Key differences between the Knapsack Problem and the Bin Packing Problem

Objective

Maximize the total value of items packed into a single knapsack.

Minimize the number of bins needed to pack all items.

Number of Containers

Only one knapsack (container) is available.

Multiple bins (containers) are available.

Capacity Constraint

The knapsack has a fixed capacity that cannot be exceeded.

Each bin has a fixed capacity, but you can use as many bins as needed.

Item Selection

Not all items have to be packed; select a subset that maximizes value within the capacity.

All items must be packed into the bins.

Focus

Choosing items with the highest value-to-weight ratio.

Efficiently packing all items to minimize the number of bins used.

Item Value

Each item has an associated weight and value.

Items have only weight; no value is associated with them.

Type of Problem

Optimization problem (maximization).

Combinatorial problem (minimization).

Typical Approach

Use dynamic programming or greedy algorithms to maximize value.

Use greedy or heuristic algorithms to minimize bin usage.

Example

Packing a backpack with the most valuable items without exceeding weight.

Packing boxes in a truck, minimizing the number of trips required.

Common Variants

0/1 Knapsack, Fractional Knapsack, Multi-dimensional Knapsack

One-dimensional Bin Packing, Two-dimensional Packing

Complexity

NP-complete for 0/1 Knapsack and bin packing.

NP-complete, especially for the one-dimensional bin packing problem.