Knapsack problem
The knapsack problem is an optimization problem that selects items subject to a capacity constraint to maximize their combined value.
Classic Computer Science Problems in Swift: Essential techniques for practicing programmers
The knapsack problem is an optimization problem that selects items with given values and weights to maximize total value without exceeding a capacity limit.
- How it is used
- Models allocating finite capacity among exhaustive options when maximizing total value.
- How deep it goes
- Develops a table-based dynamic-programming solution and backward reconstruction, while avoiding a full formal proof.
- Examples
- Uses household theft, a 75-pound item list, and a three-item table containing matches, a flashlight, and a book.
- What you can do after
- Build it from scratch with an item type, dynamic-programming table, capacity loop, and backward solution reconstruction.
Building Quantum Software with Python
The knapsack problem is an optimization problem that selects items with given weights and values to maximize total value without exceeding a weight capacity.
- How it is used
- Frames binary optimization as a quantum search example with value and weight constraints.
- How deep it goes
- Develops a worked three-item model through register encoding, circuit operations, and probability-based search, while deferring encoding details and later patterns.
- Examples
- Uses three items of weights 3, 2, 1 and values 2, 3, 1 under capacity 4, enumerating eight selections.
- What you can do after
- Apply it via solve_knapsack and encode item selections, values, and weights as quantum-register functions.
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Grokking Algorithms, Second Edition
The knapsack problem is an optimization problem that selects items within a capacity constraint to maximize their combined value.
- How it is used
- Uses a capacity-and-value choice to contrast greedy selection with dynamic programming.
- How deep it goes
- Works through an item-by-capacity grid from smaller knapsacks, beginning with a concrete cell before the formula.
- Examples
- Uses a stereo-versus-laptop-and-guitar choice, a 4-pound knapsack grid, and a time-limited travel itinerary.
- What you can do after
- Practice/perform a grid-based comparison of item choices and recognize when a greedy pick may miss the best combination.
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Advanced Algorithms and Data Structures
An optimization problem that selects each item at most once to maximize total value subject to a capacity constraint.
- How it is used
- Frames item selection as an optimization problem and motivates choosing algorithms by constraints, capacity, and acceptable computational cost.
- How deep it goes
- Contrasts brute force, dynamic programming, and greedy alternatives but stops short of detailing the dynamic-programming implementation.
- Examples
- Uses a mission crate, food items, allergy-driven combinations, a 1000-kilogram capacity, and calorie-per-kilogram selection.
- What you can do after
- Evaluate whether dynamic programming or constraint relaxation suits a capacity-limited selection problem.
Read how Advanced Algorithms and Data Structures explains knapsack problem →
Related concepts
- np-complete problemkind of · An NP-complete problem is a decision problem in NP to which every problem in NP can be reduced in polynomial time.
- dynamic programmingaddressed by · Dynamic programming is an algorithmic technique that solves problems by decomposing them into overlapping subproblems and storing or reusing their solutions.
- optimizationkind of · Optimization is the systematic selection or adjustment of values, actions, or system designs to improve a specified objective, often subject to constraints.
- greedy algorithmcontrasts with · A greedy algorithm builds a solution incrementally by choosing the locally best available option at each step, without generally accounting for all future consequences or revising prior choices.
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