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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Proving that a problem is in NP

I have an assignment in which the problem, $D$, is simple but, once found, easy to check. Is it enough to prove that a solution $x$ can be checked in polynomial time to prove that $D \in NP$? (The ...
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3-SAT where variables occur equally many times as a positive literal and as a negative literal

Let $\phi$ be a 3-CNF formula over variables $x_1,x_2,\ldots,x_n$. Every variable $x_i$, $i \in [n]$, occurs equally many times as a positive literal and as a negative literal in $\phi$. Is it NP-...
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2 Dimensional Subset Sum: looking for information

I do not know if this problems exists with a different name, if it is, I could not find it. The problem is this: Given a set $S$ of $n$ points in $\mathbb{Z}^2$, is there a subset $A\subset S$ ...
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NP-hardness of an optimization problem with real value

I have an optimization problem, whose answer is a real value, not an integer such as vertex cover and set cover. Therefore, the decision version of my problem is given an input and a real value $r$. ...
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Can't understand why the DP Subset Sum algorithm is not polynomial

I can not understand why the dynamic programming algorithm for the Subset Sum, is not polynomial. Even though the sum to find 'T' is greater than the total sum of the 'n' elements of the set , the ...
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How can I prove DP completeness?

We defined the class $\text{DP}$ like this: $$\text{DP} := \{ A \setminus B : A, B \in \text{NP} \}$$ We say a problem $P$ is $\text{DP}$ complete iff $P \in \text{DP}$ and $X \leq P \forall X \in \...
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Questions on Graph and Hamiltonian [closed]

From this book and other study in complexity theory, I have seen the following statement: The definition of NP is not symmetric with respect to yes-instances and no-instances. For example, it is ...
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When is splitting a collection coins two ways NP-complete?

Suppose we have a set $D$ of denominations of coins and a our input is a "tip jar" containing some finite number of these coins (e.g., five nickels, a dime and three quarters). In the first two ...
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Reducing from Hamiltonian Cycle to Subgraph Isomorphism

I am reading Subgraph isomorphism problem I am having trouble understanding how they prove that the subgraph isomorphism problem is NP-Complete using the Hamiltonian cycles problem in the article. ...
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Proving that finding wheel subgraphs is NP-complete

Can you help me with this problem ? Given an undirected graph $G$ and an integer $n$, prove that determining whether the graph has wheel on $n$ vertices $W_{n}$ (a wheel $W_{i}$ is such that $i$ ...
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Reducing from Hamiltonian Cycle problem to the Graph Wheel problem [duplicate]

EDIT: This question is different from the other in a sense that unlike it this one goes into specifics and is intended to solve the problem. In the previous post, the only answer was a hint. In this ...
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NP-completeness of graph isomorphism through edge contractions with an edge validity condition

Given Graphs $G=(V_1,E_1)$ and $H=(V_2,E_2)$. Can a graph isomorphic to $H$ be obtained from $G$ by a sequence of edge contractions ? We know this problem is NP-complete. What about if only a subset ...
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Karp hardness of searching for a matching cut

Follow-up question in the series: Karp hardness of searching for a matching erosion Karp hardness of searching for a matching split Maximum Matching Cut problem Input: An undirected graph $G(V, E)$ ...
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Karp hardness of searching for a matching erosion

First, read the previous question: Karp hardness of searching for a matching cut As mentioned in the supposed-to-be-comment answer in that question, without the requirement of cardinality $k$, the ...
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Topological sorting and NP-hard proof

I meet a problem. I can find a sub-optimal solution, but cannot find an optimal one and cannot prove its NPC hardness. The problem can also be described as follows. Given a sequence $X=\{x_1,x_2,...,...
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Karp hardness of an equidistant vertex set

What is the hardness of the following problem? Input: An undirected graph $G(V, E)$ and a natural number $k$ Output: YES if $G$ has an equidistant vertex set of size $k$, otherwise NO $\...
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How do I prove that SAT in coNP implies NP=coNP?

Is it true that if SAT is in coNP then its also coNP-complete (because it is NP-complete)?
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Decision Problem vs. Optimization Problem [duplicate]

Is the following statement correct? If a decision problem is NP-complete, the corresponding optimization problem can not be solved in polynomial time.
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Is the weighted transitive reduction problem NP-hard?

The transitive reduction problem is to find the graph with the smallest number of edges such that $G^t = (V,E^t)$ has the same reachability as $G=(V,E)$. When $E^t \subseteq E$ it is NP-complete. ...
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Is legislation NP-complete?

I would like to know if there has been any work relating legal code to complexity. In particular, suppose we have the decision problem "Given this law book and this particular set of circumstances, is ...
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Are there NP problems, not in P and not NP Complete?

Are there any known problems in $\mathsf{NP}$ (and not in $\mathsf{P}$) that aren't $\mathsf{NP}$ Complete? My understanding is that there are no currently known problems where this is the case, but ...
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are NP Complete languages closed under any regular operations?

I have tried looking online, but I couldn't find any definitive statements. It would make sense to me that Union and Intersection of two NPC languages would produce a language not necessarily in NPC. ...
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Rule of thumb to know if a problem could be NP-complete

This question was inspired by a comment on StackOverflow. Apart from knowing NP-complete problems of the Garey Johnson book, and many others; is there a rule of thumb to know if a problem looks like ...
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NP completeness proof of a spanning tree problem

I am looking for some hints in a question asked by my instructor. So I just figured out this decision problem is $\sf{NP\text{-}complete}$: In a graph $G$, is there a spanning tree in $G$ that ...
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Grid covering by rectangles

We have a $N_1 \times N_2$ grid. We have a collection of rectangles on this grid, each rectangle can be represented as a $N_1$-by-$N_2$ binary matrix $R$. We want to cover the grid with those ...
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Proving DOUBLE-SAT is NP-complete

The well known SAT problem is defined here for reference sake. The DOUBLE-SAT problem is defined as $\qquad \mathsf{DOUBLE\text{-}SAT} = \{\langle\phi\rangle \mid \phi \text{ has at least two ...
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What is wrong with this simple proof of P=NP?

Exactly 1 in 3 SAT ($X3SAT$) is a variation of the Boolean Satisfiabilty problem. Given an instance of clauses where each clause has three literals, is there a set of literals such that each clause ...
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Why can't we say that a Neural Network is a NP problem solver?

From this video lecture from MIT https://youtu.be/moPtwq_cVH8?t=1229 there is mention how NP complexity works with finding a "lucky" algorithm and luck can never be accounted for. The ...
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Can any NP-Complete Problem be solved using at most polynomial space (but while using exponential time?)

I read about NPC and its relationship to PSPACE and I wish to know whether NPC problems can be deterministicly solved using an algorithm with worst case polynomial space requirement, but potentially ...
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How are all NP Complete problems similar?

I'm reading few proofs which prove a given problem is NP complete. The proof technique has following steps. Prove that current problem is NP, i.e., given a certificate, prove that it can be verified ...
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How to prove that finding math proofs is $\text{NP}$-hard?

This is Exercise 2.11 of the book "Computational Complexity: A Modern Approach" by Arora and Barak. Mathematics can be axiomatized using for example the Zermelo-Frankel system, which has a ...
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What do we know about $NP \cap co-NP$?

What do we need about the intersection of $NP$ and $co-NP$ apart from the fact that $P$ is a subset of it? (beyond what these answers here say, What do we know about NP ∩ co-NP and its relation to ...
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Reduction from 3-Partition problem to Balanced Partition problem

The 3-Partition problem asks whether a set of $3n$ integers can be partitioned into $n$ sets of three integers such that each set sums up to some given integer $B$. The Balanced Partition problem asks ...
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Which NP-Complete problem has the fastest known algorithm?

In terms of worst-case asymptotic runtime, which NP-complete problem has the fastest-known (exact) algorithm and what is the algorithm? Is there something known that is faster than $O(n^2*2^n)$?
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Complexity of deciding the satisfiability of a quasi-monotone CNF formula

A quasi-monotone CNF formula is a formula where each variable appears at most once as a positive literal (and any number of times as a negative literal). What is the complexity of deciding its ...
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research on OR and AND compression in SAT formulas [closed]

this is a new/advanced paper on OR and AND compression of SAT formulas, a newer area of research that seems not covered in any textbooks so far. A simple proof that AND-compression of NP-complete ...
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Prove that Hitting Set is NP-Complete

The Hitting Set Problem (HS) is defined as follow. Let $(C,k$) $C = \{ S_1, S_2, ..., S_m \}$ collection of subset of S i.e. $ S_i \subseteq S , \forall i$ $k \in \mathbb{N}$ We want to know if ...
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NP-completeness: Reduce to or reduce from?

Very simple question, but a mistake I make often enough that I'd love to have a standard reference. I'm showing that a problem $P$ is NP-Hard by assuming I have a polynomial time algorithm to solve $...
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What is practical difference between NP and PSPACE-complete?

Here's something that has puzzled me lately, and perhaps someone can explain what I'm missing. Problems in NP are those that can be solved on a NDTM in polynomial time. Now assuming P$\,\neq\,$NP, ...
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Understanding reductions: Would a polynomial time algorithm for one NP-complete problem mean a polynomial time algorithm for all NP-complete problems?

To prove that some decision problem $A$ is NP-complete, my understanding is that it suffices to show that the problem is in NP (i.e. that one can verify or reject all statements in polynomial time), ...
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Reducing directed hamiltonian cycle to graph coloring

The 3-SAT problem can be reduced to both the graph coloring and the directed hamiltonian cycle problem, but is there any chain of reductions which reduce directed hamiltonian cycle to graph coloring ...
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Randomized and deterministic reduction

Given a problem $X$, to show it is is $\sf NP$-complete, one usually shows a deterministic reduction from an $\sf NP$-complete problem. If it is hard to show deterministic reduction, then one shows a ...
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Proof that bin packing is strongly NP-complete?

Wikipedia claims that bin packing is strongly NP-complete. Unfortunately I haven't been able to find a proof. Does anyone know where to find it?
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Reduction from set cover problem to vertex cover problem

Although the reduction from vertex cover problem to set cover problem is quite simple, I did not find anywhere the reduction in the opposite direction. From the similarity in the type of problems, I ...
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Is GAP NP-hard with at most two balls per bins?

The generalized assignment problem (GAP) [1] is given by: Instance: A pair $(B,S)$ where $B$ is a set of $m$ bins (knapsacks) and $S$ is a set of $n$ items. Each bin $j∈B$ has a capacity $c(j)$, and ...
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Hardness proof of EVEN-ODD PARTITION

The PARTITION problem is NP-complete: INSTANCE: finite set $A$ and a size $s(a) \in \mathbb{Z}^+$ for each $a \in A$ QUESTION: Is there a subset $A' \subseteq A$ such that $\sum_{a \in A'} s(a) = \...
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Why is the reduction from Vertex-Cover to Subset-Sum of polynomial time?

In the standard proof why Subset-Sum is (weakly) NP-complete, one reduces Vertex Cover to Subset-Sum by using suitable numbers with O(m+n) bits (where m is the number of edges and n the number of ...
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Solving a variant of the Exact cover problem

I am trying to solve a variant of the Exact cover problem where every element has to be covered exactly twice instead of once ( i.e. has to be in exactly two sets that are part of the cover). Now, it ...
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Are non-trivial sets formed by set operations on NPC sets still in NPC?

I know that from this answer to a question on the class NPC, that NPC is not in general closed under intersection and union. However, the answer used languages which form trivial languages under ...
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To prove 4-SAT CNF is NP-complete [closed]

To answer the question below, 4-SAT: Given a formula in Conjunctive Normal Form, where each clause contains exactly 4 literals, does it have a satisfying truth assignment? I was going to prove that ...