Is P = NP?
Abhiram Ranade
April 4, 2016
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Polynomial Time Verifier and NP-Completeness, Thesis of Computer Science
The concept of a polynomial time verifier for decision problems and its relation to the class np. It defines a polynomial time verifier as a function that takes as input strings x and y, where x is an instance of a decision problem and y has length polynomial in x. The function must return true for some y when x is true, and false for all y when x is false, while running in time polynomial in |x|. The document also states that p (the class of decision problems that can be solved in polynomial time) is a subset of np (the class of decision problems having polynomial time verifiers).
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Is P = NP?
Abhiram Ranade
April 4, 2016
Recap and Outline
The story so far..
Recap and Outline
The story so far.. I (^) We do not have polytime algorithms for many problems such as IS, VC, CSAT, CNFSAT, Knapsack, Travelling salesman problem, Graph colouring, and many more. I (^) However, we can try reducing them to each other which establishes the relative difficulty of finding polytime algorithms for them.
Recap and Outline
The story so far.. I (^) We do not have polytime algorithms for many problems such as IS, VC, CSAT, CNFSAT, Knapsack, Travelling salesman problem, Graph colouring, and many more. I (^) However, we can try reducing them to each other which establishes the relative difficulty of finding polytime algorithms for them.
Today:
Recap and Outline
The story so far.. I (^) We do not have polytime algorithms for many problems such as IS, VC, CSAT, CNFSAT, Knapsack, Travelling salesman problem, Graph colouring, and many more. I (^) However, we can try reducing them to each other which establishes the relative difficulty of finding polytime algorithms for them.
Today: I (^) Turns out these problems are similar in an intuitive sense, and this similarity can be formalized. They belong to a class ”NP”.
Recap and Outline
The story so far.. I (^) We do not have polytime algorithms for many problems such as IS, VC, CSAT, CNFSAT, Knapsack, Travelling salesman problem, Graph colouring, and many more. I (^) However, we can try reducing them to each other which establishes the relative difficulty of finding polytime algorithms for them.
Today: I (^) Turns out these problems are similar in an intuitive sense, and this similarity can be formalized. They belong to a class ”NP”. I (^) Formalizing the similarity has formal benefit too.
Recap and Outline
The story so far.. I (^) We do not have polytime algorithms for many problems such as IS, VC, CSAT, CNFSAT, Knapsack, Travelling salesman problem, Graph colouring, and many more. I (^) However, we can try reducing them to each other which establishes the relative difficulty of finding polytime algorithms for them.
Today: I (^) Turns out these problems are similar in an intuitive sense, and this similarity can be formalized. They belong to a class ”NP”. I (^) Formalizing the similarity has formal benefit too. Cook’s theorem I (^) The classes NPC and NPH.
The class NP: class of puzzle like problems
The class NP: class of puzzle like problems
”A good puzzle is one whose answer is difficult to discover, but given the answer it is check that it is indeed correct.”
Examples:
The class NP: class of puzzle like problems
”A good puzzle is one whose answer is difficult to discover, but given the answer it is check that it is indeed correct.”
Examples: I (^) Sudoku
The class NP: class of puzzle like problems
”A good puzzle is one whose answer is difficult to discover, but given the answer it is check that it is indeed correct.”
Examples: I (^) Sudoku I (^) Jigsaw puzzles
”Elegant math theorems” are also like that. Proofs are easy in hindsight.
The class NP: class of puzzle like problems
”A good puzzle is one whose answer is difficult to discover, but given the answer it is check that it is indeed correct.”
Examples: I (^) Sudoku I (^) Jigsaw puzzles
”Elegant math theorems” are also like that. Proofs are easy in hindsight.
VC, IS, TSP, CSAT, SAT are also similar!
The class NP: class of puzzle like problems
”A good puzzle is one whose answer is difficult to discover, but given the answer it is check that it is indeed correct.”
Examples: I (^) Sudoku I (^) Jigsaw puzzles
”Elegant math theorems” are also like that. Proofs are easy in hindsight.
VC, IS, TSP, CSAT, SAT are also similar!
Deciding whether a graph does have an IS of size k seems to be difficult.
But if someone gives you a proof that a graph has a size k IS, can we check it quickly?