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If I can solve Sudoku, can I solve the Travelling Salesman Problem (TSP)? If so, how?
When can a greedy algorithm solve the coin change problem?Finding the subset of $S$ that sums up to $k$ using a black box in $O(n)$ timeCan Euclidean TSP be exactly solved in time better than (sym)metric TSP?How to find partition set of a Partition Problem using its decision problemHow can we design an efficient warehouse management program?Finding vertices of a maximum clique in polynomial timeIs this an instance of a well-known problem?Is any sudoku solver an SAT solver?Applying a permutation on a sequence with multiplicationSolve this integer program (problem: Travelling salesman problem)
$begingroup$
Let us say there is a program such that if you give a partially filled Sudoku of any size it gives you corresponding completed Sudoku.
Can you treat this program as a black box and use this to solve TSP? I mean is there a way to represent TSP problem as partially filled Sudoku, so that if I give you the answer of that Sudoku, you can tell the solution for TSP in polynomial time?
If yes, how? how do you represent TSP as a partially filled Sudoku and interpret corresponding filled Sudoku for the result.
algorithms np-complete reductions traveling-salesman sudoku
edited yesterday
Rodrigo de Azevedo
702615
asked 2 days ago
Chakrapani N RaoChakrapani N Rao
11418
New contributor
Chakrapani N Rao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Let us say there is a program such that if you give a partially filled Sudoku of any size it gives you corresponding completed Sudoku.
Can you treat this program as a black box and use this to solve TSP? I mean is there a way to represent TSP problem as partially filled Sudoku, so that if I give you the answer of that Sudoku, you can tell the solution for TSP in polynomial time?
If yes, how? how do you represent TSP as a partially filled Sudoku and interpret corresponding filled Sudoku for the result.
algorithms np-complete reductions traveling-salesman sudoku
edited yesterday
Rodrigo de Azevedo
702615
asked 2 days ago
Chakrapani N RaoChakrapani N Rao
11418
New contributor
Chakrapani N Rao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
1
$begingroup$
This paper claims to give a constructive reduction from Sudoku to Hamiltonian cycle problem: sciencedirect.com/science/article/pii/S097286001630038X
$endgroup$
– C. Windolf
2 days ago
$begingroup$
@C.Windolf The question is asking for the other direction. (Indeed, there's a deleted answer that made the same mistake and cited the same paper.)
$endgroup$
– David Richerby
2 days ago
add a comment |
$begingroup$
Let us say there is a program such that if you give a partially filled Sudoku of any size it gives you corresponding completed Sudoku.
Can you treat this program as a black box and use this to solve TSP? I mean is there a way to represent TSP problem as partially filled Sudoku, so that if I give you the answer of that Sudoku, you can tell the solution for TSP in polynomial time?
If yes, how? how do you represent TSP as a partially filled Sudoku and interpret corresponding filled Sudoku for the result.
algorithms np-complete reductions traveling-salesman sudoku
edited yesterday
Rodrigo de Azevedo
702615
asked 2 days ago
Chakrapani N RaoChakrapani N Rao
11418
New contributor
Chakrapani N Rao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Let us say there is a program such that if you give a partially filled Sudoku of any size it gives you corresponding completed Sudoku.
Can you treat this program as a black box and use this to solve TSP? I mean is there a way to represent TSP problem as partially filled Sudoku, so that if I give you the answer of that Sudoku, you can tell the solution for TSP in polynomial time?
If yes, how? how do you represent TSP as a partially filled Sudoku and interpret corresponding filled Sudoku for the result.
algorithms np-complete reductions traveling-salesman sudoku
algorithms np-complete reductions traveling-salesman sudoku
edited yesterday
Rodrigo de Azevedo
702615
asked 2 days ago
Chakrapani N RaoChakrapani N Rao
11418
New contributor
Chakrapani N Rao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited yesterday
Rodrigo de Azevedo
702615
asked 2 days ago
Chakrapani N RaoChakrapani N Rao
11418
New contributor
Chakrapani N Rao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited yesterday
Rodrigo de Azevedo
702615
edited yesterday
Rodrigo de Azevedo
702615
edited yesterday
Rodrigo de Azevedo
702615
702615
asked 2 days ago
Chakrapani N RaoChakrapani N Rao
11418
New contributor
Chakrapani N Rao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
Chakrapani N RaoChakrapani N Rao
11418
asked 2 days ago
Chakrapani N RaoChakrapani N Rao
11418
11418
New contributor
Chakrapani N Rao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor
Chakrapani N Rao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Chakrapani N Rao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
$begingroup$
This paper claims to give a constructive reduction from Sudoku to Hamiltonian cycle problem: sciencedirect.com/science/article/pii/S097286001630038X
$endgroup$
– C. Windolf
2 days ago
$begingroup$
@C.Windolf The question is asking for the other direction. (Indeed, there's a deleted answer that made the same mistake and cited the same paper.)
$endgroup$
– David Richerby
2 days ago
add a comment |
1
$begingroup$
This paper claims to give a constructive reduction from Sudoku to Hamiltonian cycle problem: sciencedirect.com/science/article/pii/S097286001630038X
$endgroup$
– C. Windolf
2 days ago
$begingroup$
@C.Windolf The question is asking for the other direction. (Indeed, there's a deleted answer that made the same mistake and cited the same paper.)
$endgroup$
– David Richerby
2 days ago
1
1
$begingroup$
This paper claims to give a constructive reduction from Sudoku to Hamiltonian cycle problem: sciencedirect.com/science/article/pii/S097286001630038X
$endgroup$
– C. Windolf
2 days ago
$begingroup$
This paper claims to give a constructive reduction from Sudoku to Hamiltonian cycle problem: sciencedirect.com/science/article/pii/S097286001630038X
$endgroup$
– C. Windolf
2 days ago
$begingroup$
@C.Windolf The question is asking for the other direction. (Indeed, there's a deleted answer that made the same mistake and cited the same paper.)
$endgroup$
– David Richerby
2 days ago
$begingroup$
@C.Windolf The question is asking for the other direction. (Indeed, there's a deleted answer that made the same mistake and cited the same paper.)
$endgroup$
– David Richerby
2 days ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
For 9x9 Sudoku, no. It is finite so can be solved in $O(1)$ time.
But if you had a solver for $n^2 times n^2$ Sudoku, that worked for all $n$ and all possible partial boards, and ran in polynomial time, then yes, that could be used to solve TSP in polynomial time, as completing a $n^2 times n^2$ Sudoku is NP-complete.
The proof of NP-completeness works by reducing from some NP-complete problem R to Sudoku; then because R is NP-complete, you can reduce from TSP to R (that follows from the definition of NP-completeness); and chaining those reductions gives you a way to use the Sudoku solver to solve TSP.
answered 2 days ago
D.W.♦D.W.
102k12127290
$endgroup$
1
$begingroup$
Could you please explain how? Yes lets assume I have general sudoku solver which acts as a black box. So how can you use it? How do you represent TSP as a partially filled Sudoku
$endgroup$
– Chakrapani N Rao
2 days ago
2
$begingroup$
@ChakrapaniNRao, see updated answer. Yes, I understand it is a black box. To work out the details, find the proof of NP-completeness for Sudoku and understand how the reduction works.
$endgroup$
– D.W.♦
2 days ago
8
$begingroup$
@ChakrapaniNRao It doesn't answer the question completely but the full answer would be ridiculously long, be full of intricate details and wouldn't give you any more enlightenment than the sketch here. It's possible that a reduction of some NP-complete problem to $n^2times n^2$ sudoku might be interesting but, unless the proof that sudoku is NP-complete was actually by reduction from TSP (unlikely), the answer is still going to end "and then chain those two reductions together".
$endgroup$
– David Richerby
2 days ago
8
$begingroup$
@ChakrapaniNRao You are asking how to solve problem X using a black box for problem Y. That is literally asking for a reduction. That's what "reduction" means. And, as this answer explains, the answer to your yes/no question is yes.
$endgroup$
– David Richerby
2 days ago
2
$begingroup$
@SolomonUcko, well, no, not necessarily. The questions asks: if we have a Sudoku solver, can we use it to solve TSP? The answer is yes, we can. I explain how. This will give you a way to solve TSP about as fast as the Sudoku solver will solve Sudoku. If the Sudoku solver runs in polynomial time, this will give you a way to solve TSP in polynomial time. If the Sudoku solver runs in subexponential time, this will give you a way to solve TSP in subexponential time. And so on.
$endgroup$
– D.W.♦
yesterday
|
show 5 more comments
$begingroup$
It is indeed possible to use a general Sudoku solver to solve instances of TSP, and if this solver takes polynomial time then the whole process will as well (in complexity terminology, there is a polynomial-time reduction from TSP to Sudoku). This is because Sudoku is NP-complete and TSP is in NP. But as is usually the case in this area, looking at the details of the reduction isn't particularly illuminating. If you want, you can piece it together by using the simple reduction from Latin square completion to Sudoku here, the reduction from triangulating uniform tripartite graphs to Latin square completion here, the reduction from 3SAT to triangulation here, and a formulation of TSP as a 3SAT problem. However, if you want to understand the idea behind reducing from Sudoku to TSP I think you would be better off studying Cook's theorem (showing that SAT is NP-complete) and a couple of simple reductions from 3SAT (e.g. to 3-dimensional matching) and being satisfied in the knowledge that the TSP-Sudoku reduction is just the same kind of thing but longer and more fiddly.
answered 2 days ago
rlmsrlms
32114
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rlms is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
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votes
$begingroup$
For 9x9 Sudoku, no. It is finite so can be solved in $O(1)$ time.
But if you had a solver for $n^2 times n^2$ Sudoku, that worked for all $n$ and all possible partial boards, and ran in polynomial time, then yes, that could be used to solve TSP in polynomial time, as completing a $n^2 times n^2$ Sudoku is NP-complete.
The proof of NP-completeness works by reducing from some NP-complete problem R to Sudoku; then because R is NP-complete, you can reduce from TSP to R (that follows from the definition of NP-completeness); and chaining those reductions gives you a way to use the Sudoku solver to solve TSP.
answered 2 days ago
D.W.♦D.W.
102k12127290
$endgroup$
1
$begingroup$
Could you please explain how? Yes lets assume I have general sudoku solver which acts as a black box. So how can you use it? How do you represent TSP as a partially filled Sudoku
$endgroup$
– Chakrapani N Rao
2 days ago
2
$begingroup$
@ChakrapaniNRao, see updated answer. Yes, I understand it is a black box. To work out the details, find the proof of NP-completeness for Sudoku and understand how the reduction works.
$endgroup$
– D.W.♦
2 days ago
8
$begingroup$
@ChakrapaniNRao It doesn't answer the question completely but the full answer would be ridiculously long, be full of intricate details and wouldn't give you any more enlightenment than the sketch here. It's possible that a reduction of some NP-complete problem to $n^2times n^2$ sudoku might be interesting but, unless the proof that sudoku is NP-complete was actually by reduction from TSP (unlikely), the answer is still going to end "and then chain those two reductions together".
$endgroup$
– David Richerby
2 days ago
8
$begingroup$
@ChakrapaniNRao You are asking how to solve problem X using a black box for problem Y. That is literally asking for a reduction. That's what "reduction" means. And, as this answer explains, the answer to your yes/no question is yes.
$endgroup$
– David Richerby
2 days ago
2
$begingroup$
@SolomonUcko, well, no, not necessarily. The questions asks: if we have a Sudoku solver, can we use it to solve TSP? The answer is yes, we can. I explain how. This will give you a way to solve TSP about as fast as the Sudoku solver will solve Sudoku. If the Sudoku solver runs in polynomial time, this will give you a way to solve TSP in polynomial time. If the Sudoku solver runs in subexponential time, this will give you a way to solve TSP in subexponential time. And so on.
$endgroup$
– D.W.♦
yesterday
|
show 5 more comments
$begingroup$
For 9x9 Sudoku, no. It is finite so can be solved in $O(1)$ time.
But if you had a solver for $n^2 times n^2$ Sudoku, that worked for all $n$ and all possible partial boards, and ran in polynomial time, then yes, that could be used to solve TSP in polynomial time, as completing a $n^2 times n^2$ Sudoku is NP-complete.
The proof of NP-completeness works by reducing from some NP-complete problem R to Sudoku; then because R is NP-complete, you can reduce from TSP to R (that follows from the definition of NP-completeness); and chaining those reductions gives you a way to use the Sudoku solver to solve TSP.
answered 2 days ago
D.W.♦D.W.
102k12127290
$endgroup$
1
$begingroup$
Could you please explain how? Yes lets assume I have general sudoku solver which acts as a black box. So how can you use it? How do you represent TSP as a partially filled Sudoku
$endgroup$
– Chakrapani N Rao
2 days ago
2
$begingroup$
@ChakrapaniNRao, see updated answer. Yes, I understand it is a black box. To work out the details, find the proof of NP-completeness for Sudoku and understand how the reduction works.
$endgroup$
– D.W.♦
2 days ago
8
$begingroup$
@ChakrapaniNRao It doesn't answer the question completely but the full answer would be ridiculously long, be full of intricate details and wouldn't give you any more enlightenment than the sketch here. It's possible that a reduction of some NP-complete problem to $n^2times n^2$ sudoku might be interesting but, unless the proof that sudoku is NP-complete was actually by reduction from TSP (unlikely), the answer is still going to end "and then chain those two reductions together".
$endgroup$
– David Richerby
2 days ago
8
$begingroup$
@ChakrapaniNRao You are asking how to solve problem X using a black box for problem Y. That is literally asking for a reduction. That's what "reduction" means. And, as this answer explains, the answer to your yes/no question is yes.
$endgroup$
– David Richerby
2 days ago
2
$begingroup$
@SolomonUcko, well, no, not necessarily. The questions asks: if we have a Sudoku solver, can we use it to solve TSP? The answer is yes, we can. I explain how. This will give you a way to solve TSP about as fast as the Sudoku solver will solve Sudoku. If the Sudoku solver runs in polynomial time, this will give you a way to solve TSP in polynomial time. If the Sudoku solver runs in subexponential time, this will give you a way to solve TSP in subexponential time. And so on.
$endgroup$
– D.W.♦
yesterday
|
show 5 more comments
$begingroup$
For 9x9 Sudoku, no. It is finite so can be solved in $O(1)$ time.
But if you had a solver for $n^2 times n^2$ Sudoku, that worked for all $n$ and all possible partial boards, and ran in polynomial time, then yes, that could be used to solve TSP in polynomial time, as completing a $n^2 times n^2$ Sudoku is NP-complete.
The proof of NP-completeness works by reducing from some NP-complete problem R to Sudoku; then because R is NP-complete, you can reduce from TSP to R (that follows from the definition of NP-completeness); and chaining those reductions gives you a way to use the Sudoku solver to solve TSP.
answered 2 days ago
D.W.♦D.W.
102k12127290
$endgroup$
For 9x9 Sudoku, no. It is finite so can be solved in $O(1)$ time.
But if you had a solver for $n^2 times n^2$ Sudoku, that worked for all $n$ and all possible partial boards, and ran in polynomial time, then yes, that could be used to solve TSP in polynomial time, as completing a $n^2 times n^2$ Sudoku is NP-complete.
The proof of NP-completeness works by reducing from some NP-complete problem R to Sudoku; then because R is NP-complete, you can reduce from TSP to R (that follows from the definition of NP-completeness); and chaining those reductions gives you a way to use the Sudoku solver to solve TSP.
answered 2 days ago
D.W.♦D.W.
102k12127290
edited 15 hours ago
answered 2 days ago
D.W.♦D.W.
102k12127290
answered 2 days ago
D.W.♦D.W.
102k12127290
answered 2 days ago
D.W.♦D.W.
102k12127290
102k12127290
1
$begingroup$
Could you please explain how? Yes lets assume I have general sudoku solver which acts as a black box. So how can you use it? How do you represent TSP as a partially filled Sudoku
$endgroup$
– Chakrapani N Rao
2 days ago
2
$begingroup$
@ChakrapaniNRao, see updated answer. Yes, I understand it is a black box. To work out the details, find the proof of NP-completeness for Sudoku and understand how the reduction works.
$endgroup$
– D.W.♦
2 days ago
8
$begingroup$
@ChakrapaniNRao It doesn't answer the question completely but the full answer would be ridiculously long, be full of intricate details and wouldn't give you any more enlightenment than the sketch here. It's possible that a reduction of some NP-complete problem to $n^2times n^2$ sudoku might be interesting but, unless the proof that sudoku is NP-complete was actually by reduction from TSP (unlikely), the answer is still going to end "and then chain those two reductions together".
$endgroup$
– David Richerby
2 days ago
8
$begingroup$
@ChakrapaniNRao You are asking how to solve problem X using a black box for problem Y. That is literally asking for a reduction. That's what "reduction" means. And, as this answer explains, the answer to your yes/no question is yes.
$endgroup$
– David Richerby
2 days ago
2
$begingroup$
@SolomonUcko, well, no, not necessarily. The questions asks: if we have a Sudoku solver, can we use it to solve TSP? The answer is yes, we can. I explain how. This will give you a way to solve TSP about as fast as the Sudoku solver will solve Sudoku. If the Sudoku solver runs in polynomial time, this will give you a way to solve TSP in polynomial time. If the Sudoku solver runs in subexponential time, this will give you a way to solve TSP in subexponential time. And so on.
$endgroup$
– D.W.♦
yesterday
|
show 5 more comments
1
$begingroup$
Could you please explain how? Yes lets assume I have general sudoku solver which acts as a black box. So how can you use it? How do you represent TSP as a partially filled Sudoku
$endgroup$
– Chakrapani N Rao
2 days ago
2
$begingroup$
@ChakrapaniNRao, see updated answer. Yes, I understand it is a black box. To work out the details, find the proof of NP-completeness for Sudoku and understand how the reduction works.
$endgroup$
– D.W.♦
2 days ago
8
$begingroup$
@ChakrapaniNRao It doesn't answer the question completely but the full answer would be ridiculously long, be full of intricate details and wouldn't give you any more enlightenment than the sketch here. It's possible that a reduction of some NP-complete problem to $n^2times n^2$ sudoku might be interesting but, unless the proof that sudoku is NP-complete was actually by reduction from TSP (unlikely), the answer is still going to end "and then chain those two reductions together".
$endgroup$
– David Richerby
2 days ago
8
$begingroup$
@ChakrapaniNRao You are asking how to solve problem X using a black box for problem Y. That is literally asking for a reduction. That's what "reduction" means. And, as this answer explains, the answer to your yes/no question is yes.
$endgroup$
– David Richerby
2 days ago
2
$begingroup$
@SolomonUcko, well, no, not necessarily. The questions asks: if we have a Sudoku solver, can we use it to solve TSP? The answer is yes, we can. I explain how. This will give you a way to solve TSP about as fast as the Sudoku solver will solve Sudoku. If the Sudoku solver runs in polynomial time, this will give you a way to solve TSP in polynomial time. If the Sudoku solver runs in subexponential time, this will give you a way to solve TSP in subexponential time. And so on.
$endgroup$
– D.W.♦
yesterday
1
1
$begingroup$
Could you please explain how? Yes lets assume I have general sudoku solver which acts as a black box. So how can you use it? How do you represent TSP as a partially filled Sudoku
$endgroup$
– Chakrapani N Rao
2 days ago
$begingroup$
Could you please explain how? Yes lets assume I have general sudoku solver which acts as a black box. So how can you use it? How do you represent TSP as a partially filled Sudoku
$endgroup$
– Chakrapani N Rao
2 days ago
2
2
$begingroup$
@ChakrapaniNRao, see updated answer. Yes, I understand it is a black box. To work out the details, find the proof of NP-completeness for Sudoku and understand how the reduction works.
$endgroup$
– D.W.♦
2 days ago
$begingroup$
@ChakrapaniNRao, see updated answer. Yes, I understand it is a black box. To work out the details, find the proof of NP-completeness for Sudoku and understand how the reduction works.
$endgroup$
– D.W.♦
2 days ago
8
8
$begingroup$
@ChakrapaniNRao It doesn't answer the question completely but the full answer would be ridiculously long, be full of intricate details and wouldn't give you any more enlightenment than the sketch here. It's possible that a reduction of some NP-complete problem to $n^2times n^2$ sudoku might be interesting but, unless the proof that sudoku is NP-complete was actually by reduction from TSP (unlikely), the answer is still going to end "and then chain those two reductions together".
$endgroup$
– David Richerby
2 days ago
$begingroup$
@ChakrapaniNRao It doesn't answer the question completely but the full answer would be ridiculously long, be full of intricate details and wouldn't give you any more enlightenment than the sketch here. It's possible that a reduction of some NP-complete problem to $n^2times n^2$ sudoku might be interesting but, unless the proof that sudoku is NP-complete was actually by reduction from TSP (unlikely), the answer is still going to end "and then chain those two reductions together".
$endgroup$
– David Richerby
2 days ago
8
8
$begingroup$
@ChakrapaniNRao You are asking how to solve problem X using a black box for problem Y. That is literally asking for a reduction. That's what "reduction" means. And, as this answer explains, the answer to your yes/no question is yes.
$endgroup$
– David Richerby
2 days ago
$begingroup$
@ChakrapaniNRao You are asking how to solve problem X using a black box for problem Y. That is literally asking for a reduction. That's what "reduction" means. And, as this answer explains, the answer to your yes/no question is yes.
$endgroup$
– David Richerby
2 days ago
2
2
$begingroup$
@SolomonUcko, well, no, not necessarily. The questions asks: if we have a Sudoku solver, can we use it to solve TSP? The answer is yes, we can. I explain how. This will give you a way to solve TSP about as fast as the Sudoku solver will solve Sudoku. If the Sudoku solver runs in polynomial time, this will give you a way to solve TSP in polynomial time. If the Sudoku solver runs in subexponential time, this will give you a way to solve TSP in subexponential time. And so on.
$endgroup$
– D.W.♦
yesterday
$begingroup$
@SolomonUcko, well, no, not necessarily. The questions asks: if we have a Sudoku solver, can we use it to solve TSP? The answer is yes, we can. I explain how. This will give you a way to solve TSP about as fast as the Sudoku solver will solve Sudoku. If the Sudoku solver runs in polynomial time, this will give you a way to solve TSP in polynomial time. If the Sudoku solver runs in subexponential time, this will give you a way to solve TSP in subexponential time. And so on.
$endgroup$
– D.W.♦
yesterday
|
show 5 more comments
$begingroup$
It is indeed possible to use a general Sudoku solver to solve instances of TSP, and if this solver takes polynomial time then the whole process will as well (in complexity terminology, there is a polynomial-time reduction from TSP to Sudoku). This is because Sudoku is NP-complete and TSP is in NP. But as is usually the case in this area, looking at the details of the reduction isn't particularly illuminating. If you want, you can piece it together by using the simple reduction from Latin square completion to Sudoku here, the reduction from triangulating uniform tripartite graphs to Latin square completion here, the reduction from 3SAT to triangulation here, and a formulation of TSP as a 3SAT problem. However, if you want to understand the idea behind reducing from Sudoku to TSP I think you would be better off studying Cook's theorem (showing that SAT is NP-complete) and a couple of simple reductions from 3SAT (e.g. to 3-dimensional matching) and being satisfied in the knowledge that the TSP-Sudoku reduction is just the same kind of thing but longer and more fiddly.
answered 2 days ago
rlmsrlms
32114
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add a comment |
$begingroup$
It is indeed possible to use a general Sudoku solver to solve instances of TSP, and if this solver takes polynomial time then the whole process will as well (in complexity terminology, there is a polynomial-time reduction from TSP to Sudoku). This is because Sudoku is NP-complete and TSP is in NP. But as is usually the case in this area, looking at the details of the reduction isn't particularly illuminating. If you want, you can piece it together by using the simple reduction from Latin square completion to Sudoku here, the reduction from triangulating uniform tripartite graphs to Latin square completion here, the reduction from 3SAT to triangulation here, and a formulation of TSP as a 3SAT problem. However, if you want to understand the idea behind reducing from Sudoku to TSP I think you would be better off studying Cook's theorem (showing that SAT is NP-complete) and a couple of simple reductions from 3SAT (e.g. to 3-dimensional matching) and being satisfied in the knowledge that the TSP-Sudoku reduction is just the same kind of thing but longer and more fiddly.
answered 2 days ago
rlmsrlms
32114
New contributor
rlms is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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add a comment |
$begingroup$
It is indeed possible to use a general Sudoku solver to solve instances of TSP, and if this solver takes polynomial time then the whole process will as well (in complexity terminology, there is a polynomial-time reduction from TSP to Sudoku). This is because Sudoku is NP-complete and TSP is in NP. But as is usually the case in this area, looking at the details of the reduction isn't particularly illuminating. If you want, you can piece it together by using the simple reduction from Latin square completion to Sudoku here, the reduction from triangulating uniform tripartite graphs to Latin square completion here, the reduction from 3SAT to triangulation here, and a formulation of TSP as a 3SAT problem. However, if you want to understand the idea behind reducing from Sudoku to TSP I think you would be better off studying Cook's theorem (showing that SAT is NP-complete) and a couple of simple reductions from 3SAT (e.g. to 3-dimensional matching) and being satisfied in the knowledge that the TSP-Sudoku reduction is just the same kind of thing but longer and more fiddly.
answered 2 days ago
rlmsrlms
32114
New contributor
rlms is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
It is indeed possible to use a general Sudoku solver to solve instances of TSP, and if this solver takes polynomial time then the whole process will as well (in complexity terminology, there is a polynomial-time reduction from TSP to Sudoku). This is because Sudoku is NP-complete and TSP is in NP. But as is usually the case in this area, looking at the details of the reduction isn't particularly illuminating. If you want, you can piece it together by using the simple reduction from Latin square completion to Sudoku here, the reduction from triangulating uniform tripartite graphs to Latin square completion here, the reduction from 3SAT to triangulation here, and a formulation of TSP as a 3SAT problem. However, if you want to understand the idea behind reducing from Sudoku to TSP I think you would be better off studying Cook's theorem (showing that SAT is NP-complete) and a couple of simple reductions from 3SAT (e.g. to 3-dimensional matching) and being satisfied in the knowledge that the TSP-Sudoku reduction is just the same kind of thing but longer and more fiddly.
answered 2 days ago
rlmsrlms
32114
New contributor
rlms is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 2 days ago
rlmsrlms
32114
New contributor
rlms is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 2 days ago
rlmsrlms
32114
answered 2 days ago
rlmsrlms
32114
32114
New contributor
rlms is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
rlms is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
rlms is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
add a comment |
Chakrapani N Rao is a new contributor. Be nice, and check out our Code of Conduct.
Chakrapani N Rao is a new contributor. Be nice, and check out our Code of Conduct.
Chakrapani N Rao is a new contributor. Be nice, and check out our Code of Conduct.
Chakrapani N Rao is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
This paper claims to give a constructive reduction from Sudoku to Hamiltonian cycle problem: sciencedirect.com/science/article/pii/S097286001630038X
$endgroup$
– C. Windolf
2 days ago
$begingroup$
@C.Windolf The question is asking for the other direction. (Indeed, there's a deleted answer that made the same mistake and cited the same paper.)
$endgroup$
– David Richerby
2 days ago