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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complexity of a variant of the subset sum problem

We have a set of positive integers $N=\{a_1,...,a_n\}$, we want to select a subset $N'$ of $N$ with maximum total sum of integers such that this sum should not exceed a given integer $B$. What is the ...
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What is the name of this problem (the dual of the asymmetric k-center problem)

I know $k-center$ problem is, given $n$ cities with specified distances, one wants to build $k$ warehouses in different cities and minimize the maximum distance of any city to a warehouse. In this ...
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2answers
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On the hardness of satisfying K number of linear constraints

Background: Normally in linear programing we have some objective function $$\text{maximize}\sum_{i = 1}^n a_i x_i $$ $$\text{subject to} \sum_{i =1}^n b_{ji}x_i \leq c_j \text{ for all } 1 \leq j \leq ...
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Complexity class of finding a siphon containing a trap?

I have found that finding a minimal siphon containing a set of places in general Petri Net is NP-complete. However, I am curious if this problem lands into a smaller complexity class if we only are ...
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1answer
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What are the differences between NP-Complete and NP-Hard? [duplicate]

What are the differences between NP, NP-Complete and NP-Hard? I am aware of many resources all over the web. I'd like to read your explanations, and the reason is they might be different from what's ...
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A question about NP and coNP

It is an open question if NP $\neq$ Co-NP but if the conjecture were proved, this would mean that P $\neq$ NP because P is closed under complement. Now a fact that fails to enter my head is the ...
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1answer
70 views

Concrete example of Vertex Cover to Subset Sum reduction

In Computational Intractability, we often come across a need to reduce Vertex Cover (VC) problem to a Subset Sum problem, mostly to prove Subset Sum is NP-Complete. I also see a reduction in the line ...
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Convex quadratic approximation to binary linear programming

Munapo (2016, American Journal of Operations Research, http://dx.doi.org/10.4236/ajor.2016.61001) purports to have a proof that binary linear programming [1] is solvable in polynomial time, and hence ...
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Why haven't I solved the Travelling Salesman problem with the following argument using djikstras algorithm?

I claim to have solved the travelling salesman problem as follows. (You will have to be familiar with djikstra's algorithm for this.) 1) I am about to start using djikstra's algorithm on any given ...
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125 views

Is this equivalent to any famous NP-complete problem?

Given the following problem. Given an $n\times n$ matrix $A := \{a_{ij}\}$. Find an $n\times n$ matrix $X := \{x_{ij}\}$, where $x_{ij} \in \{-1, 1\}$ for $i, j \in [n]$, that minimizes the ...
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Is resampling random variables to maximize value NP-hard?

Setup Let $S = {X_1, ..., X_n}$ be a set of independent binary random variable, i.e. $X_i \in \{0, 1\}$, each with prior $P(X_i = 1) = p_i$. The $X_i$ are not iid, so $p_i, p_j$ need not be equal if $...
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Variant of TSP: allow each vertex to be visited at most twice

We are given a finite set $V$ and a set of distance $d : V\times V \rightarrow R\ge 0$ and we wish to compute a tour. Suppose we allow each vertex to be visited at most twice in the tour. How can we ...
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2answers
119 views

Is finding a minimal set of seed variables for a complete deduction of a system of equations NP-complete?

Suppose we have a set of variables $V$. We also have a set of equations $E$, which are sets of at least two variables. We don't know anything about these equations, except if we know all but one of ...
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83 views

TSP Variant — Colored Path

Recently I came up with a traveling-salesman-esque problem. As usual, we have $n$ vertices, and a weighted edge between any two vertices. However, each vertex is associated with a color, which may be ...
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Is the reduction for HAMPATH to HAMCYCLE and UHAMPATH to UHAMCYCLE the same?

HAMPATH/UHAMPATH is A directed / undirected graph G and 2 nodes s and t and is there a hamilton path from s to t? Likewise with HAMCYCLE/UHAMCYCLE but has a hamilton cycle on $G'$ The reduction ...
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1answer
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UNDIRECTED HAMCYCLE to HAMPATH reduction

I'll define the problems UHAMPATH Input: A undirected graph G and 2 nodes, s and t Question: Is there a hamiltonian path from s to t in G? UHAMCYCLE Input: A undirected graph G ...
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1answer
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A doubt on converting NOT gate to CNF formula

For a NOT gate if $x_1$ is input and $x_2$ is the corresponding output, I see the equivalent CNF (conjunctive normal form) is $(x_1 \lor x_2) \land (\overline x_1 \lor \overline x_2)$. My ...
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How to use c-gap problems to prove inapproximability?

Suppose there is a specific set function with some properties - $f=2^V\to \mathcal{R}$. It is known that the following problem is NP-Hard: Find $S\subseteq V, |S|\leq k$ such that $f(S)$ is maximized....
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1answer
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Is this variation of set-cover NP-hard to approximate?

The classic set-cover problem is described as follows: Let $S = \{s_1, ..., s_n\}$ be a target set, and let $\Lambda = \{A_1, ..., A_m: A_i \subset S\}$ be a collection of subsets of $S$. The ...
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Given a system in $\mathbb{F}_2$ in RREF, how do I find a solution of minimal norm?

I have a $12 \times 12$ (so not really large) system of linear equations in $\mathbb{F}_2$ which I got to RREF through the usual row reduction. Suppose the system has multiple solutions, and call the ...
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How Reduction works in proving NP-Hard?

A problem $X$ is $NP$-Hard if for all $Y \in NP$, $Y \leq_P X$. Further, if a problem $Z$ is $NP$-Complete, and $Z \leq_P X$, then I can prove (rather mechanically) that $X$ is $NP$-Hard. I also ...
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1answer
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Simple Hamiltonian cycle reduction

HAMPATH Input: An undirected graph $G$ and 2 nodes $s, t$ Question: Does G contain a Hamiltonian path from $s$ to $t$? HAMCYCLE Input: A undirected graph $G$ and a nodes $s$ ...
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1answer
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Verifying Hamiltonian Cycle solution in O(n^2), n is the length of the encoding of G

In the textbook of CLRS, 'ch. 34.2 Polynomial-time verification' it says the following: Suppose that a friend tells you that a given graph G is hamiltonian, and then offers to prove it by giving ...
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56 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
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1answer
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Techniques for proving NP-completeness

I've been reading Garey-Johnson book Computers and Intractability and I am focusin on Section 3.2, Techniques for proving NP-completeness. In these definitions and explanations nothing is formally ...
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1answer
51 views

Are there any “complete” languages in $coNP -NP$?

Suppose $coNP \neq NP$ language B would be called "complete" in $coNP-NP$ if: $B\in coNP - NP$ $A\in coNP-NP \implies A\leq_pB$ Are there any "complete" languages in $coNP - NP$?
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How to prove the optimization version problem (whose decision version is NP-complete) can be solved in poly-time iff P=NP?

I have proved the decision version of my problem be $\mathcal{NP}$-complete. And I know that if I can solve the optimization version in poly-time, then I can just to compare the obtained minimum (or ...
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Proving NP-completeness of an extension in List Coloring Problem

In the List Coloring Problem (LCP), one is given an undirected graph $G(V,E)$, each vertex $v \in V$ is given a list of permissible colors $L(v) \subseteq \{1,2,\dots,k\}$, we want to find a coloring $...
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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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1answer
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Reducing 3 SAT to 3 SET PACKING

I'm trying to prove NP-hardness of 3 SET PACKING, which is a following problem: given a family of sets where each set contains 3 elements, decide whether the family contains k sets that are pairwise ...
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1answer
35 views

Complexity of “half cycle”

I'm working on the following problem: HALFCYCLE (HALFC): Input: A directed graph $G = (V,E)$. Output: Whether the longest cycle in $G$ has length $ \lfloor |V|/2 \rfloor$. Prove that if $\...
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1answer
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Prove Subset-Sum is NP-complete - Alternative reduction?

It is well known the Subset-Sum problem is NP-complete. This can be shown using a reduction from the 3SAT problem. I am wondering: is there any other NP-Complete problem that could be reduced to the ...
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2answers
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Proving a problem as NP-complete

According to this article, A problem X can be proved to be NP-complete if an already existing NP-complete problem (say Y) can be polynomial-time reduced to current problem X. The problem also needs ...
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1answer
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pseudo-polynomial reduction from 3-Partition to Partition

A problem $\Pi'$ is pseudo-polynomially reducible to the problem $\Pi$ ($\Pi' \leq_{pp} \Pi$) if, for any instance $I'$ of $\Pi'$, an instance $I$ of $Π$ can be constructed in pseudo-polynomially ...
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1answer
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Proving NP completeness of maximal length path

I have this question to answer: For each node i in an undirected network $G = (N,E)$, let $N(i) = \{j \in N : \{i, j\} \in E\}$ denote the set of neighbors of node $i$ and let $c_e\geq0$ denote the ...
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1answer
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Time complexity of subset sum problem with reals instead?

It is well known that the conventional subset sum problem with integers is NP-complete. What if the array elements can be any real numbers and also target sum can be any real number? Is it NP-complete ...
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1answer
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Prove/Disprove: Every two non-trivial NP-complete problems are decreasing reducible?

We say that two languages $L_1,L_2$ are decreasing reducible if there exists a polynomial time reduction $f:\Sigma^*\to\Sigma^* $ and there exists $n\in\mathbb{N}$ such that for every $x\in\Sigma^*$ ...
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Why isn't the Generalized Super Mario Bros. obviously in NP?

It is shown in the paper: "Classic Nintendo Games are (Computationally) Hard" by Greg Aloupis, Erik D. Demaine, Alan Guo, and Giovanni Viglietta that the Generalized Super Mario Bros. (SMB, for short) ...
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1answer
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Weak NP-completeness

I know that the Knapsack problem is weakly NP-complete. I also notice that on Wikipedia: "A problem is said to be strongly NP-hard if a strongly NP-complete problem has a polynomial reduction to it ...
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1answer
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Reduction from NP-complete problem to unknown complexity problem and vice-versa

Suppose I have two problems: $B$, which is NP-complete, and $A$, of unknown complexity. Question: If I show that $B \le A$ I can state that $A$ is also NP-complete because the two required ...
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NP-completeness of Induced disjoint paths between a set of sources and a set of sinks

In a given undirected graph $G(V,E)$, a set of $k$ paths is said to be induced if: They are vertex-disjoint. Each one is itself an induced path. No edge connects two vertices of two different paths. ...
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3answers
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How to solve the optimization of bin packing using the decision version

Let us say the optimization version of the bin packing problem asks you to give a packing using the fewest bins possible and the decision version asks if it is possible to pack the bins into $k$ bins. ...
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1answer
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NP-completeness of Induced disjoint paths

In graph theory, it is well-known to be NP-complete the problem of given a set of $k$ pairs of source-sink, deciding whether there exists $k$ vertex-disjoint paths connecting these pairs. Our problem ...
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1answer
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What are the current state of art approximation algorithm for NP-Hard problems? [closed]

I came cross some works try to use deep learning to approximate NP-Hard https://arxiv.org/pdf/1810.10659.pdf Though the paper seems to have very good results but based on the citations. I'm quit ...
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1answer
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Parity Hamiltonian path problem

Wikipedia says that Hamiltonian path problem is NPC, but Parity Hamiltonian path problem (i.e., is there an odd amount of hamiltonian path) is P. Does a reduction from, e.g., SAT, to HPP, unavoidably ...
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1answer
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Reduction from minimum dominating set to the set cover

To solve the min dominating set problem of a graph G, we can reduce it to a set cover problem. For example to find the MDS of the graph G: We can create an instance of the Set Cover problem by: ...
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1answer
72 views

Subset with modified condition, is it still NP-complete? [closed]

So I know the conditions required for a problem to be NP-Complete is that it has to lie within NP and has to be NP-hard. The given problem I have is subset sum. However, the conditions have been ...
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Is it NP-complete to test if a graph contains $t$ $k$-cliques?

Let $(G,t,k)$ - a graph with $t$ cliques with $k$ vertices (there are $t$ cliques of size $k$ in graph $G$), for $t,k > 100$. How to prove that $(G,t,k)$ is NP-complete? It is obvious that it is ...
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Matrix covering by squares

I wonder about the following decision problem : Instance: We consider a $n\times p$ matrix $M$ of zeros and ones, and two integers $N$ and $k$. Question: is it possible to cover all the ones of the ...
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1answer
40 views

Reducing CLIQUE to Super Connector problem

I am trying to show that our problem is NP-Complete by reducing the known problem CLIQUE to our problem. Regular CLIQUE problem: Input: An undirected graph $G$ and a positive integer $K$. ...