I want to summarize some problem on Complexity. Which of them can be solved in poly-time?
I) finding maximal sub complete graph of given graph = Clique Problem
II) select some elements among n objects in which value and weights
are given, such that sum of weights of selected elements is not bigger
than an specific bound and sum of value being maximum
III) finding all cycles of a graph
IV) Finding a path that visit each vertex exactly once = Determine a graph is Hamiltonian
I think IV is Hamiltonian path that is NP-Complete, III is NP-Hard and NP-Complete, II is NP-Complete, and I is NP-Complete. so 0 of these solved in poly-time.
Who can more clearer me about NP-Hard and NP-Complete of these problem in a nice way? Am I right?
As you've noted, parts (1), (2), and (4) are all famous NP-hard problems (max clique, knapsack, and Hamiltonian path). These problems are not in NP, though, because NP consists of decision problems (questions for which the answer is either "yes" or "no") and these are not decision problems.
Part (3) is more nuanced. This problem is a counting problem - the goal is to determine how many objects of some type exist - rather than a decision problem, so it can't be in NP. To the best of my knowledge, it's not really known how hard this problem is. It's known that if it can be solved in polynomial time, then P = NP (see this link for details), and the specific proof shows that it's NP-hard as well.
If P ≠ NP, then none of these can be solved in polynomial time. If any of these can be solved in polynomial time, then P = NP. They are all NP-hard.
Hope this helps!
Because I got asked about a reference, I am posting my comment as the answer:
II) select some elements among n objects in which value and weights
are given, such that sum of weights of selected elements is not bigger
than an specific bound and sum of value being maximum
This is a knapsack problem which is poly-time if weights are not a part of input size, i.e the solution is polynomial in terms of n only.
It runs in O(n * W) where W is the maximum allowed weight. Of course this can be not polynomial if W is related to n, for example if W = 2^n.
You can read about it here:
http://en.wikipedia.org/wiki/Knapsack_problem#Dynamic_programming_in_advance_algorithm
http://www.personal.kent.edu/~rmuhamma/Algorithms/MyAlgorithms/Dynamic/knapsackdyn.htm
Related
Few definitions first. The co-NP problem is a decision problem where the answer "NO" can be verified in polynomial time. The perfect matching in a bipartite graph is a set of pairs of nodes (a pair is an edge in the graph) and where every node occurs in this set exactly once.
I am given an n x n bipartite graph, and I am trying to find out if the problem of finding whether k different perfect matchings exist in the graph, where k= polynomial(n), is a co-NP problem.
Work done so far
To initially simplify the problem, I believe that if k=2, then this is a co-NP problem. I think this is true, because the bipartite graph does not have 2 different perfect matchings, if there does not exist an exchange of neighbors between 2 nodes. I define the exchange of neighbors as the following. Let G1 be the first set in the graph, and G2 be the second set in the graph. The exchange occurs when we have a subset of G1, S1={A,B}, and a second subset of G2, S2={X,Y}, where {(A,X),(A,Y),(B,X),(B,Y)} belongs to the set of edges E. I call it exchange because if A was initially matched with X, and B with Y, then when A gets paired with Y, and B with X, A and B have exchanged their neighbors. I believe that the only way to have 2 different perfect matchings is to have at least one such exchange.
Now, we can verify that no such exchange exist in polynomial time. This is true since getting all the possible subsets S1 and S2 has O(n^4) time complexity. This because we need (n choose 2) from G1 multiplied by (n choose 2) from G2, and this gives us an upper bound of n^4.
I am not sure if this is a co-NP problem, but it is NP for certain. I think you have a little mixed up the definition of "verifying an answer". In complexity theory verify an answer means that you provide a certificate that proves that your answer is correct, and such certificate may be checked (verified) in polynomial time.
For example, in the case of your problem, if you have a set k different perfect matchings, that will be a good certificate, verifying it means checking that it is indeed a set of perfect matchings in your input graph. You can check this in polynomial time by checking that all edges are in you graph and in each matching no two edges share a vertex, and all of them are different. Since the number of edges in a matching is linear, then verifying each matching can be done in polynomial time, then, since k is polynomial, we verify that property for all matchings also in polynomial time. Finaly, checking that all are different can be done in k square times something polynomial on n, yielding a polynomial complexity. So yes, your problem can be verified in polynomial time, and thus it is in NP.
Now, if you can find such certificate in polynomial time that will be proof enough that you problem is in P, and all problems in P are in NP and in co-NP. So I see two possible ways to solve this, you may prove that your problem is in P, that will yield a yes answer to your question, or you may prove that your problem is NP-complete, that will prove that your answer is no, since all NP-complete problems are not in co-NP (unless P = NP).
Any other way of proving that your problem is or is not in co-NP, might be very difficult and confusing, in fact the work you have done so far was moving towards proving that you can decide negative cases in polynomial time which is a different thing as verifying them, that would prove that it is co-NP, but because you proved that it is in P.
We know that the minimum vertex cover is NP complete, which means that it is in the set of problems that can be verified in polynomial time.
As I understand it, the verification process would require the following:
Verify that the solution is a vertex cover at all
Verify that the solution is the smallest possible subset of the source graph that satisfies condition #1
I'm finding it hard to establish that step #2 can be done in polynomial time. Can anyone explain how it is?
The minimum vertex cover is NP-hard. It is only NP-complete if it is restated as a decision problem which can be verified in polynomial time.
The minimum vertex cover problem is the optimization problem of finding a smallest vertex cover in a given graph.
INSTANCE: Graph G
OUTPUT: Smallest number k such that G has a vertex cover of size k.
If the problem is stated as a decision problem, it is called the vertex cover problem:
INSTANCE: Graph G and positive integer k.
QUESTION: Does G have a vertex cover of size at most k?
Restating a problem as a decision problem is a common way to make problems NP-complete. Basically you turn an open-ended problem of the form "find the smallest solution k" into a yes/no question, "for a given k, does a solution exist?"
For example, for the travelling salesman problem, verifying that a proposed solution the shortest path between all cities is NP-hard. But if the problem is restated as only having to find a solution shorter than k total distance for some k, then verifying a solution is easy. You just find the length of the proposed solution and check that it's less than k.
The decision problem formulation can be easily used to solve the general formulation. To find the shortest path all you have to do is ratchet down the value of k until there are no solutions found.
I have problem with scheduling. I need to prove that the problem is NP complete. What can be the methods to prove it NP complete?
To show a problem is NP complete, you need to:
Show it is in NP
In other words, given some information C, you can create a polynomial time algorithm V that will verify for every possible input X whether X is in your domain or not.
Example
Prove that the problem of vertex covers (that is, for some graph G, does it have a vertex cover set of size k such that every edge in G has at least one vertex in the cover set?) is in NP:
our input X is some graph G and some number k (this is from the problem definition)
Take our information C to be "any possible subset of vertices in graph G of size k"
Then we can write an algorithm V that, given G, k and C, will return whether that set of vertices is a vertex cover for G or not, in polynomial time.
Then for every graph G, if there exists some "possible subset of vertices in G of size k" which is a vertex cover, then G is in NP.
Note that we do not need to find C in polynomial time. If we could, the problem would be in `P.
Note that algorithm V should work for every G, for some C. For every input there should exist information that could help us verify whether the input is in the problem domain or not. That is, there should not be an input where the information doesn't exist.
Prove it is NP Hard
This involves getting a known NP-complete problem like SAT, the set of boolean expressions in the form:
(A or B or C) and (D or E or F) and ...
where the expression is satisfiable, that is there exists some setting for these booleans, which makes the expression true.
Then reduce the NP-complete problem to your problem in polynomial time.
That is, given some input X for SAT (or whatever NP-complete problem you are using), create some input Y for your problem, such that X is in SAT if and only if Y is in your problem. The function f : X -> Y must run in polynomial time.
In the example above, the input Y would be the graph G and the size of the vertex cover k.
For a full proof, you'd have to prove both:
that X is in SAT => Y in your problem
and Y in your problem => X in SAT.
marcog's answer has a link with several other NP-complete problems you could reduce to your problem.
Footnote: In step 2 (Prove it is NP-hard), reducing another NP-hard (not necessarily NP-complete) problem to the current problem will do, since NP-complete problems are a subset of NP-hard problems (that are also in NP).
You need to reduce an NP-Complete problem to the problem you have. If the reduction can be done in polynomial time then you have proven that your problem is NP-complete, if the problem is already in NP, because:
It is not easier than the NP-complete problem, since it can be reduced to it in polynomial time which makes the problem NP-Hard.
See the end of http://www.ics.uci.edu/~eppstein/161/960312.html for more.
In order to prove that a problem L is NP-complete, we need to do the following steps:
Prove your problem L belongs to NP (that is that given a solution you can verify it in polynomial time)
Select a known NP-complete problem L'
Describe an algorithm f that transforms L' into L
Prove that your algorithm is correct (formally: x ∈ L' if and only if f(x) ∈ L )
Prove that algo f runs in polynomial time
First, you show that it lies in NP at all.
Then you find another problem that you already know is NP complete and show how you polynomially reduce NP Hard problem to your problem.
Get familiar to a subset of NP Complete problems
Prove NP Hardness : Reduce an arbitrary instance of an NP complete problem to an instance of your problem. This is the biggest piece of a pie and where the familiarity with NP Complete problems pays. The reduction will be more or less difficult depending on the NP Complete problem you choose.
Prove that your problem is in NP : design an algorithm which can verify in polynomial time whether an instance is a solution.
"Prove that it is NP-Complete to determine given input G and k whether G has both a clique of size k and an independent set of size k. Note that this is 1 problem, not 2; the answer is yes if and only if G has both of these subsets."
We were given this problem in my algorithms course and a large group of students could not figure it out. Here is what we have so far...
We know that both the clique and independent set problems are NP-Complete in of themselves. We also know that the verification of this problem, given some "certificate" is in NP.
The problem is somehow performing a reduction on the above problem (which contains both independent sets and cliques) to either a problem consisting entirely of cliques or independent sets (at least that's what we think we need to do). We don't know how to perform this reduction without losing information needed to reduce the reduction back to its original form.
Hint: Reduce CLIQUE to this problem, by adding some vertices.
Thanks to "Moron" and "Rafal Dowgird" for the hints! Based on that I think I've got a solution. Please correct me if I am incorrect:
Since we already know the the clique and independent-set problems are NP-Complete, we can use that as a foundation for proving our problem. Let's call our problem the Combination Clique Independent Set problem (CCIS).
Suppose we are given a graph G which has a clique C of size k. We can reduce this graph into a graph G' (read: G prime) which has both a clique C' of size k' and independent-set I of size k' by attaching k vertices to each vertex in C. This reduction occurs in polynomial time since the addition of the vertices takes O(n*k) time (n vertices in the graph and k vertices attached to each node).
Note that C=C' and k=k'.
Now suppose we are given a graph G' which has a clique C' of size k' and independent-set I of size k' which is determined to be true. The reduction to the clique problem is trivial since we don't need to modify the graph at all to find only a clique.
Some months ago there was a nice question regarding a "1:n matching problem" and there seems to be no poly-time algorithm.
I would like to add constraints to find a maximum matching for the 1:n matching problem with a polynomial algorithm. I would like to say: "For vertex A1 choose either {B1,B2,B5} or {B2,B3} if the vertices are not already taken from another A-vertex" i.e. I would not allow all possible combinations.
This could be expressed if we introduce helper vertices H for each choice and substitute edges with trees => we get a problem similar to the ordinary bipartite matching. Every vertex of A or B can have only one edge in the matching. The edges to or from vertices in H are either all in the matching or none of them is present in the matching. Imagine the following tri-partite graph:
Now define h_ij="tree rooted that contains H_ij" to express the matching easily:
Then in the example M={h12,h22} would be one 'maximum' matching, although not all vertices from B are involved
The set {h12,h23} is not a matching because then B3 would have be choosen twice.
Would this problem then be solvable in polynomial time? If yes, is there a polytime solution for the weighted (w(h_ij)) variant? If no, could you argue or even proof it for a "simple-man" like me or suggest other constraints to solve the 1:n matching problem?
E.g. could the graph transformed to a general graph which then could be solved with the weighted matching for general graphs? Or could branchings or even matching forests help here?
PS: not a homework ;-)
There is a difference between maximal and maximum. I have assumed you meant maximum for the below writeup.
You don't seem to have defined your problem very clearly, but if I have understood your intent correctly, It seems like your problem is NP complete (and 'equivalent' to Set Packing).
We can assume that the allowed sets sizes is the same (k) for all A_i to find a [1:k] matching, as any other set size can be ignored. To find max k, we just run the algorithm for [1:k] for k = 1,2,3.. etc.
So your problem is (I think...):
Given m set families F_i = {S_1i, .., S_n(i)i} (|F_i| = size of F_i = n(i), need not be same as |F_j|), each set of size k, you have to find one set from each family (say S_i) such that
S_i and S_j are disjoint for any i neq j.
number of S_i's is maximum.
We can show that it is NP-Complete for k=3 in two steps:
The NP-Complete problem Set Packing can be reduced it. This shows that it is NP-Hard.
Your problem is in NP and can be reduced to Set Packing. This and 1) implies your problem is NP-Complete. It also helps you leverage any approximation/randomized algorithms already existing for Set-Packing.
Set Packing is the problem:
Given n sets S_1, S_2, ..., S_n, find the maximum number of pairwise disjoint sets among these.
This problem remains NP-Complete even if |S_1| = |S_2| = ... = |S_n| = 3 and is called the 3-Set packing problem.
We will use this to show that your problem is NP-Hard, by providing an easy reduction from 3-Set packing to your problem.
Given S_1, S_2, .., S_n just form the families
F_i = {S_i}.
Now if your problem had a polynomial time solution, then we get a set of Sets {S_1, S_2, ..., S_r} such that
S_i and S_j are disjoint
Number of S_i is maximum.
This easy reduction gives us a solution to the 3-set Packing problem and thus your problem is NP-Hard.
To see that this problem is in NP, we reduce it to Set-Packing as follows:
Given F_i = {S_1i, S_2i, ..., S_ni}
we consider the sets T_ji = S_ji U {i} (i.e. we add an id of the family into the set itself) and run them through the Set-Packing algorithm. I will leave it to you to see why a solution to Set-Packing gives a solution to your problem.
For a maximal solution, all you need is a greedy algorithm. Just keep picking up sets till you can pick no more. This would be polynomial time.