how is Sudoku an np-complete problem? according to wiki, to be classed as an np-complete problem it must satisfy 2 conditions
problem must be in np
every other problem in np must be reducible to given problem in polynomial time
how is the second condition satisfied? can you give an example? for instance, I don't see any correlation between Sudoku problem and the travelling salesman problem or knapsack problem
(kindly forgive poor formatting as I'm typing this question on my mobile device)
NP-completeness of SUDOKU notes in part:
This result was first shown in this master’s thesis by reduction from
the NP-complete problem LATIN SQUARE COMPLETION. Sudoku wikipedia
page.
Here is how it works (simplified, without reference to
ASP-completeness, which I don’t cover in this course).
Suppose we have a n×n instance of LATIN SQUARE COMPLETION. We
construct a n2×n2 instance of SUDOKU, that encodes the instance of
LATIN SQUARE COMPLETION. Moreover, the encoding is very direct.
http://www-imai.is.s.u-tokyo.ac.jp/~yato/data2/MasterThesis.pdf being a link to the thesis in PDF.
Related
I'm having a bit of trouble understanding how one would verify if there is no solution to a given instance of the subset sum problem in polynomial time.
Of course you could easily verify the positive case: simply provide the list of integers which add up to the target sum and check they are all in the original set. (O(N))
How do you verify that the answer "false" is the correct one in polynomial time?
It’s actually not known how to do this - and indeed it’s conjectured that it’s not possible to do so!
The class NP consists of problems where “yes” instances can be verified in polynomial time. The subset sum problem is a canonical example of a problem in NP. Importantly, notice that the definition of NP says nothing about what happens if the answer is “no” - maybe it’ll be easy to show this, or maybe there isn’t an efficient algorithm for doing so.
A counterpart to NP is the class co-NP, which consists of problems where “no” instances can be verified in polynomial time. A canonical example of such a problem is the tautology problem - given a propositional logic formula, is it always true regardless of what values the variables are given? If the formula isn’t a tautology, it’s easy to verify this by having someone tell you how to assign the values to the variables such that the formula is false. But if the formula is always true, it’s unclear how you’d show this efficiently.
Just as the P = NP problem hasn’t been solved, the NP = co-NP problem is also open. We don’t know whether problems where “yes” answers have fast verification are the same as problems where “no” answers have fast verification.
What we do know is that if any NP-complete problem is in co-NP, then NP = co-NP. And since subset sum is NP-complete, there’s no known polynomial time algorithm to verify if the answer to a subset sum instance is “no.”
I've seen online that one can write the travelling salesman problem as a linear expression and compute it using software such as CPLEX for java.
I have a 1000 towns and need to find a short distance. I plan on partitioning these 1000 towns into clusters of ~100 towns and performing some linear programming algorithm on these individual clusters.
The question I have is, how exactly do I represent this as a linear expression.
So I have 100 towns and I'm sure everyone's aware of how TSP works.
I literally have no clue how I can write linear constraints, objectives and variables which satisfy the TSP.
Could someone explain to me how this is done or send me a link which explains it clearly, because I've been researching a lot and can't seem to find anything.
EDIT:
A bit of extra information I found:
We label the cities with numbers 0 to n and define the matrix:
Would this yield the following matrix for 5 towns?
The constraints are:
i) Each city be arrived at from exactly one other city
ii) From each city there is a departure to exactly one other city
iii) The route isn't broken up into separate islands.
Again, this makes complete sense to me, but I'm still having trouble writing these constraints as a linear expression. Apparently it's a simple enough matrix.
Thanks for any help !
According to this Wikipedia article the travelling salesman problem can be modelled as an integer linear program, which I believe to be the key issue of the question. The idea is to have decision variables of permitted values in {0,1} which model selected edges in the graph. Suitable constraints must ensure that the selected edges cover every node, the selected edges form a collection of cycles and there is only one connected component (which in total means that there is exactly one cycle which contains every node). Note that the article also gives an explicit formulation and explains the interpretations of the constraints.
As the travelling salesman problem is NP-hard, it cannot be solved via (non-integral) linear programming unless P=NP.
The TSP problem is a rather complex integer programming problem due to its combinatorial nature.
There are several (exact and approximated) techniques to solve it. The model in Wikipedia is just one of them: it has constraints to ensure there is only one incoming and outgoing arc in each node and the constraints with u variables are for preventing sub-cycles in the solution. There is also several ways to write this sub-cycle preventing constraints, some of them can be found on section 4.1 of this article. The section presents some models for the TSP problem (all of them can be solved using CPLEX, Gurobi, MATLAB or other Integer/Linear Programming Software.
Hope I could be of any help. (=
What is the most accurated algorithm to Knapsack when weight and values are positive?
not sure what language your using but Wikipedia has a great page on information and algorithms to solve it. If you want more sample code to understand how to do it, check this site out: http://rosettacode.org/wiki/Knapsack_problem/Unbounded/Python_dynamic_programming (this is all in python but there's more languages).
Basically it depends on what your doing but the most common way to solve this is dynamic programming.
I read the article on wikipedia but could not understand what exactly are NP problems. Can anyone tell me about them and also what is relation of them with P Problems?
NP problems are problems that given a proposed solution, you can verify the solution in a polynomial time. For example, if you have a list of University courses and need to create a schedule so that courses won't conflict, it would be a really difficult task (complexity-wise). However, given a proposed schedule, you can easily verify its correctness.
Another important example from the field of encryption: given a number which is the result of multiplying two very large prime numbers, it's very difficult to find those primes based only on the result. However, given two numbers, it's very easy to check the solution (multiply them, compare).
I have intentionally chose examples that are in NP and not in P (i.e. problem that are hard to find the solution for) so you can understand the difference. All problems that are easy to solve, are also easy to verify - just solve and compare. That is, P is a subset of NP.
Not really an answer, because Piccolo's link is more useful, but a HP researcher claims having proven P != NP, here is the paper.
www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp12pt.pdf
It was not accepted yet, but I wish him good luck for the 1M$.
This problem comes out of the dynamic programming chapter in The Algorithm Deisgn Manual by Skiena.
Give an algorithm to determine whether you can generate a given string by pasting cutouts from a magazine. You are given a function that will identify the character and its position on the reverse side of the page for any given character position.
I solved this with with backtracking, but since it's in the dynamic programming chapter I think there must be a recurrence I can't figure out. Can anyone give me a hint?
You can solve it with maximum bipartite matching.
Each character L of the given string forms the left set. (Note, you repeat the characters if the string has repeated characters).
Each pair of characters (R1,R2) of the magazine forms the right set.
L is connected to (r1,r2) iff L=R1 or L=R2.
Find a maximum matching in the resulting graph. If all left vertices are part of the matching, you have the answer. If not, such a string is not possible.
See Maximum Bipartite Matchings for an algorithm.
Not sure if this is optimal though and sorry for not answering exactly as asked.
If you have a recursive backtracking solution, you may be able to apply memoization, which is one way to do dynamic programming.