Questions tagged [np-hard]

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

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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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1answer
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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 $$\text{SAT-TM}:=\{⟨M,p,1^k⟩ \; | \;∃c,\; |c|\leq p(k), \;\text{such that M accepts c in ≤k steps}\}$$ 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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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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If a solution to Partition is known to exist, can it be found in polynomial time?

In the Partition problem, there is a set of integers, and the goal is to decide whether it can be partitioned into two sets of equal sum. This problem is known to be NP-complete. Suppose we are given ...
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Cyclic tour minimizing total weight

I asked the following question on math.se but it wasn't really answered so moved it over here as I feel it's more relevant. I saw the question below on an old stack exchange question when looking to ...
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Difference longest path problem and underlying decision problem [duplicate]

I am studying the longest path problem with the final objective to show that it is NP-complete. On wikipedia I read that the problem itself is NP-hard but the underlying decision problem is NP-...
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Trapdoor functions that depends only on NP-hard?

As far as I can tell, there are no examples of trapdoor functions which hardness assumptions is merely hardness of some NP-hard problem in worst case. That is, a trapdoor function in which, it is ...
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NP-hardness for one-dimensional facility location problem with entrance fee for each customer

We have $n$ customers, $(x_1, \dots, x_n)$, sorted on the read line. For convenience, we also use $x_i$ to denote its coordinate on the line. We need to locate $m$ facilities on the real line. We note ...
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Is the Maximum Independent Set problem NP-complete? [duplicate]

It can be read on Wikipedia that MIS is NP-hard. However, is it also NP-complete? This article says: "Thus, the Maximum Clique Problem(MCP) and the Maximum Independent Set(MIS) Problem are ...
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Are there any tetrational-time problems?

I know there exists problems decidable in polynomial-time, exponential-time, etc. I couldn't find any tetrational-time problems, however. Are there any and if not, why?
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NP-hardness language

How to prove that language $L = \{G \; | \; \omega(G) \geq \frac{9}{10}\cdot n\}$, where $n$ - number of vertices in the graph, NP-hard? $\omega$ - is a clique number.
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How can I prove the following problem is NP?

Because of the covid-19 pandemic, our firm works semi-remote working. We want to that: Every day 2/5 of staff should be in the office. Everyone should go to the office 2 times a week. Teams want to ...
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Is it possible to determine an instance of an NP-hard problem is easy or hard by the optimization?

I have an NP-hard problem and an optimization to deal with the problem. I want to know that is it possible to distinguish between easy and difficult instances of the problem by the parameters of the ...
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What parameter of optimizations, like time solving, can be used to show a phase transition in NP-hard problems?

Before asking the question, I should say that I am not sure here is a proper community to ask this question or not. I have an NP-hard problem and an optimization to deal with the problem. Recently, I ...
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NP-hardness of language of graphs with $\alpha(G)=n/3$

How to prove the NP-hardness of the language $\{G \mid \alpha(G) =\frac{1}{3} |V(G)|\}$? Here $G$ is a graph, $V(G)$ is its vertex set, and $\alpha(G)$ is the independence number of $G$, which is the ...
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3Col reduction Variation, Special edges

I have a question concerning NP reduction. My question asks me to show that if I have a graph with Edges that connect 3 nodes together instead of 2, (Y style I assume). I need to prove that finding ...
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Proving NP-completeness for a not so cheesy problem

Let's say we have a matrix M[1..B, 1..B] (i.e., a square matrix) and a mouse in the upper left corner (1,1). We also have an integer A, which tells how many pieces of cheese there are in the matrix. ...
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Subset selection with maximum sum and minimum variance?

So I am trying to tackle a combinatorial optimization problem and would like some insights on how to approach it. The problem statement is as follows: Consider a set of elements of size N, how do I ...
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Why do I only need to reduce from one problem in NP to prove NP-Hardness?

Suppose I wish to show that my decision problem $Q$ is NP-Hard. Why do I need to reduce from one problem $Q'$ of known hardness ? Consider for instance the following situation: Here, I have my set NP ...
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3-Sat reduction to facility location problem

I'm learning about NP problems and I this problem which is a bit challenging for me. You are given an undirected, simple graph G = (V,E) and an integer k where nodes represent cities and edges ...
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Reducing 3-SAT to restricted 3-SAT

I am trying to show that the following problem is NP-hard. Input: A boolean function in CNF (conjunctive normal form) such that every clause has at most three literals and every variable appears in at ...
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Are “pipes” puzzles NP-hard?

Is the decision version (i.e. does a solution exist) of this puzzle NP-hard (for an nxn puzzle)? It feels like it has very local strategies which allow easily solving the instances, but it's not ...
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Modify a binary matrix to minimize the sum of values of the rows

I have a matrix of zeros and ones, i.e., $\mathbf{X}=[x_{ij}]$ with $x_{ij}\in\{0,1\}$ for all $i=1,2,\ldots,m$ and $j=1,2,\ldots,n$. Associated with each row $i$ of the matrix $\mathbf{X}$ a set of ...
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Is Partition Problem with non-integer input NP-complete?

The Partition Problem supposes that we have a set $S=\{s_{i} \in \mathbb{R^{+}}\}_{i=1}^{n}$, is there a subset $T$ such that $\sum_{s \in T} s = \sum_{s \notin T}s$? I have read a lot of proofs using ...
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Are most machine learning problems the same problem? (NP-hard)

I am trying to teach P vs NP to some primarily Machine Learning folks. I wanted to come up with an introductory fact to grab their attention. Reasoning for Question: Most problems in Machine Learning ...
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What is a good reference for NP hardness in the machine learning/optimization/operations research context?

I am reading some papers in machine learning and at the very beginning (introduction) you will see statements of theorems that says, for example: Theorem 1.1. For any constant ϵ > 0, it is NP-hard ...
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If a poly-time solution exists for an NP-Complete (Decision) problem, then there exists a poly-time solution for the NP-hard (Optimization) flavor?

This question boils down to: If a polynomial time solution exists for a decision problem, is there also a polynomial time solution for the same problems optimization flavor? Let's take the Traveling ...
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Algorithm for assigning people to groups

Given a list $L = [1, 2, .., n]$ and a list $C = [(L_i, L_j), ....]$ form a group of pairs $G = g_1, ..., g_{n/2}$ such that: every element of $L$ is assigned to exactly one group $g_k = (L_i, L_j) \...
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How difficult is this matching-like problem?

Let $A$ and $B$ be two sets of integers with $|A|>|B|$. Given a map $f: A \rightarrow B$ and $i \in A, j \in B$, let us use the shorthand "$i$ is matched to $j$" if $f(i)=j$. I am ...
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How to find a cut in a graph with additional constraints?

I have a complete undirected graph $G=(V,E)$ with positive non-null rational weights $c:E \to \mathbb{Q}^+_{*}$ on the edges, such that $c(v,v) = 0$ for all $v$, and a subset $C \subset V$. I would ...
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Hardness of maximizing difference of functions

Suppose that the problem of maximizing a real function $f$ over a certain domain $D$ is NP_HARD. What can be said about the problem of maximizing $f-g$, with $g$ being another function over $D$? Is it ...
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Finding a boolean submatrix isomorphic to a specific fixed set of other boolean matrices

Given a matrix $M$ of certain size $h\times w$, where $h\leq w$, for example $5\times 6$, are also given the following set $B$ of additional all-ones matrices, that I like to call target (b)oxes. $$ \...
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Is there an NP-hard problem for which no Fixed-Parameter Tractable algorithm exists?

Question Is there an NP-hard problem for which we can add a parameter1 to create a "natural"2 parametrised problem for which no FPT algorithm exists? The adding a parameter is needed ...

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