Questions tagged [np-hard]

decision problems that are at least as hard as NP-complete problems

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How to prove that the generalized assignment problem (GAP) is NP-hard?

Specifically, what NP-hard problem can we reduce (the decisions version of) GAP to and how do we prove its correctness? The decision version of the generalized assignment problem is to determine ...
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How to sample the most unique vectors from a very large set efficiently?

While this question already exists and does talk about a heuristic with the Farthest Point First technique, I would like to approach the problem in a more efficient way. I do agree that this is an NP ...
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Are approximations to $#P$ gibberish? [closed]

approximations to #P are gibberish the model count in satisfiability (#P) implies straightforward access to a vast empire inside the logic of truth including N Boolean variables in a formula with N ...
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NP-Hardness of $\{ (S,k) | \exists S' \subset S \text{ s.t } \forall x \neq y \in S' \gcd(x,y)=1 \text{ and } \sum_{s \in S'} s \geq k \}$

I have been practicing NP-Hardness reductions and have been particularly interested in the language $L = \{ (S,k) | \exists S' \subset S \text{ s.t } \forall x \neq y \in S' \gcd(x,y)=1 \text{ and } \...
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Reducing subsetsum to {<G, l, u> | G is a weighted graph that has a spanning tree with weight between l and u}

How can I reduce Subsetsum (or maybe other np-complete problem) problem to the problem below? input : a weighted graph $G$ and numbers $l$ and $u$. output : Does $G$ has spanning tree, $S$, such that $...
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38 views

SUBSET SUM reduction to PARTITION

This is the PARTITION problem: Given a multiset S of positive integers, decide if it can be partitioned into two equal-sum subsets. This is the SUBSET SUM problem: Given a multiset S of integers ...
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1answer
34 views

Why this problem is NP-Hard?

I'm asking about the question described here: Knapsack Problem with exact required item number constraint Can't we iterate over $\binom{n}{L}$ options (which is polynomial), and for each option check ...
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Understanding the complexity class of a problem formulation

I'll keep the reasoning abstract. If I start from a mathematical formulation of a problem $A$ known to be $NP$-hard, I add a set of constraints which creates a problem $A'$. However, I do know that ...
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36 views

Is this variation of Max-Coverage NP-hard?

Setup An instance of Max-Coverage is typically defined by a collection of $n$ sets $S = \{s_1, s_2, \dots, s_n\}$, and a budget $k$, where the objective is to select a subset $U\subset S$ such that $$\...
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VS Problem algorithm

In CLOSED VERTEX COVER, we are given a graph G In CLOSED VERTEX COVER, we are given a graph 𝐺 where each vertex π‘£βˆˆπ‘‰(𝐺) has self-utility π‘’π‘£βˆˆβ„• and self-pollution π‘π‘£βˆˆβ„•, and π‘˜,π‘ˆβ‹†,π‘ƒβ‹†βˆˆβ„•. For each ...
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Algorithm for Variant of 0-1 Knapsack Problem

Variant of 0-1 Knapsack Problem is when you can choose exactly $k$ items from $n$ items, and $k$ is positive integer parameter that came in the input. Is there an algorithm with running time ...
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26 views

Is it NP-hard to check whether for a $k$ there exist both a Cut and a Bisection of value $k$?

Input: An undirected, unweighted graph $G=(V,E)$. A cut is defined as a partition $V=A\dot\cup B$. A bisection is defined as a partition $V=A\dot\cup B$ with $|A|=|B|$ if $|V|$ is even (or $|A|= |B|+1$...
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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 ...
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Computational complexity of dividing a set of constraints into a minimum number of satisfiable clusters

I am looking for the computational complexity of the following problem. Divide a given set of constraints into a minimum number of satisfiable clusters such that the constraints within the same ...
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How far would complexity hierarchies collapse if $L\in CoNP$ is $L\in NPH$?

Let $L\in CoNP$. Assuming that $L\in NPH$, what would we get? So, as $L\in NPH$ then every language $A\in NP$ has a reduction $A \leq L$. This would mean that $\overline{L} \leq L$ as well. By ...
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NP-Hardness of Half-SAT (at least half clauses)

I'm solving Problem 14.14 of What can be computed?. 14.14 Consider the computational problem HALFSAT defined as follows. The input is a Boolean formula B in CNF. If it is impossible to satisfy at ...
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What computational hardness concept corresponds to strongly-polynomial time algorithms?

Consider the computational problems in which the input is a set of $n$ integers with maximum magnitude $M$. According to Erik Demaine's lecture notes, assuming $P\neq NP$, the following are true: If ...
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1answer
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Inapproximability of graph problems on a restricted setting

I am considering the following problem $\mathcal{P}$. $\mathcal{P}$: Given an undirected graph $G$, and an integer $k$, find a set of vertices $S \subseteq V(G)$, with $|S| = k$, such that the number ...
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Shortest walk in a grid with turnstiles and waypoint

This video shows a grid maze with a type of tile that I'd call turnstiles. The mechanics of those tiles are described below. A turnstile tile has two bars on adjacent sides of the tile. You cannot ...
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Is the problem “find the sequence of $N$ numbers between 1 and $D$ with least cost”, NP-hard?

Consider sequences $p=(p_1,\dots,p_N)$ (the order matters) of length $N$, where $p_i\in\{1,\dots,D\}$ for fixed $D$. Moreover, consider a cost function $c:\{1,\dots,D\}^N\to\mathbb{R}$ which comply $c(...
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Is this variation of the traveling salesman problem NP-hard

Consider the following setting. You have $n$ cities, and there is a cost to travel from a city $i$ to a city $j$ given by $c_{ij}>0$ where $c_{ij}\neq c_{ji}$. Moreover, if you are traveling to ...
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1answer
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How to find long trails in a multidigraph

I have a directed multigraph (a multigraph is a graph that can have more than one edge between any two nodes). In Wikipedia's terminology, this is a directed multigraph (edges without own identity). I ...
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maximal independent set on grid-based graph proof of approximation ratio

We have a G = (V, E, w), in form of a grid graph with a single diagonal line in each grid in form of below. Where w is the V weight. We use a greedy algorithm that takes in each step maximum weighted ...
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1answer
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maximal independent set on grid graph proof

I'm trying to figure out proof of maximum independent set from: this link. (1b part). And I'm bit confused why exactly sum of $w(v)$ is less than or equal to sum of $w(v')$. Shouldn't it be other way ...
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Is there any NP-hard problem which was proven to be solved in polynomial time or at least close to polynomial time?

I know this could be a strange question. But was there any algorithm ever found to compute an NP-problem, whether it be hard or complete, in polynomial time. I know this dabbles into the "does P=...
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Proving a problem is NP Hard

Consider the following problem: Given a weighted directed graph $G$, determine if $G$ has a cycle whose total weight is $k$. All edge weights are integer but might be negative. $k$ is not an inputted ...
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Is the following problem NP-hard? (or have you seen it before?)

I genuinely don't know if the following problem is NP-hard. I have never seen it mentioned online, but it's hard to even search for exact problems like this. I have been trying to find an efficient ...
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Given N positive integers are they the integers 1-N : What is the relationship between this problem and NP?

Here’s the decision problem: Suppose I have N positive integers encoded in base-2 (as oppose to unary) Are these integers precisely the integers 1-N in some order? This is related to the Hamiltonian ...
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k disjoint triangles with graph splitting to two distinct groups

Please note that this question is different than this question. The $k$-disjoint triangles problem is as follows: Input: A graph $G=(V,E)$ and an integer $k\in \mathbb{N}$ Output: Are there $k$ ...
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Are problems that are fractions of constraints of NP-complete problems also NP-complete?

We know that Hamiltonian path, clique and independent sets are NP-complete but what about half or the square root of each problem or a fraction of $n$ ? That is, for a graph, $G$ of $n$ nodes, is ...
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How to prove NP-completeness of this mail-problem?

Let's say we have n postmen that are bringing people mail in a neighbourhood, and that they start and end at the same post office. This situation can be decribed with a directed graph, where the ...
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Clarification for binary search in solving optimal TSP when a polynomial algorithm with a budge exists

Below is Question 8.1 in Algorithms by Dasgupta et al. There's a solution to this problem that uses binary search from here. Pasting the answer for posterity. My questions are: When they say input ...
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Proving NP-hardness of Hamiltonian Cycle problem variant

I need to prove that determining whether a graph has a relaxed-Hamiltonian cycle (definition given ahead) is NP-hard. A relaxed-Hamiltonian cycle in $G$ is a closed walk $C$ that visits every vertex ...
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Does NP-hard problems have to be decision problems? (What the fact please) (contradicting answers)

Let me explain my trouble by another example. The wiki page says that Lattice problems are an example of NP-hard problems However, by clicking NP-hard, i find this definition A decision problem H ...
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Clique is NP hard to approximate up to $n^{a}$ for some $a \in (0,1)$

Given that $\mathsf{NP}=\mathsf{PCP}_{[\frac{1}{n},1]}\left(O\left(\log n\right),\left(O(\log n\right)\right)$, show that it is NP-hard to approximate clique up to factor of $n^a$ for some $a \in (0,1)...
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Is n-dimentional assignment problem for points NP-hard?

We have $n$ sets of $k$ points in $\mathbb R^d$ and we are trying to partition them to $k$ clusters of $n$ points such that from each set every point is mapped to a different cluster and the sum of ...
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Sub-exponential time algorithm to compute playoff chances

There are 10 teams, Team A through Team J, playing in a triple round robin pool (each team plays thrice against each other team, for a total of a 27 games per team). After the round robin pool, the ...
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Minimum set cover with incompatible sets

I'm interested in a variant of minimum set cover where some sets are ``incompatible'' (they can't be chosen simultaneously). To state it more formally: We have a finite base set $X$ and a family $\...
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Does having a similar constraint while reducing a problem to similar problem to prove np hard means they are same?

I have been trying to find the computational complexity of my optimization problem and found that it is Np-Hard. To prove it to Np-Hard, I try reducing it Nurse Scheduling Problem. I am quite confused ...
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Bin packing when items can be broken

In the bin packing problem, there are some $m$ items of size less than $1$, and they have to be packed into as few as possible bins of size $1$. The problem is NP-hard, but if we are allowed to break ...
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The concept of the creation of a trapdoor in NP-complete or NP-hard problems

I am reading the book An Introduction to Mathematical Cryptography. In its chapter 7, there is the following statement: In real world scenarios, cryptosystems based on NP-hard or NP-complete problems ...
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Why is crossing paths bad in Traveling Salesman?

I'm learning about Traveling Salesman in an online course (sorry I can't share the link it's paid only) and the first step to solving it then just state "as a heuristic we avoid crossed paths&...
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Must an optimization problem with a greedy algorithm belong to P?

If it is known that for some optimization problem there is a greedy algorithm that solves it and the solution includes sorting of input at the preliminary stage, is it necessarily true that the ...
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Algorithm for solving a mixed integer programming problem in polynomial time?

I have the following mixed integer programming (MIP) problem: $$ \begin{array}{rll} \text{Maximize } & z=k \\ \text{subject to } & a_ik - m_i \geq 0 & (i=1,\dots,n) \\ & b_ik - m_i \...
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Reducing the Hamiltonian cycle to the travelling salesman problem and self loops

If this is my adjacency matrix for the hamiltonian cycle: $$\begin{pmatrix}0&1&0&1\\ 1&0&1&0\\ 0&1&0&1\\ 1&0&1&0\end{pmatrix}$$ Then a reduction ...
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Hardness of a problem which is the sum of two NP-Hard problems

Consider the problem of computing an exponential sum over a certain function $g(x)=f(x)+h(x)$, that is computing $$\sum_{x}g(x)=\sum_{x}f(x)+\sum_{x}h(x)$$ now if we know that $\sum_{x}f(x)$ and $\...
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P/NP - Proof that SAT-TM is NP-complete uses certificate

To prove that SAT-TM (Turing machine emulating the satisfiability problem) is NP-Hard $$\text{SAT-TM}:=\{⟨M,p,1^k⟩ \; | \;βˆƒc,\; |c|\leq p(k), \;\text{such that M accepts c in ≀k steps}\}$$ my ...
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any hope to solve np hard problem using deep learning? [duplicate]

I know some basic machine learning and deep learning. Now a days deep learning solve many types of problem. I working working optimization problem like np, np hard problem. Is there any hope to solve ...
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228 views

Is Bitcoin mining NP-Hard?

I can't find this anywhere online. Is bitcoin mining NP-Hard? If so, how would we be able to prove a reduction from a known NP-hard problem? I am a bit lost.
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A multidimensional “moving van problem”: a mix of a knapsack and a bin-packing problem

This problem is a mix of the bin-packing and the knapsack problems. I call it "the moving van problem": there is a moving van with a limit on the weight it can transport, and a set of boxes ...

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