# Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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### NP-hardness on Cayley graphs

What is known about complexity of NP-hard problems on Cayley graphs? Suppose that the graph is given explicitly as the multiplication table of the group and the list of generators. So the input ...
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### Complexity of finding 2 vertex-disjoint $(|V|/2)$-cycles in cubic graphs?

I posted this on mathoverflow but with no luck: Finding a connected 2-factor is $NP$-complete since it is equivalent to the Hamiltonian cycle problem. I'm interested in the complexity of finding two ...
388 views

### NP-hardness of an optimization problem

While studying a problem in algorithmic game theory I got interested in the complexity of the following optimization question: Problem Given: ground set $U = [n] = \{1,\ldots,n\}$ given by $n$, $m$ ...
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### What do we know about checking real-stability of multivariate complex polynomials?

Given a polynomial $p : \mathbb{C}^n \rightarrow \mathbb{C}$ it is to be called "real-stable" if (1) all its coefficients are real and (2) if it has no roots such that all the coordinates of the root ...
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### NP-hardness proof: looking for some good restricted np-hard problems

To show the NP-hardness of a problem, one need to choose a known NP-hard problem and find a polynomial reduction from the known problem to his problem in hand. Theoretically, any NP-hard problem can ...
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### On which classes of graphs is resource constrained shortest path (RCSP) NP-hard?

I'm looking to link a problem I'm working on to a known NP-hard problem. I think I can model my problem as a resource constrained shortest path problem. However, the structure of my graph is not ...
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### A partition problem with order constraints

In the OrderedPartition problem, the input is two sequences of $n$ positive integers, $(a_i)_{i\in [n]}$ and $(b_i)_{i\in [n]}$. The output is a partition of the ...
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### Is that particular case of the “minimum weight solution to linear equations” still NP-complete?

We in our research group are working in the application of heuristic methods to the inverse illumination problem (that is, given a set of constraints about the illumination conditions in a scene, find ...
293 views

### BQP algorithm for two graph bisection problems and its implications on NP $\subseteq$ BQP

I read the paper Ahmed Younes, "A Bounded-error Quantum Polynomial Time Algorithm for Two Graph Bisection Problems", 2015. doi:10.1007/s11128-015-1069-y which is published in Springer's journal ...
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### Max-weight connected subgraph problem in planar graphs

The maximum-weight connected subgraph problem is as follows: Input: a graph $G=(V,E)$ and a weight $w_i$ (possibly negative) for each vertex $i \in V$. Output: a maximum-weight subset $S$ of vertices ...
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### The complexity of the puzzle game Net

Net (known also as FreeNet, or as NetWalk) is a puzzle game played on a $n \times n$ grid with the following objects: there are $m$ computers ; each computer occupies one cell and has one link cable; ...
388 views

### Planar Exact Cover by even-size sets

Major edit on June 6, 2019: Replaced the target problem with a simpler (but equivalent) one. Is the following problem NP-complete? Planar Exact Cover by even-size sets Input: A set $U$, a ...
393 views

### complexity of a constraint satisfaction promise problem

(This is the "upper end" of my question from over 10 months ago on cs.stackexchange. That question and the "lower end" I asked here over 8 months ago, which I also have a bounty ...
288 views

### NP-hardness of a Set Cover specialization

Is the following problem NP-hard? Given a set of $N$ real numbers (targets) $x_1,\dotsc,x_N$ and a "trident" defined by two distances $a$, $b$ from the center of the trident, what is the minimum ...
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### Vertex isoperimetric number of a graph - NP-hard?

The vertex isoperimetric number of a graph $G=(V,E)$ is $i_V(G) = \min\{\frac{|N(S)|}{|S|} : S \subseteq V, 1\le |S|\le \frac{|V|}{2}\}$. Several academic papers state that the problem of computing ...
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### NP-complete decision problems on deterministic automata

Do you know any NP-complete decision problems on deterministic automata? Most NP-complete problems that come to my mind are either (see, or here) graph theoretical, or involve some string rewriting or ...
365 views

### Complexity of a variant of the max word problem. NP-complete?

I'd like to be able to state that the following problem is NP hard. I am wondering whether anybody have any pointers to related/recent work? The problem: Given a finite set of transition matrices $A$ ...
351 views

### Is the problem of finding the chromatic number of a modified interval graph NP-Complete

Few days ago I was working on interval graphs to solve the known problem of resource allocation, as we know there is a greedy approach that solves this problem (chromatic number) in polynomial time ...
860 views

### Non-rooted MST of directed graph

I've found a problem that boils down to this: I need to find the non-rooted MST of a directed weighted graph. In other words, I need to find the minimal set of edges such that from any one node in the ...
478 views

### Graph classes in which CLIQUE is known to be NP-hard?

Given a graph $G$ and a positive integer $k$, the CLIQUE problem asks if $G$ contains a clique (complete subgraph) on at least $k$ vertices. This problem is long known to be NP-complete --- in fact, ...
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### A relaxed Steiner Tree Problem

Given a weighted graph $G(V,E,w)$ where $w$ is the weight function on edges and a subset of vertices $S\subseteq Q$ called terminals, a Steiner Tree is a connected subgraph which connects all vertices ...
585 views

### Is this vertex ordering optimization NP-Hard?

Could you help me to prove that the following problem is NP-hard? Problem. Given an undirected graph $G=(V,E)$, find an ordering of the nodes such that $\sum\limits_{v\in V}|succ(v)|\times|pred(v)|$ ...
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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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### Techniques for proving NP completeness for a specific sequence of instances

Most NP-completeness proofs I have seen pertain to proving that a problem on a class of instances is NP-complete. E.g., satisfiability is NP-complete on the class of instances with clauses having ...
208 views

### Is Clique completion for intersection models hard?

The following seems like a natural problem and I'm surprised I can't find any literature on it... but maybe it's because I don't know the name for it. Given a list of sets $S_1, S_2, S_3, \ldots$ ...
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### What are some problems in $P$ which have lower bounds assuming that $P \neq NP$ or the ETH?

Last year I had watched this talk online about problems in $P$ for which an algorithm which runs in some subclass of $P$, say in subquadratic time, would imply $P = NP$, or violate the ETH (...
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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 are the hardness results known for CSP over $\mathbb{F}_q$?

I found two related papers, There is a UGC hardness result here, https://www.cs.cmu.edu/~venkatg/pubs/papers/qaryCSP.pdf A kind of a stronger result might be found in these two other papers, http://...
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### Simple reduction to unbounded knapsack?

Does anyone know (or can anyone think of) a simple reduction from (for example) PARTITION, 0-1-KNAPSACK, BIN-PACKING or SUBSET-SUM (or even 3SAT) to the UBK problem (integral knapsack with unlimited ...
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### Is it NP-hard to _play_ minesweeper perfectly?

This paper shows that it is NP-hard "to determine if there is some pattern of mines in the blank squares that give rise to the numbers seen." If there is a way to "lead a perfect player into" such ...
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### Complexity of selective network improvement problem

We have a network flow problem with a given directed graph $G=(V,E)$, for each arc $(i,j) \in E$, there is a cost $c_{ij}$ and upper and lower capacity $u_{ij}$ and $l_{ij}$ for the flow $f_{ij}$ ...
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### Complexity of constructing minimum depth decision trees

I am interested in the computational complexity of Problem 1: Given a finite, non-empty set $J$, given $A, B \subseteq \{0,1\}^J$ such that $A \cap B = \emptyset$, and given $n \in \mathbb{N}$, does ...
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### How hard is PromiseFlowFree?

Playing more Flow Free, I think I've realized why I'm so amazingly brilliant at this game: The objective is to connect all pairs while covering the entire board, but in every puzzle there is always ...
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### Reduction from Vertex Cover to Max-Cut? [closed]

I am referring to Computational Complexity by Arora and Barak for my course. In the section on NP-completeness reductions, the book has a diagram that is represents how one NP-complete problem ...
I had some hard times trying to formulate the question, so I'll start with some examples: Suppose you are given a Dominating Set instance, $<G,k>$. Now suppose I give you a set of vertices $D$ ...