Questions tagged [np-complete]

Questions about the hardest problems in NP, i.e. of those that can be solved in polynomial time by nondeterministic Turing machines.

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37
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2answers
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How do I construct reductions between problems to prove a problem is NP-complete?

I am taking a complexity course and I am having trouble with coming up with reductions between NPC problems. How can I find reductions between problems? Is there a general trick that I can use? How ...
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6answers
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Dealing with intractability: NP-complete problems

Assume that I am a programmer and I have an NP-complete problem that I need to solve it. What methods are available to deal with NPC problems? Is there a survey or something similar on this topic?
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Are there subexponential-time algorithms for NP-complete problems?

Are there NP-complete problems which have proven subexponential-time algorithms? I am asking for the general case inputs, I am not talking about tractable special cases here. By sub-exponential, I ...
29
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5answers
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"NP-complete" optimization problems

I am slightly confused by some terminology I have encountered regarding the complexity of optimization problems. In an algorithms class, I had the large parsimony problem described as NP-complete. ...
28
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1answer
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Is the k-clique problem NP-complete?

In this Wikipedia article about the Clique problem in graph theory it states in the beginning that the problem of finding a clique of size K, in a graph G is NP-complete: Cliques have also been ...
20
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2answers
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Can one show NP-hardness by Turing reductions?

In the paper Complexity of the Frobenius Problem by Ramírez-Alfonsín, a problem was proved to be NP-complete using Turing reductions. Is that possible? How exactly? I thought this was only possible by ...
7
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1answer
5k views

Find a 3-colouring using the 3-colourability decision problem

I was learning about NP problems. I read that for many problems, like Clique, we can easily convert its decision problem to derive a solution of search problem. (For Clique problem, you only need to ...
62
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3answers
29k views

Knapsack problem -- NP-complete despite dynamic programming solution?

Knapsack problems are easily solved by dynamic programming. Dynamic programming runs in polynomial time; that is why we do it, right? I have read it is actually an NP-complete problem, though, which ...
21
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3answers
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If P = NP, why wouldn't $\emptyset$ and $\Sigma^*$ be NP-complete?

Apparently, if ${\sf P}={\sf NP}$, all languages in ${\sf P}$ except for $\emptyset$ and $\Sigma^*$ would be ${\sf NP}$-complete. Why these two languages in particular? Can't we reduce any other ...
3
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1answer
2k views

Decisional problems vs Optimization problems : NP-COMPLETE vs NP-HARD

I'm studying some of computation theory and i encounter a big question mark. I have an optimization problem and i have to proof that is NP-HARD. I know that my problem can be reduced to another np-...
36
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5answers
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How can I verify a solution to Travelling Salesman Problem in polynomial time?

So, TSP (Travelling salesman problem) decision problem is NP complete. But I do not understand how I can verify that a given solution to TSP is in fact optimal in polynomial time, given that there is ...
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5answers
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How can I reduce Subset Sum to Partition?

Maybe this is quite simple but I have some trouble to get this reduction. I want to reduce Subset Sum to Partition but at this time I don't see the relation! Is it possible to reduce this problem ...
27
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3answers
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Teaching NP-completeness - Turing reductions vs Karp reductions

I'm interested in the question of how best to teach NP-completeness to computer science majors. In particular, should we teach it using Karp reductions or using Turing reductions? I feel that the ...
14
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Prove NP-completeness of deciding satisfiability of monotone boolean formula

I am trying to solve this problem and I am really struggling. A monotone boolean formula is a formula in propositional logic where all the literals are positive. For example, $\qquad (x_1 \lor x_2) ...
29
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1answer
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Is regex golf NP-Complete?

As seen in this recent XKCD strip and this recent blog post from Peter Norvig (and a Slashdot story featuring the latter), "regex golf" (which might better be called the regular expression separation ...
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1answer
785 views

If a problem is Cook-NP hard, and this problem is in NP, does it prove that the problem is Karp-NP-complete?

I got recently confused when I noticed that there exists 2 types of polytime reductions, which lead to 2 different concepts of NP-hardness. This is well explained in the answer to this question: Can ...
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1answer
174 views

If a language is X-complete, is its complement is X-complete as well?

I'm looking for an information about closure of complexity complete classes. Is it true that any language, if the language is X-complete, then its complement is X-complete? Why? I was thinking ...
15
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Types of reductions and associated definitions of hardness

Let A be reducible to B, i.e., $A \leq B$. Hence, the Turing machine accepting $A$ has access to an oracle for $B$. Let the Turing machine accepting $A$ be $M_{A}$ and the oracle for $B$ be $O_{B}$. ...
18
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2answers
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A polynomial reduction from any NP-complete problem to bounded PCP

Text books everywhere assume that the Bounded Post Correspondence Problem is NP-complete (no more than $N$ indexes allowed with repetitions). However, nowhere is one shown a simple (as in, something ...
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2answers
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Reducing the integer factorization problem to an NP-Complete problem

I'm struggling to understand the relationship between NP-Intermediate and NP-Complete. I know that if P != NP based on Ladner's Theorem there exists a class of languages in NP but not in P or in NP-...
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2answers
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Do any decision problems exist outside NP and NP-Hard?

This question asks about NP-hard problems that are not NP-complete. I'm wondering if there exist any decision problems that are neither NP nor NP-hard. In order to be in NP, problems have to have a ...
21
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3answers
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HALF CLIQUE - NP Complete Problem

Let me start off by noting this is a homework problem, please provide only advice and related observations, NO DIRECT ANSWERS please. With that said, here is the problem I am looking at: Let HALF-...
13
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1answer
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Finding optimal sequence of questions to minimize total student time

Suppose there is a tutorial session at a university. We have a set of $k$ questions $Q = \{ q_1 \ldots q_k \}$ and a set of $n$ students $S = \{ s_1 \ldots s_n \}$. Each student has a doubt in a ...
9
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1answer
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Why is the clique problem NP-complete? [duplicate]

Possible Duplicate: Is the k-clique problem NP-complete? I've been lately reading about the clique problem, specifically, the variety of the clique problem of deciding whether a given graph $G$ ...
4
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1answer
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Is finding the longest path of a graph NP-complete?

The problem of finding the largest subgraph of a graph that has a Hamiltonian path can be restated as finding the longest path of a graph. Is this NP-complete? Also, is finding the $k$-length path of ...
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2answers
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Modeling the problem of finding all stable sets of an argumentation framework as SAT

As a continuation of my previous question i will try to explain my problem and how i am trying to convert my algorithm to a problem that can be expressed in a CNF form. Problem: Find all stable sets ...
3
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2answers
760 views

Is this problem NP-hard? Maximizing selected sets so that their union is less than k?

There is an NP-hard problem called Minimum k-Union where we are given a set system with $n$ sets and are asked to select $k$ sets in order to minimize the size of their union. I'm currently ...
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1answer
434 views

Is selecting some of the sets such that their intersection is of size k NP-hard?

Given collection of sets S1,...,Sn and a positive number k. Is the problem of selecting some of the sets such that their intersection is of size k NP-hard? So far, I tend to believe it is so I tried ...
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5answers
225 views

NP-hard or not: partition with irrational input or parameter

See some related questions in Cont: NP-hard or not: partition with irrational input or parameter Given a set $N=\{a_1,...,a_{n}\}$ with $n$ positive numbers and $\sum_i a_i=1$, find a subset $S\...
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4answers
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Does the complexity of strongly NP-hard or -complete problems change when their input is unary encoded?

Does the difficulty of a strongly NP-hard or NP-complete problem (as e.g. defined here) change when its input is unary instead of binary encoded? What difference does it make if the input of a ...
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2answers
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How is the Subset Sum Problem NP-Complete?

You can't find a solution online for it that doesn't run in polynomial time complexity, when using dynamic programming. Have all these sites secretly solved P=NP, and no one knows about it?
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3answers
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Reduce Vertex Cover with size k to Vertex Cover with size n/2

Disclaimer: This is a homework question. I would like to reduce vertex cover problem to the following problem: $$L = \{G \mid G\text{ has a vertex cover of size } |V(G)|/2\}\,.$$ I have divided the ...
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3answers
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Why is the class NP-Complete important compared to NP-hard?

I'm studying computational complexity and I was wondering why the NP-Complete (NPC) problems is an important class at all. I find it obvious why we're interested in showing a given NP problem is NP-...
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2answers
881 views

Why doesn't Godel's Second Incompleteness Theorem rule out a formalizable proof of P!=NP?

I'm sure there must be something wrong with the following reasoning because otherwise a lot of P vs. NP research would be curtailed but I cannot determine my error: For any fixed integer $k>0$ ...
14
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3answers
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Is 2-SAT with XOR-relations NP-complete?

I'm wondering if there is a polynomial algorithm for "2-SAT with XOR-relations". Both 2-SAT and XOR-SAT are in P, but is its combination? Example Input: 2-SAT part: ...
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2answers
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Can we construct a Karp reduction from a Cook reduction between NP problems?

We have had several questions about the relation of Cook and Karp reductions. It's clear that Cook reductions (polynomial-time Turing reductions) do not define the same notion of NP-completeness as ...
9
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1answer
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Negative numbers in Subset-Sum

If I have a set $A$ with positive and negative numbers, and a number to find C. It is possible to reduce the problem to one with only positive numbers in set $A$? I mean, it is possible to find a ...
8
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1answer
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3-SAT for variables appearing 3 times

I've been trying to investigate 3-SAT for variables appearing 3 times and so far I'm getting some mixed answers as to its complexity. For example, https://people.maths.ox.ac.uk/scott/Papers/...
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Is there an efficient test for if an NFA accepts a subset of another NFA?

So, I know that testing if a regular language $R$ is a subset of regular language $S$ is decidable, since we can convert them both to DFAs, compute $R \cap \bar{S}$, and then test if this language is ...
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2answers
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Relationship between Independent Set and Vertex Cover

Directly from Wikipedia, a set of vertices $X \subseteq V(G)$ of a graph $G$ is independent if and only if its complement $V(G) \setminus X$ is a vertex cover. Does this imply that the complement of ...
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4answers
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What is an instance of NP complete problem?

I don't understand this definition of an "instance" of a problem. Quoting from the CLRS book on page 1054 on abstract problems (Chapter 34.1): We define an abstract problem $Q$ to be a binary ...
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3-sat to 2-sat reduction

It is known that 3-SAT belong to - NP-Complete complexity problems, while 2-SAT belong to P as there is known polynomial solution to it. So you can state that there is no such reduction from 3-SAT to ...
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2answers
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Why does Schaefer's theorem not prove that P=NP?

This is probably a stupid question, but I just don't understand. In another question they came up with Schaefer's dichotomy theorem. To me it looks like it proves that every CSP problem is either in P ...
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3answers
139 views

Spanning tree whose sum of edge weights are between two boundries

I saw this problem: $\langle G,w,k_1,k_2 \rangle \in L$ iff Graph $G$ has a spanning tree whose sum of edge wights are less than $k_2$ and greater than $k_1$. The problem says that we can prove this ...
14
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1answer
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Why do Shaefer's and Mahaney's Theorems not imply P = NP?

I'm sure someone has thought about this before or immediately dismissed it, but why does Schaefer's dichotomy theory along with Mahaney's theorem on sparse sets not imply P = NP ? Here's my ...
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3answers
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All NP problems reduce to NP-complete problems: so how can NP problems not be NP-complete?

My book states this If a decision problem B is in P and A reduces to B, then decision problem A is in P. A decision problem B is NP-complete if B is in NP and for every problem in A in ...
5
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1answer
697 views

NP-complete decision problems - how close can we come to a solution?

After we prove that a certain optimization problem is NP-hard, the natural next step is to look for a polynomial algorithm that comes close to the optimal solution - preferrably with a constant ...
4
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1answer
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Proving NP Completeness of a subset-sum problem - how?

So I'm trying to understand P/NPC problems. The one I'm trying to tackle now is subset sum (we have a collection of integers $S$ and a $k$ param: is there a subset of $S$ that sum of all it's elements ...
4
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1answer
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If P = NP, why does P = NP = NP-Complete? [duplicate]

If P = NP, why does P = NP also then equal NP-Complete? I.e. Why would it then be the case that ...
2
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2answers
552 views

Reduce subset sum to 3SAT

How to do it? I'm not asking the solution for the proof of why subset sum is NPC, but rather the opposite reduction