# 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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### Reduce Subset-Sum to Sat

Is there a reduction from SUBSET-SUM to SAT? Just general SAT, not 3-SAT. Also the given multiset S only has positive integers. SUBSET-SUM is defined as follows: Input: a multiset S = { x1 , ... , xn }...
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### Prove that the following problem is NP-complete

I am studying computational complexity and trying to prove that the following problem is NP-complete. Ursula is planning to make all the city’s kids sick the day before Thanksgiving, by overfeeding ...
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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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### Find the mistake in this demonstration

Let L={0^n 1^n | n>=0}, we show that L≤p VERTEX-COVER through this reduction: f(<G,k>) = { 01 if G has a k-cover; 0 otherwise} Since VERTEX-COVER ∈ NPC, L≤p VERTEX-COVER and L ∈ P. We get P=...
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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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### Reduce 3-SAT to EQ-GF2 in polynomial time

The EQ-GF2 problem is the following: given a system of polynomial equations over integers modulo 2, determine if there is an assignment that satisfies all the equations simultaneously. For example, {x ...
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### Which is the best theory of computation/automata/formal language book for practice problems(unsolved)? [closed]

I have Introduction to automata theory, languages and computation by hopcroft, rajeev motwani, jeffrey d ullman Elements of the theory of computation HR lewis Christos H papadimitriou Theory of ...
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### 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 $i,j\in I$. ...
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### Simple measurement of complexity in Game Of Life

For a project of mine I need to measure the complexity of a game “game of life”(it is a cellular automata) after a given number of iterations. I’m not searching for a complicated way of measuring it ...
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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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### 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 ...
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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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### 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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### Rearranging strings so that the Hamming distance between them is 1

This is a question from CodeFights.com: Given an array of equal-length strings, check if it is possible to rearrange the strings in such a way that after the rearrangement the strings at ...
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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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### 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 ...
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### How to prove that the generalized assignment problem (GAP) is NP-hard?

Specifically, what NP-hard problem can we reduce (the decisions version of) GAP to and how do we prove its correctness? The decision version of the generalized assignment problem is to determine ...
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### Is this variant of the pinwheel scheduling NP-Hard?

I wonder if the following variant of the pinwheel scheduling is NP-Hard. Given a set of n radars S = {s$_1$, s$_2$... s$_n$} and a set of m areas A = {a$_1$, a$_2$, ... a$_m$}. Each radar s $\in$ S ...
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### P-NP related 3 sub-problems

This is a question on a practice final. Which of the following statements are true? If it is false, what is the underlying reason behind that? I. If 3-CNF-SAT is in P, then Clique is also in P. II. ...
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### Number partition subjected to the cardinality of subset

I hope someone can take some time to consider the following problem and welcome to discuss together. Number partition problem is one of well-known NP-hard problems. Now I am considering the hardness ...
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### 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 ...
Take this example mentioned here: NP-hard problems with very fast exponential-time algorithms We can create such problem by padding assuming ETH‌. Take an NP-complete problem $L$ such that $L$ is ...
Karp reducibility between NP-complete problems $A$ and $B$ is defined as a polynomial-time computable function $f$ such that $a \in A$ if and only if $f(a) \in B$. I am interested in polynomial-time ...