Confusion about NP-hard and NP-Complete in Traveling Salesman issues

Question

Optimization of the traveling salesman (TSP-OPT) is NP-difficult, and Traveling Salesman Search (TSP) is NP-complete. However, the TSP-OPT can be reduced to TSP, because if the TSP can be solved in polynomial time, then the TSP-OPT (1) can be enabled. I thought that to reduce A to BB, it should be just as tough, if not harder than A. As I see in the links below, TSP-OPT can be reduced to TSP. TSP-OPT should be harder than TSP. I'm confused...

Links: (1) Algorithm, Dasgupta, Papadimitriou, Vazirani Exercise 8.1 http://algorithmics.lsi.upc.edu/docs/Dasgupta-Papadimitriou-Vazirani.pdf https://cseweb.ucsd.edu/classes/sp08/cse01 /hw/hw6soln.pdf

http://cs170.org/assets/disc/dis10-sol.pdf

Answer 1

I quickly looked over the links you gave, and I have to admit that I really dislike your textbook (1st pdf): they consider NP-completeness, barely mentioning the solution problems. The provided definition of an NP-complete problem is also slightly different from what I expect from the tutorial. I guess it was a conscious decision to make the introduction more attractive ...

I will give a short answer, followed by a more detailed explanation of related concepts.

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Short version

Intuitively (and informally) the problem is in NP , if its solutions are easily verified .

, NP-hard, , .

NP-, NP, NP-hard. , NP-. , .

, - . , , . TSP, TSP-OPT NP-, . , pdf 8.1 , , TSP TSP-OPT .

TSP TSP-OPT , (, ) , . , TSP , NP-. , , . TSP-OPT, , NP, , NP-. ( .)

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Tl; dr - , TSP-OPT , TSP . , , , , .

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, , . NP-, NP-, NP, P .. , .

.

- , YES, NO.

TSP

: G, b

: G b? (/)

TSP

: G, b

: G b, .

TSP

: G

: G .

TSP . , TSP, - . TSP TSP-OPT .

, . , , . , .

? NP- , . , , , NP-complete/NP-hard /, , . , - , NP-, .

, , , . , - , 8.1 8.2 . , , , , .

Answer 2

NP-, , , TSP_DECIDE .

NP-, TSP_OPTIMIZE (HAM) , - C, , , ( ).

, , TSP

( TSP) . , , .

, NP-hard. : " x G , , x?", NP-. NP- .

x, x . , , .

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, TSP NP . , TSP NP . :

TSP - , . , , , . . , , .

. , . , . n , n-1 = (n-1) (n-2)... 3.2.

- NP, ( ) , .

.

, TSP, :

• ,
• , ,

! - , . 2 . , TSP NP, .

TSP NP, . , TSP NP .

, TSP NP , , (/ ) NP :

TSP_DECISION. L, , L?

NP , ( ) , "" TSPDECISION.