Added branch_and_bound Algorithm

Author: Dhanush D Prabhu

knapsack/knapsack_branch_and_bound.py

@@ -0,0 +1,122 @@
+"""
+Branch and Bound solution for the 0/1 Knapsack problem.
+
+This implementation uses a best-first search strategy and prunes
+non-promising branches using an upper bound calculated via the
+fractional knapsack (greedy) approach.
+
+References:
+https://en.wikipedia.org/wiki/Branch_and_bound
+https://en.wikipedia.org/wiki/Knapsack_problem
+"""
+
+from dataclasses import dataclass
+from typing import List, Tuple
+import heapq
+
+
+@dataclass
+class Item:
+    weight: int
+    value: int
+
+
+@dataclass
+class Node:
+    level: int
+    profit: int
+    weight: int
+    bound: float
+
+
+def calculate_bound(
+    node: Node, capacity: int, items: List[Item]
+) -> float:
+    """
+    Calculate the upper bound of profit for a node using
+    the fractional knapsack approach.
+    """
+    if node.weight >= capacity:
+        return 0.0
+
+    profit_bound = float(node.profit)
+    total_weight = node.weight
+    index = node.level + 1
+
+    while index < len(items) and total_weight + items[index].weight <= capacity:
+        total_weight += items[index].weight
+        profit_bound += items[index].value
+        index += 1
+
+    if index < len(items):
+        profit_bound += (
+            (capacity - total_weight)
+            * items[index].value
+            / items[index].weight
+        )
+
+    return profit_bound
+
+
+def knapsack_branch_and_bound(
+    capacity: int, weights: List[int], values: List[int]
+) -> int:
+    """
+    Solve the 0/1 Knapsack problem using the Branch and Bound technique.
+
+    >>> knapsack_branch_and_bound(50, [10, 20, 30], [60, 100, 120])
+    220
+    """
+    items = [Item(weight=w, value=v) for w, v in zip(weights, values)]
+    items.sort(key=lambda item: item.value / item.weight, reverse=True)
+
+    priority_queue: List[Tuple[float, Node]] = []
+
+    root = Node(level=-1, profit=0, weight=0, bound=0.0)
+    root.bound = calculate_bound(root, capacity, items)
+
+    heapq.heappush(priority_queue, (-root.bound, root))
+    max_profit = 0
+
+    while priority_queue:
+        _, current = heapq.heappop(priority_queue)
+
+        if current.bound <= max_profit:
+            continue
+
+        next_level = current.level + 1
+        if next_level >= len(items):
+            continue
+
+        # Include next item
+        include_node = Node(
+            level=next_level,
+            profit=current.profit + items[next_level].value,
+            weight=current.weight + items[next_level].weight,
+            bound=0.0,
+        )
+
+        if include_node.weight <= capacity:
+            max_profit = max(max_profit, include_node.profit)
+
+        include_node.bound = calculate_bound(include_node, capacity, items)
+        if include_node.bound > max_profit:
+            heapq.heappush(
+                priority_queue, (-include_node.bound, include_node)
+            )
+
+        # Exclude next item
+        exclude_node = Node(
+            level=next_level,
+            profit=current.profit,
+            weight=current.weight,
+            bound=0.0,
+        )
+
+        exclude_node.bound = calculate_bound(exclude_node, capacity, items)
+        if exclude_node.bound > max_profit:
+            heapq.heappush(
+                priority_queue, (-exclude_node.bound, exclude_node)
+            )
+
+    return max_profit