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Complexity: P, NP, NP-completeness, Reductions
Master the art of NP-completeness with MIT's Erik Demaine, who proves problems from Super Mario to jigsaw puzzles are intractable.
Lecture guide · Written by AI from the lecture. For the lecture’s own words, open the Transcript tab.
Erik Demaine introduces computational complexity theory, covering P, NP, and NP-completeness through reductions. He demonstrates how NP-hardness applies across various domains, starting from 3SAT and reducing it to Super Mario Brothers, 3-dimensional matching, subset sum, partition, rectangle packing, and jigsaw puzzles.
This lecture assumes
- Polynomial Time Algorithmtaught in Mathematical Optimization & Why Convexity Is the Watershed at 28:33
- NP-completenesstaught in Mathematics and Computation: The Landscape of P vs. NP at 10:52
- Dynamic programmingtaught in Dynamic Programming I: Fibonacci, Shortest Paths at 0:40 and 1 more
Key Takeaways
- P represents decision problems solvable in polynomial time, while NP represents those solvable in non-deterministic polynomial time.
- A problem is NP-complete if it belongs to NP and every problem in NP can be reduced to it in polynomial time.
- Proving NP-hardness requires reducing a known NP-complete problem, such as 3SAT, to the target problem.
- Gadgets are structural constructs used in reductions to map features of an input problem to an output problem.
Concepts Explained22
P—The class of problems that can be solved in polynomial time.
NP—The class of decision problems solvable in non-deterministic polynomial time.
Decision Problems—Problems where the answer is a single bit, either yes or no.
3SAT—The problem of determining if a Boolean formula in conjunctive normal form with exactly three literals per clause is satisfiable.
Examples, Proofs & Derivations
Works & Sources Cited1
Book
Computers and Intractability: A Guide to the Theory of NP-Completeness—Book from the late 70s by Garey and Johnson on NP-completeness.