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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Minimum number of intervals to cover all possible colors

Given $n$ points in $\mathbb{R}$ each colored with one of following three colors $$C=\{c_1, c_2, c_3\}.$$ In polynomial time, Choose the minimum number of intervals of length $1$ each containing some ...
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Difficulty in finding a counter example for a polynomial reduction

I am doing some exercises on proving NP Complete problems and the first problem was directed bipartite graphs with a Hamiltonian cycle and the second one was undirected bipartite graphs with a ...
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How to reduce 3-SAT to Set Splitting

I've been reading through Garey & Johnson's "Computers and intractability", and a problem SP4 caught my attention. It is stated as following: Given a collection $C$ of subsets of a ...
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Similar problem to Subset Sum?

I've been trying to search for a problem which I think could be similar to Subset Sum. The definition of the problem would be as follows: Given k $\in$ $\mathbb{Z}$ and S = {$s_1$,...,$s_n$} s.t. $s_i ...
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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 ...
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Reduction from SAT to EXACTSAT

PROBLEM: EXACTSAT INPUT: A boolean formula $\phi$ in CNF with $n$ variables, and a natural number $k \le n$. OUTPUT: "Yes" if and only if there is truth assignment $\theta$ which sets ...
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Can we DISPROVE that a problem is NP-complete

So I basically had an exam question in which we were given a problem and we had to prove or disprove that it was in NP-complete. I tried to prove it but could not because apparently it could not be ...
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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 ...
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Is it known whether PARTITION is NP-complete via first order reductions?

The PARTITION decision problem is defined as follows (taken from COMPUTERS AND INTRACTABILITY from Garey and Johnson): Instance: A finite set $A$ and a size $s(a) \in \mathbb{Z}^{+}$ for each $a \in A$...
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ILP - Maximize the number of pairs of variables with the same value

I have a 0-1 integer linear program for a set of $2n$ variables $S = \{x_1, ..., x_n, y_1, ..., y_n\}$. My objective is to maximize the number of pairs $(x_i, y_i)$ such that $x_i = y_i$, $i = 1, ..., ...
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How to reduce SUBSET-SUM with integers to SUBSET-SUM with non-negative integers?

The subset sum problem is as follows: Given a sequence of integers $\mathcal S=(a_1, ..., a_n)$ with cardinality $n$ and an integer $T$, determine whether there is a subsequence of $\mathcal S$ whose ...
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Prove TILING is NP-Complete

I have a homework task to show that $\mathrm{TILING} = \{(T, 1^N) \mid \text{it is possible to cover } N \times N \text{ square with tiles from }T\}$, where $t\in T$ is $C^4$ for some color set $C$, ...
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How do I prove that the clique problem is polynomial-time reducible to the odd cycle transversal problem?

I have the following problem: Let $H=(W, F)$ a graph and $k \in \mathbb{N^*}$ be an instance for problem $\textbf{CMP}$ (i.e. the clique problem). Let $W'$ a set of new vertices, $|W'|=|H|=n$. We ...
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Is Self-Modifying Turing Machine equivalent to NTM or TM?

Let SMTM be Turing Machine, but the commands recorded in which can change to others in some random way (for example, choose with a 50/50 probability the command to move to the right or move to the ...
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About NP-completeness and reduction

Probably this is a basic question but I'm not sure how to finish this proof. I have a problem $X$ and I want to prove that it is possible to reduce $X$ to another problem $Y$. I know that $Y$ is NP-...
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Hitting set problem and Vertex Cover

In Chapter 8 Question 5 of Kleinberg and Tardos, the problem is as follows: Consider a set $A = \{a_1, \ldots , a_n\}$ and a collection $B_1, B_2, \ldots , B_m$ of subsets of $A$ (i.e., $B_i \...
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A be an NP-complete problem, and if B be an NP-hard problem. If A is polynomial time solvable then is B is polynomial time solvable?

if A be an NP-complete problem, and if B be an NP-hard problem. If A is polynomial time solvable then is B is polynomial time solvable? on the contrary, if A be an NP-complete problem, and B be an NP-...
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Prove that T3SAT is NP-Complete

Instance: A boolean formula f(x1, . . . , xn) in 3CNF form, with m clauses labelled C1, . . . , Cm. Is there an assignment to x1, . . . , xn such that every third clause is False and all other clauses ...
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Is this variant of Subset Product NP-hard? [duplicate]

Given a set $Y$ with whole number positive divisors of $N$, is there a combination of divisors that have a product equal to $N$? Does Subset Product remain NP-hard when whole number divisors are only ...
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Is integer multicommodity flow problem is NP-hard?

As Wikipedia states the time complexity of Integer Linear programming(ILP) is NP-hard, so this means integer multicommodity flow problem is also NP-hard?
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How exactly is the process of showing a problem to be NP-Complete a proof by contradiction?

The steps involved in proving that a problem is NP-Complete are fairly straightforward to follow, it's the logic behind why the proof is valid that's really throwing me for a loop. Okay so an easy one:...
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Is this set covering problem NP-Hard?

Consider this variant of set covering problem. Input: a collection of sets $S = \{s_1, s_2, \ldots, s_n\}$ and a universal set $U$, in which $s_k \subseteq U$ for all $k$. The problem is, divide $S$ ...
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chromatic number is np-hard

I'm referring to a question in this book: Algorithms by Jeff Erickson, link:http://jeffe.cs.illinois.edu/teaching/algorithms/notes/J-approx.pdf, in particular, pg.21, Q11. The author mentioned that ...
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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,...
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Variant of "Exact Cover by 3-Set "

Exact cover by 3-sets is 𝖭𝖯-complete: Instance: Given a finite set $X = \{x_1, x_2, …, x_{3n}\}$ of $3n$ elements and a collection $C = \{(x_{i_1}, x_{i_2}, x_{i_3})\}$ of 3-elements subsets of $X$; ...
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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 ...
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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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Reduce subset sum to simple path

I have a similar question as this post:Given a complete, weighted and undirected graph $G$, complexity of finding a path with a specific cost My question: Given a weighted and directed graph $G$, it ...
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Is Knapsack-optimization problem NP-hard while Knapsack-search problem NP-complete?

I am learning Computational Complexity. Is Knapsack-optimization problem (find an arrangement to maximize the value) known to be NP-hard, while Knapsack-search problem (find an arrangement so that ...
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Are there any FNP-complete problems with a unique solution?

Are there any FNP-complete problems where there's only one possible solution? For example, the travelling salesman problem can have multiple routes all shorter than $X$. There's only one shortest ...
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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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NP-Completeness of SAT with given hamming weight k [duplicate]

I think that the following problem is NP-Complete but I don't have any idea of how doing the reduction. Input: A propositional formula $\varphi$ and a number $k$. Output: Yes if exists an valuation $\...
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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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Exact and approximate agorithms for independent set probem in large graphs

I have a problem which could be stated as follows: Given an unweighted undirected graph $G=(V, E)$ and positive integer $k \leq |V|$, I need to find a subset of vertices $R \subseteq V$ such that $|R| ...
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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 ...
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Transitions of Turing machine in Cook Levin theorem proof

I am looking at the proof of the Cook-Levin theorem in Computers and Intractability: A Guide to the Theory of NP-Completeness. In particular, I find one thing ...
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Smallest 3-SAT problem that no one has been able to solve?

In number theory progress is sometimes guided by people stating a specific Diophantine equation that they don't know how to solve. Is there anything similar in the field of Boolean satisfiability? ...
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Check if the given satisfying assignment of CNF formula is lexicographically the first

If there is a CNF Boolean formula in $n$ variables then the potential satisfying assignments are the binary strings of length $n$. Given a CNF Boolean formula and a satisfying assignment how ...
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Quasilinear time algorithm for 3-SAT

Is it consistent with the current knowledge that there is an algorithm solving a 3-SAT instance in $n$ clauses in quasilinear time in $n$?
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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 ...
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Reduction of np to npc

Given that $A$ is $NPC$ problem. And I need to check "if $D$ belongs to $NP$ and $D\leq_p^\mathsf{}A$ then $D$ is $NPC$" is true or not? My approach: Since $D\leq_p^\mathsf{}A$, therefore $...
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NP-completeness of satisfiability of formula over 50 variables

Given a boolean formula $F$ of length $n$ defined over a fixed number of variables (say 50), is it NP-complete to decide whether $F$ is satisfiable?
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Can one show NP-completeness by showing a reduction to 3SAT?

The standard technique to show NP-completeness of $L$ seems to be to show that $L$ is in NP, and then to show that some NP-complete language can be reduced to it. What if one tried to show it the ...
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Is there an instance of 3-SAT in less than 100 variables that no one has been able to solve?

In number theory, progress is sometimes guided by people stating a specific Diophantine equation that they don't know how to solve. Is there anything similar in the field of Boolean satisfiability? ...
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Complexity class of computation of Homfly polynomial

It is claimed that "the problem of the computation of the homfly polynomial is NP-hard." but is it known if it is NP-complete? By the definition of NP-completeness, wouldn't it be enough to ...
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Proofs of reduction of any hard problem

Approach:1 To prove any unknown problem $B$ is NPH then take any known NPH problem $A$ (e.g. $3$-sat) which reduces to $R$ in polynomial time. If I take any instance example $I_1$ of $A$, then ...
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If $A\leq_m^\mathsf{}B$ and $B$ is a regular language, does that imply that $A$ is a regular language?

Corollary:1 We know that if $A\leq_m^\mathsf{}B$ and $B$ is decidable then $A$ is also decidable. This is because if there exists a specific algorithm for solving $B$ and we can also reduce $A$ to $B$ ...
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Given a graph and specific VC instance, find number of variables when reducing from VC to SAT

I have question already answered from past exam, and I'm trying to figure where my logic fails. Given a graph find vertex cover of size 2. The question is how many variables are there going to be for ...
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How hard is random SAT?

There is plenty of research into the so-called "random SAT" problem, where we basically try to solve SAT instances with clauses chosen "at random" in some sense. There are all ...

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