83 lines
3.8 KiB
Markdown
83 lines
3.8 KiB
Markdown
---
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type: theoretical
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---
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## Definition
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- Class P
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- The set of decision problems that can be solved by a deterministic Turing machine in polynomial time[^1]. These are problems considered to be efficiently solvable.
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- $O(\log n)$
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- The set of decision problems for which a given solution can be verified in polynomial time by a deterministic Turing machine[^2]. NP stands for "nondeterministic polynomial time."
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- :LiLoaderPinwheel: Does $\mathbf{P = NP}$?
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- 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.
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## Understanding P and NP
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### Class P Problems
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- Solvable in polynomial time.
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- Algorithms exist that can find a solution efficiently as the input size grows.
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- **As seen in**:
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- Prime Testing: Determining if a number is prime.
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- Shortest Path: Finding the shortest path in a graph (e.g., Dijkstra's algorithm).
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- Sorting Algorithms: Such as Quick Sort and Merge Sort.
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### Class NP Problems
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- Solutions can be verified in polynomial time.
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- No known polynomial-time algorithms to solve all NP problems.
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- **As in**:
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- Subset Sum: Determining if a subset of a given set of integers sums up to a target integer.
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- 3-SAT: Determining if a Boolean formula in conjunctive normal form with at most three literals per clause is satisfiable.
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- Hamiltonian Cycle: Determining if a Hamiltonian cycle exists in a graph.
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## NP-Complete Problems
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- The hardest problems in NP. A problem is NP-Complete if:
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- It is in NP.
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- Every problem in NP can be reduced to it in polynomial time[^3].
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- Implications: If any NP-Complete problem is solvable in polynomial time, then $P = NP$.
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- Traveling Salesperson Problem (Decision Version): Determining if there's a tour shorter than a given length.
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- Clique Problem: Finding a complete subgraph (clique) of a certain size in a graph.
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- Vertex Cover: Determining if there exists a set of vertices covering all edges.
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## Problems
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### Subset Sum Problem
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- **Input**: A set of integers and a target sum.
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- **Question**: Is there a subset whose sum equals the target?
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- **Approach**:
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- **Exponential Time**: Checking all possible subsets.
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- **Dynamic Programming**: Pseudo-polynomial time algorithm when numbers are small.
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### 3-SAT Problem
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- **Input**: A Boolean formula in 3-CNF (Conjunctive Normal Form).
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- **Question**: Is there a truth assignment that satisfies the formula?
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- **Importance**: The first problem proven to be NP-Complete (Cook-Levin theorem).
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## Strategies
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- Find solutions close to optimal in polynomial time.
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- Practical methods that find good-enough solutions without guaranteeing optimality (heuristics).
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- Restricting the problem to a subset where it becomes solvable in polynomial time.
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## NP vs. NP-Complete Differences
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|**Aspect**|**NP**|**NP-Complete**|
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|**Definition**|Problems whose solutions can be verified quickly.|Hardest problems in NP. All NP problems reduce to them.|
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|**Relation to P**|Contains P (P ⊆ NP).|If any NP-complete problem is in P, P = NP.|
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|**Ease of Solution**|Some problems may have unknown solution methods.|Believed to be computationally difficult to solve.|
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|**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.
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[^2]: a theoretical model that, unlike a deterministic Turing machine, can make "guesses" to find solutions more efficiently.
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[^3]: a method of converting one problem to another in polynomial time, preserving the problem's computational complexity.
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[^4]: the complements of NP-Complete problems, where verifying a "no" instance is in NP.
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