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Intermediate Topic 26 of 51 Algorithms

Knapsack Problem

Choosing items under a capacity to maximize value.

How it works

In the 0/1 knapsack problem, each item has a weight and a value, and you choose a subset whose total weight fits the capacity W while maximizing total value. Dynamic programming fills a table dp[i][w] with the best value using the first i items and capacity w, running in O(nW) time, which is pseudo-polynomial because W is a number, not a length. The problem is NP-hard in general.

Where it is used

Budget allocation, cargo loading, selecting investments, cutting stock, resource scheduling and choosing features within effort limits use knapsack models. Variants include unbounded and fractional versions.

Common pitfall

Iterating capacity in the wrong direction in the one-dimensional table lets items be used more than once. Large capacities make the table huge. The fractional version is solved greedily by value-to-weight ratio, but the same greedy fails for the 0/1 version. Reconstructing the chosen items needs a back-pointer step.

Where it fits in the course

Knapsack Problem is topic 26 of 51 in Algorithms, rated Intermediate. It follows Dynamic Programming. The next topic is Longest Common Subsequence.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Algorithms quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Knapsack Problem?
Choosing items under a capacity to maximize value. It belongs to Algorithms and is an intermediate-level topic.
Where do people use Knapsack Problem?
Budget allocation, cargo loading, selecting investments, cutting stock, resource scheduling and choosing features within effort limits use knapsack models.
What is the most common mistake with Knapsack Problem?
Iterating capacity in the wrong direction in the one-dimensional table lets items be used more than once.
What should I learn before this topic?
Start with Dynamic Programming, the topic just before it in Algorithms.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Algorithms quiz. You can also explore DP Table (LCS) in the Visual Lab.
Is Knapsack Problem hard to learn?
It is an Intermediate topic. Be comfortable with the Beginner topics in this area first, then read the common pitfall carefully, since it is where most mistakes happen.
How long does it take to learn Knapsack Problem?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Knapsack Problem?
Dynamic Programming, Longest Common Subsequence, Greedy Algorithms are closely connected. The full list is in the related topics section of this page.
What should I study after Knapsack Problem?
Continue with Longest Common Subsequence, the next topic in Algorithms.

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