Minimum weight hamiltonian path on a weighted (0 and 1) tournament graph

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Suppose we have a weighted tournament graph. (A directed graph in which every pair of distinct vertices is connected by a single directed edge.)



The weights are constrained to be 0 and 1.



I know that every tournament contains atleast one hamiltonian path and can be found in time complexity $O(n^2)$.



However, I have two questions:



a) Is there a way to find the minimum weight hamiltonian path if we know that all weights are constrained to be either 0 or 1?



b) Is there an efficient algorithm to find ALL hamiltonian paths in a tournament graph?? This would solve a) automatically if true.




combinatorics graph-theory computer-science




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  • Cross-posted on Computer Science: cs.stackexchange.com/questions/95616/….
    – Yuval Filmus
    Jul 25 at 20:21














up vote
0
down vote

favorite












Suppose we have a weighted tournament graph. (A directed graph in which every pair of distinct vertices is connected by a single directed edge.)



The weights are constrained to be 0 and 1.



I know that every tournament contains atleast one hamiltonian path and can be found in time complexity $O(n^2)$.



However, I have two questions:



a) Is there a way to find the minimum weight hamiltonian path if we know that all weights are constrained to be either 0 or 1?



b) Is there an efficient algorithm to find ALL hamiltonian paths in a tournament graph?? This would solve a) automatically if true.




combinatorics graph-theory computer-science




share|cite|improve this question



















  • Cross-posted on Computer Science: cs.stackexchange.com/questions/95616/….
    – Yuval Filmus
    Jul 25 at 20:21












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Suppose we have a weighted tournament graph. (A directed graph in which every pair of distinct vertices is connected by a single directed edge.)



The weights are constrained to be 0 and 1.



I know that every tournament contains atleast one hamiltonian path and can be found in time complexity $O(n^2)$.



However, I have two questions:



a) Is there a way to find the minimum weight hamiltonian path if we know that all weights are constrained to be either 0 or 1?



b) Is there an efficient algorithm to find ALL hamiltonian paths in a tournament graph?? This would solve a) automatically if true.




combinatorics graph-theory computer-science




share|cite|improve this question











Suppose we have a weighted tournament graph. (A directed graph in which every pair of distinct vertices is connected by a single directed edge.)



The weights are constrained to be 0 and 1.



I know that every tournament contains atleast one hamiltonian path and can be found in time complexity $O(n^2)$.



However, I have two questions:



a) Is there a way to find the minimum weight hamiltonian path if we know that all weights are constrained to be either 0 or 1?



b) Is there an efficient algorithm to find ALL hamiltonian paths in a tournament graph?? This would solve a) automatically if true.





combinatorics graph-theory computer-science





share|cite|improve this question










share|cite|improve this question




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asked Jul 25 at 16:28









Vinayak Suresh

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  • Cross-posted on Computer Science: cs.stackexchange.com/questions/95616/….
    – Yuval Filmus
    Jul 25 at 20:21
















  • Cross-posted on Computer Science: cs.stackexchange.com/questions/95616/….
    – Yuval Filmus
    Jul 25 at 20:21















Cross-posted on Computer Science: cs.stackexchange.com/questions/95616/….
– Yuval Filmus
Jul 25 at 20:21




Cross-posted on Computer Science: cs.stackexchange.com/questions/95616/….
– Yuval Filmus
Jul 25 at 20:21















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