remainingWeight >= item.weight) {` `                ``bound += item.value;` `                ``remainingWeight -= item.weight;` `            ``}` `            ``else` `{` `                ``bound` `                    ``+= (``int``)(remainingWeight * item.ratio);` `                ``break``;` `            ``}` `        ``}`   `        ``return` `bound;` `    ``}` `}`

## Java

 `Item[] items = {` `  ``new` `Item(``60``, ``10``),` `  ``new` `Item(``100``, ``20``),` `  ``new` `Item(``120``, ``30``)` `};` `Knapsack knapsack = ``new` `Knapsack(``50``, items);` `int` `bestValue = knapsack.solve();` `System.out.println(``"Best value: "` `+ bestValue);`

## Javascript

 `const items = [` `  ``new` `Item(60, 10),` `  ``new` `Item(100, 20),` `  ``new` `Item(120, 30)` `];` `const knapsack = ``new` `Knapsack(50, items);` `const bestValue = knapsack.solve();` `console.log(``"Best value: "` `+ bestValue);`

## Python3

 `items ``=` `[` `    ``Item(``60``, ``10``),` `    ``Item(``100``, ``20``),` `    ``Item(``120``, ``30``)` `]` `knapsack ``=` `Knapsack(``50``, items)` `bestValue ``=` `knapsack.solve()` `print``(``"Best value: "` `+` `str``(bestValue))`

## C++

 `int` `main() {` `    ``vector items = {` `        ``Item(60, 10),` `    ``Item(100, 20),` `    ``Item(120, 30)` `    ``};` `    ``Knapsack knapsack(50, items);` `    ``int` `result = knapsack.solve();` `    ``cout << ``"Best value: "` `<< result << endl;` `    ``return` `0;` `}`

## C#

 `class` `Program {` `    ``static` `void` `Main(``string``[] args) {` `        ``List items = ``new` `List {` `            ``new` `Item(60, 10),` `            ``new` `Item(100, 20),` `            ``new` `Item(120, 30)` `        ``};` `        ``Knapsack knapsack = ``new` `Knapsack(50, items);` `        ``int` `bestValue = knapsack.Solve();` `        ``Console.WriteLine(``"Best value: "` `+ bestValue);` `    ``}` `}`

output:

Best value: 220

Time Complexity: O(N), as only one path through the tree will have to be traversed in the beat case and its worst time complexity is still given as O(2N) .

Source:

Above images and content is adopted from following nice link. http://www.cse.msu.edu/~torng/Classes/Archives/cse830.03fall/Lectures/Lecture11.ppt   Branch and Bound | Set 2 (Implementation of 0/1 Knapsack)