# Questions tagged [np-hard]

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

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### Why proofs of Cook's Theorem assume k is given (n^k for NTM)?

A typical proof of Cook-Levin's Theorem proceeds like this: Suppose problem X is in NP. Then there is an NTM M deciding X in time n^k, for some k. Given a word w, NTM M, and k, we construct a Boolean ...
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### Proving correctness of Polynomial reduction

Given a problem A is NP-Hard and A ≤𝑝 B, is there a way to prove that B is also NP-Hard?
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### Is it NP-hard to find different roots of different matrices simultaneously?

Consider the following problem: input: pairwise distinct natural numbers $k_1,\dots,k_m$ that are all $\leq n$, and matrices $A_1,\dots,A_m \in \Bbb Q^{n \times n}$ where $m \leq n$. output: a ...
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### Partition columns into m groups to maximize absolute value sums

The Task You are given $n$ columns each of length $m$. All values are either $-1$ or $1$. Find an assignment $s$ of each of columns to 1 of $m$ groups in order to maximize the sum of all the absolute ...
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### How do we prove this NP hard decision problem?

How can we prove this statement? Based on a formal decision problem it can be solved, right, but how exactly we can define this problem to be proved. I am not getting this concept properly. A ∈ P if ...
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### polynomial time approximation algorithm problem

How can we actually define a polynomial-time 4-approximation algorithm for vertex cover or knapsack problem? For say we have 2 approximation problems which less than equal 2C*. But when we have a ...
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### Need Help Solve an NP problem with an Approximation Algorithm

I have an algorithm problem which I do not know how to solve and I think it is NP-complete. Let me try to explain with a general example. Given $n$ objects, each with $k$ possible properties, ...
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### Does reducing a NP Hard problem to a NP problem make that NP hard problem a NP Complete problem?

I was asked a question in my algorithms exam which had this as the core question after simplifying. I had written that it would be NP-Hard but I got it wrong my professor is saying that it would be NP-...
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### Find the class of the problem PP1 and PP2 using the information given below

Assume that P1, P2,..., Pn are all NP-class problems. PP1 and PP2 are unknown problems (i.e., we don't know whether they belong to the P or NP classes). If "P1, P2,...., Pn" problems can be ...
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### Subset Sum With Interval Integer Target

Define the subset sum with interval integer target problem (SSIITP) as follows: SSIITP Input: A multiset $S = \{a_1, …, a_p\}$ of positive integers $a_i$. An integer $T$. SSIITP Output: True, if ...
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### Vertex cover in a special graph

We say that an undirected graph $G=(V, E)$ is special if for every vertex $v\in V$ and edge $\{u, w\}\in E$, it holds that $\{v, u\}\in E$ or $\{v, w\}\in E$. In other words, a graph is special if for ...
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### reference that minimum makespan on identical machines is NP-hard?

I need to cite a reference that the minimum makespan problem on ($>2$) identical machines is NP-hard. I've seen Garey and Johnson cited as a reference, but it I'm not sure which of the problems is ...
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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 ...