Questions tagged [np-complete]

Questions about the hardest problems in NP, i.e. of those that can be solved in polynomial time by nondeterministic Turing machines.

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11
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Covering a complete graph with n copies of an arbitrary graph: NP-complete?

Given a complete graph $G$, an arbitrary graph $H$, and a positive integer $n$, are there subgraphs $A_1,\dots,A_n$ of $G$ (not necessarily disjoint) such that their union is $G$, and each of them ...
7
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0answers
47 views

P=NP turns 50. 1971 STOC conference

Stephen Cook presented his seminal paper "The complexity of theorem-proving procedures" at the 1971 STOC (Symposium on Theory of Computing) conference which was held May 3-5, 1971 at Case ...
7
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1answer
249 views

Polynomial time algorithm for finding a maximal monotone subset

Input: Some fixed $k>1$, vectors $x_i,y_i\in\mathbb R^k$ for $1\le i\le n$. Output: A subset $I\subset\{1,\dots,n\}$ of maximal size such that $(x_i-x_j)^T(y_i-y_j) \ge 0$ for all $i,j\in I$. ...
7
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153 views

Complexity of finding a Eulerian path such that the image under a bijection is also a Eulerian path

Problem input: undirected graphs $G$, $H$ and a bijection $f: E(G) \to E(H)$ Question: Is there a Eulerian path $p: \{1,\dots,|E(G)|\} \to E(G)$ in $G$ such that $f \circ p$ is a Eulerian path in $H$? ...
7
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1answer
935 views

Fastest known algorithm for $3$-$\mathrm{Partition}$ problem

$3$-$\mathrm{Partition}$ problem is $\mathsf{NP}$-Complete in a strong sense meaning there is no pseudo-polynomial time algorithm for it unless $\mathsf{P=NP}$. I am looking for the fastest known ...
6
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537 views

What could we say about that conjecture that yields P != NP?

Let $F$ be the set of all Boolean formulae. We say that a Boolean formula $\varphi$ is positive (=monotone) if $\forall \alpha\in F,i\leq n$, if $\alpha\wedge\neg x_i\models\varphi$, then $\alpha\...
5
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0answers
90 views

Optimization problem with discrete and continuous components

Suppose we have a sequence of $m$ tokens $(T_1, T_2, \ldots, T_m)$. We can split this sequence considering two parameters $w$ (which is the width of the window) and $x$ which is the overlap between ...
5
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0answers
61 views

Complexity of still life extentions in Game of Life

The game of life is one of the most famous cellular automata in 2D. It has a variety of objects, some of them are moving like gliders, some have an oscillating behavior and others do not change at all,...
5
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1answer
210 views

Why the Cook-Levin reduction is not a parsimonious reduction?

In the textbook Arora-Barak, after presenting the Cook-Levin theorem's proof the authors say that the proof can be modified slightly to make the reduction parsimonious. The parsimonious reduction is ...
5
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295 views

Reduce factoring to solving quadratic equations

The problem of solving quadratic equations is as follows: Suppose you are given a set of quadratic equations and are asked to find $0$-$1$ values for the variables such that all equations are ...
5
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88 views

Find a minimum-cost pair of arc-disjoint paths, both within a given restricted distance

Is there a polynomial algorithm that can find a pair of arc-disjoint paths in a directed graph that has a minimum total cost, subject to the condition that both paths are within the same distance. ...
5
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643 views

2 Dimensional Subset Sum: looking for information

I do not know if this problems exists with a different name, if it is, I could not find it. The problem is this: Given a set $S$ of $n$ points in $\mathbb{Z}^2$, is there a subset $A\subset S$ ...
4
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33 views

Strongly connected subgraph that contains no negative cycles

Is there an efficient algorithm that solves the following decision problem: Given a strongly connected weighted directed graph $G$, defined by its transition matrix, is there a strongly connected ...
4
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0answers
61 views

Convex quadratic approximation to binary linear programming

Munapo (2016, American Journal of Operations Research, http://dx.doi.org/10.4236/ajor.2016.61001) purports to have a proof that binary linear programming [1] is solvable in polynomial time, and hence ...
4
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137 views

Is this equivalent to any famous NP-complete problem?

Given the following problem. Given an $n\times n$ matrix $A := \{a_{ij}\}$. Find an $n\times n$ matrix $X := \{x_{ij}\}$, where $x_{ij} \in \{-1, 1\}$ for $i, j \in [n]$, that minimizes the ...
4
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144 views

Is Hamiltonian cycle problem on graphs with out-degree at most 3 NP hard?

I am trying to show a different form of Hamiltonian cycle problem is NP Hard. The problem is as follows. We have a directed graph and each node can have at most 3 outgoing edges. Determine if this ...
4
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99 views

Need help figuring out a planning/assignment problem

I'm looking to solve this planning problem. Any pointers or ideas are much appreciated! You have a number of i individuals i = { 1, 2, ..., n } that need to perform tasks. Tasks are performed in ...
4
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2k views

$NP = PSPACE$ and what that would mean about $PH$

So, a paper showed up on arXiv: https://arxiv.org/abs/1609.09562 The above states in the abstract that it contains a proof that $NP = PSPACE$ Since $NP \subseteq PH \subseteq PSPACE$, that would ...
4
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70 views

PTAS vs. exact-time sub-exponential algorithms

I have recently summarized several algorithms for the maximum disjoint set problem. This problem is NP-hard, but it has both PTAS and sub-exponential algorithms. These algorithms seem to me closely ...
3
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0answers
64 views

Coloring nodes and edges in node-weighted graph

I have a graph $G$ with $n$ nodes and $O(n)$ edges. Each of the nodes has a positive integral weight at least 2, such that sum of all weights is at most $O(n)$. We want to color the nodes and edges ...
3
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44 views

NP-complete problem

Is the following line true? Consider three problems A, B and C. If $A$ $<p$ $B$ and $B$ $<p$ $C$ and $B$ is NP-complete problem, then $C$ is also NP-complete. If B is NP-complete, then C would ...
3
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1answer
119 views

NP completeness of deciding whether a set of examples, consisting of strings and states, has a corresponding DFA?

I'm working on a textbook problem, 7.36 in Sipser 3rd edition. It claims that if we are given an integer $N$ and set of pairs $(s_i, q_i)$, where $s_i$ are binary strings and $q_i$ are states (we are ...
3
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1answer
99 views

Approximation factor preserving reduction

The definition of approximation factor preserving reduction from the book by Vijay V. Vazirani, page 365: Let $\Pi_1$ and $\Pi_2$ be two minimization problems, an approximation factor preserving ...
3
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0answers
28 views

Complexity of finding an alternating Hamiltonian (x,y)-path in edge bicolored complete graphs

Let $G$ be a simple complete graph with an edge-2-coloring. An alternating Hamilton (x,y)-path is a Hamiltonian path which starts at vertex $x$ and ends at vertex $y$ such that the colors of its ...
3
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99 views

Proving NP-completeness of an extension in List Coloring Problem

In the List Coloring Problem (LCP), one is given an undirected graph $G(V,E)$, each vertex $v \in V$ is given a list of permissible colors $L(v) \subseteq \{1,2,\dots,k\}$, we want to find a coloring $...
3
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137 views

A special case of the SUBSET SUM problem

Consider the following special case of SUBSET SUM Inputs: Positive integers $a$ and $b$ with $a \ne b$, and positive integers $k$ and $t$, with $k$ specified in unary. Encoding: These inputs (...
3
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0answers
51 views

Karp hardness of a module in a graph

DEFINITION: A set of vertices $A\subseteq V$ in a graph $G(V,E)$ is called a module if it satisfies the following property: For every $v\in V\setminus A$, either $A\subseteq N(v)$ or $A\cap N(v)=\...
3
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0answers
260 views

How to minimize the sum of edge weight in the graph while keep the all-pair shortest path greater than a constant?

For example, if we have a graph G = (V, E) and a subset of vertices $U \subset V$. We can set $w(e)$ where $e \in E$ to be a non-negative real number. We want to minimize the total edge weight, but ...
3
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0answers
170 views

Complete problems in NP∩coNP

I often read in Complexity literature that NP∩coNP is unlikely to have any complete problems. Is that unlikelihood "proved" ? By proved, I mean that there would be a theorem that would relate the ...
3
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46 views

Misunderstanding SS22 Sequencing to Minimize Tardy Tasks

I was reading "COMPUTERS AND INTRACTABILITY. A Guide to the Theory of NP-Completeness", and I am stuck at this part (page 236, SS22): In the second paragraph, 3rd line, the authors said that: "The ...
3
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146 views

String version of even-odd partition problem

Motivated by Hardness proof of EVEN-ODD PARTITION post I came up with a string version. String even-odd partition INPUT: $(x_{1,0},x_{1,1}),\dots,(x_{n,0},x_{n,1})$, i.e., $n$ pairs of strings over ...
3
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0answers
175 views

Showing a problem on a specific class of graphs is NP-hard

We know that a set of problems like minimum clique cover problem, coloring problem, vertex cover, ... are NP-hard for general graphs, but may be polynomial-time solvable for some classes of specific ...
3
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0answers
72 views

What is the time complexity of Summing Triples with duplicates?

Summing Triples problem is strongly $NP$-complete as shown by McDiarmid. Summing Triples problem: Input: list of 3N distinct positive integers Question: Is there a partition of the list into N ...
3
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0answers
330 views

Maximum Number of Edge Disjoint Paths of Length k in DAG

Is it known if the problem of finding the maximum number of edge disjoint paths of length k in a DAG is in P? Or has it shown to be NP-Complete? If so, are there approximation algorithms known for it? ...
3
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0answers
230 views

Is this modification of the subset-sum problem NP-complete?

Suppose we have input $s_1,\dots,s_n \in \mathbb Z$ and $t \in \mathbb Z$. We want to know if there exist variables $x_1,\dots,x_n$ in which each $x_i=1/2^k$, where $k \in \{0,1,2,3,4,\dots,\infty\}$, ...
2
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3answers
89 views

Proving that a class equals NP ∩ coNP

We say that a non-deterministic Turing machine is nice if for every input x the following holds: • Every computation path returns either ’accept’, ’reject’ or ’quit’. • There is at least one non-quit ...
2
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32 views

Independent Feedback Vertex Set

In Independent Feedback Vertex Set, we are given an undirected graph $G$, and an integer $k \in \mathbb{N}$. The objective is to decide whether there exists a feedback vertex set S of G of size at ...
2
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0answers
28 views

Is unary NP-Completeness equivalent to strong NP-Completeness?

I try to prove the equivalence between the two following properties of an NP-Complete problem $P$: (A) $P$ is unary NP-Complete if it is NP-Complete even if we encode the integers of the inputs with ...
2
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0answers
34 views

Under ETH: $\exists$ Problem unsolvable in $2^{o(n)}$ $\Leftrightarrow^?$ 3-SAT can be represented in linear bits

It is a popular open question if there is a problem unsolvable in $2^{o(n)}$ on inputs with $n$ bits, assuming ETH. I recommend reading that question first. That question states that, assuming the ETH ...
2
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0answers
81 views

An unknown combinatorial optimization problem

I have $N$ available sensors and $M$ devices. Each device needs $a$ sensors. One sensor cannot be used on multiple devices. Each sensor has two properties defined by $H$ and $R$. Let $\sigma_{i\_H}$ ...
2
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0answers
33 views

Sub-graph Selection Algorithm Problem (Dynamic Programming or NP)

We have an algorithm problem in hand, can you please write your ideas about this, thank you! There are N many nodes with K different colors. Some of the nodes have direct connection between each other ...
2
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0answers
56 views

3-cycle cover decision problem for directed graphs: best known algorithm and maximum size of tractable problems

I know that the 3-cycle cover decision problem for directed graphs (3-DCC), defined as finding whether a directed graph has a disjoint vertex cycle cover in which every cycle has at least 3 edges, is ...
2
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0answers
74 views

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 ...
2
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0answers
69 views

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 ...
2
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0answers
42 views

Is there a super-linear lower bound on the time complexity of all solutions of NP complete problems?

$P \ne NP$ would imply that any polynomial is a lower bound on the time complexity of any NP complete problem. Is some non-trivial lower bound known at all?
2
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0answers
66 views

Examples of exponential time linear space exact algorithms

I am looking for examples of NP-complete problem-solving exact algorithms with a linear space complexity and an exponential time complexity. Algorithms which solve the k-SAT problem exactly (such as ...
2
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0answers
158 views

Reducing Dominant Set Problem to SAT

I am trying to solve a problem and I am really struggling, I would appreciate any help. Given a graph $G$ and an integer $k$ , recognize whether $G$ contains dominating set $X$ with no more than $k$ ...
2
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0answers
74 views

Is the 3D-matching still NP-complete if all elements occur in exactly three triples?

The 3D-matching problem is known to be NP-complete even if all elements occur at most three times (see Garey&Johnson). My question: Is it also NP-complete if all elements occur in exactly three ...
2
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0answers
96 views

Reduction of Max-cut

How to show that Max-cut$_{dec}$ is NP-complete using Mon-NAE-3SAT ? Mon-NAE-3SAT definition : An instance is a m clauses of three positive literals (no complemented variable) $x \vee y \vee z$ The ...
2
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0answers
201 views

Example of *small* non monotone circuit such that any equivalent monotone circuit has greater size?

A "general" Boolean (combinatoiral) circuit is a labeled (with the labels: AND, OR, NOT, IN, OUT), directed, acyclic graph, that satisfies: fan-in=2 for the AND and OR nodes fan-n=1 for the NOT ...

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