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# Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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### NP complete graph problems about structural properties

(This question is a bit of a "survey".) I'm currently working on a problem where I'm trying to partition the edges of a tournament into two sets, both of which are required to fulfill some structural ...
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### The weakly NP-complete problems and their associated counting problem

Are there weakly NP-complete problems whose associated counting problem can be computed in pseudo-polynomial time? And if one were to be found (and assuming it is #P-complete), what would be the ...
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### Binary matrix column subset selection complexity

Given an $m \times n$ matrix ($m$ rows) containing only $0$'s and $1$'s, what is the complexity of finding an $m \times k$ submatrix (of $k$ columns) such that within the chosen submatrix there is no ...
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### Can you identify the sum of two permutations in polynomial time?

There were two questions asked recently on cs.se which were either related to or had a special case equivalent to the following question: Suppose you have a sequence $a_1, a_2, \ldots a_n$ of $n$ ...
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### What is the minimum required depth of reductions for NP-hardness of SAT?

As everyone knows, SAT is complete for $\mathsf{NP}$ w.r.t. polynomial-time many-one reductions. It is still complete w.r.t. $\mathsf{AC^0}$ many-one reductions. My questions is what is the minimum ...
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### The computational complexity of spectral norm of a matrix

How hard is computing the spectral norm of a matrix? This paper says, ... it suffices to say that, except for few particular cases, the Matrix Norm problem is NP-hard. I expected that the ...
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### What is complexity of this max-edge subgraph problem?

While discussing the question I had asked here, @NealYoung and I encounter another problem, which is to judge complexity of the problem below: Given a connected undirected graph, finding a maximum-...
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### Could there be an extremely large hidden subset of Polynomially solvable problems within NP-Complete problems?

Suppose P != NP. We know that we can make easy instances of 3-SAT at any time. We can also generate what we believe to be hard instances (because our algorithms can't solve them quickly). Is there ...
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### Complexity of interval cover problem

Consider the following problem $Q$: We are given an integer $n$, and $k$ intervals $[l_i,r_i]$ with $1\leq l_i\leq r_i\leq 2n$. We are also given $2n$ integers $d_1,…,d_{2n}\geq 0$. The task is to ...
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### Complexity of the densest $k$-subgraph problem on planar graphs

In the densest $k$-subgraph problem, one is given an undirected graph $G$ and wants to find a set of vertices $N$ with $|N| = k$ such that the number of edges in the subgraph of $G$ induced by $N$ is ...
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### PTAS (polynomial time approximatin scheme) for euclidean TSP/Minimum-Cost k-Connected subgraph problem

Problem 1 I have read "On Approximation of the Minimum-Cost k-Connected Spanning Subgraph Problem" (by A. Czumaj, A. Lingas), and even in the abstract are 2 statements "We present a polynomial time ...
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### Decision problem related to coloring

Given a $k$-colorable graph $G$ and vertices $u$ and $v$ of $G$, what is the complexity of deciding if every $k$-coloring of $G$ must assign the same color to both $u$ and $v$? It does not seem ...
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### What is the complexity of this edge coloring problem?

Recently, I have encountered the following variant of edge coloring. Given a connected undirected graph, find a coloring of the edges that uses the maximum number of colors while also satisfying ...
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### Reference for NP-hardness of 3-colouring?

I have a historical question. I’m trying to determine the reference for the fact that 3-colourability of graphs (alternatively, $k$-colourability for given $k\geq 3$) is NP-hard. The tempting answer ...
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### Is there any reference on the hardness of approximation for 2-partition problem?

I tried to look for some references but could not find any. I knew it is proved to be NP-complete via a transformation from Knapsack or 3DM problem. But I couldn't find a way to apply PCP theorem to ...
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### Reduction from OR-SAT to Exact CNF-SAT, keeping the number of variables polynomially bounded?

Let me define both the problems first: $OR$-$SAT$: $m$ Boolean formulae are given in $CNF$, $\phi_1$,$\phi_2$, $\ldots$, $\phi_m$, each on variable set, $x_1, x_2, \ldots, x_n$. ($m$ $<$ $2^n$, ...
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### Complexity reductions to Hamiltonian Path?

I am looking for a NP-hardness reduction from an arbitrary problem to the Hamiltonian Path problem such that the reduced no-instances of Hamiltonian path are "far" from having a Hamiltonian path. Do ...
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### P vs NP: Instructive example of when Brute Force search can be avoided

To be able to explain the P vs NP problem to non-mathematicians I would like to have a pedagogical example of when Brute Force-search can be avoided. The problem should ideally be immediately ...
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### Intermediate $\mathsf{NP}$-complete problems?

Partition problem is weakly NP-complete since it has polynomial (pseudo-polynomial) time algorithm if input integers are bounded by some polynomial. However, 3-Partition is strongly NP-complete ...
### What are the consequences of a ${\bf O}$(m) algorithm for SAT
We are given a Boolean formula $F$ in conjunctive normal form with $n$ variables and $m$ clauses and we would like to know if there exists at least one assignment to the $n$ variables that makes $F$ ...