Notes/Advanced Algorithms/P vs. NP.md

type
theoretical

Definition

• Class P
  • The set of decision problems that can be solved by a deterministic Turing machine in polynomial time¹. These are problems considered to be efficiently solvable.
• O(\log n)
  • The set of decision problems for which a given solution can be verified in polynomial time by a deterministic Turing machine². NP stands for "nondeterministic polynomial time."
• :LiLoaderPinwheel: Does \mathbf{P = NP}?
  • This is one of the most important open questions in computer science. It asks whether every problem whose solution can be quickly verified can also be quickly solved.

Understanding P and NP

Class P Problems

• Solvable in polynomial time.
• Algorithms exist that can find a solution efficiently as the input size grows.
• As seen in:
  • Prime Testing: Determining if a number is prime.
  • Shortest Path: Finding the shortest path in a graph (e.g., Dijkstra's algorithm).
  • Sorting Algorithms: Such as Quick Sort and Merge Sort.

Class NP Problems

• Solutions can be verified in polynomial time.
• No known polynomial-time algorithms to solve all NP problems.
• As in:
  • Subset Sum: Determining if a subset of a given set of integers sums up to a target integer.
  • 3-SAT: Determining if a Boolean formula in conjunctive normal form with at most three literals per clause is satisfiable.
  • Hamiltonian Cycle: Determining if a Hamiltonian cycle exists in a graph.

NP-Complete Problems

• The hardest problems in NP. A problem is NP-Complete if:
• It is in NP.
• Every problem in NP can be reduced to it in polynomial time³.
• Implications: If any NP-Complete problem is solvable in polynomial time, then P = NP.
  • Traveling Salesperson Problem (Decision Version): Determining if there's a tour shorter than a given length.
  • Clique Problem: Finding a complete subgraph (clique) of a certain size in a graph.
  • Vertex Cover: Determining if there exists a set of vertices covering all edges.

Problems

Subset Sum Problem

• Input: A set of integers and a target sum.
• Question: Is there a subset whose sum equals the target?
• Approach:
  • Exponential Time: Checking all possible subsets.
  • Dynamic Programming: Pseudo-polynomial time algorithm when numbers are small.

3-SAT Problem

• Input: A Boolean formula in 3-CNF (Conjunctive Normal Form).
• Question: Is there a truth assignment that satisfies the formula?
• Importance: The first problem proven to be NP-Complete (Cook-Levin theorem).

Strategies

• Find solutions close to optimal in polynomial time.
• Practical methods that find good-enough solutions without guaranteeing optimality (heuristics).
• Restricting the problem to a subset where it becomes solvable in polynomial time.

NP vs. NP-Complete Differences

Aspect	NP	NP-Complete
Definition	Problems whose solutions can be verified quickly.	Hardest problems in NP. All NP problems reduce to them.
Relation to P	Contains P (P ⊆ NP).	If any NP-complete problem is in P, P = NP.
Ease of Solution	Some problems may have unknown solution methods.	Believed to be computationally difficult to solve.
Examples	Subset Sum, 3-SAT	TSP, Clique, Vertex Cover

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1. a theoretical computing machine that uses a predetermined set of rules to determine its actions. ↩︎

2. a theoretical model that, unlike a deterministic Turing machine, can make "guesses" to find solutions more efficiently. ↩︎

3. a method of converting one problem to another in polynomial time, preserving the problem's computational complexity. ↩︎