# Questions tagged [reductions]

In computability and complexity, finding mappings between problems that allow solving one problem using a solution of another one. For reduction in programming language theory (e.g. beta-reduction), see [lambda-calculus] or [term-rewriting].

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### Find an undecidable language that is mapping-reducible to its complement

As the title suggests. Also, such a language must satisfy that neither it nor its complement are semi-decidable. I already know that $All_{TM}, EQ_{TM}, T$ (that is the set of all deciders) satisfy ...
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### Optimization problem vs decision problem - reduction

Assume we have an optimization problem with function $f$ to maximize. Then, the corresponding decision problem 'Does there exist a solution with $f\ge k$ for a given $k$?' can easily be reduced to ...
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### Showing that the set of TMs which visit the starting state twice on the empty input is undecidable

I'm trying to prove that $L_1=\{\langle M\rangle \mid M \text{ is a Turing machine and visits } q_0 \text{ at least twice on } \varepsilon\} \notin R$. I'm not sure whether to reduce the halting ...
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### Is MAX-SAT NP-hard?

Is the MAX-SAT problem NP-hard? From the Wikipedia page: The MAX-SAT problem is NP-hard, since its solution easily leads to the solution of the boolean satisfiability problem, which is NP-complete ...
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### A polynomial reduction from any NP-complete problem to bounded PCP

Text books everywhere assume that the Bounded Post Correspondence Problem is NP-complete (no more than $N$ indexes allowed with repetitions). However, nowhere is one shown a simple (as in, something ...
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### What complexity class does this variation of traveling salesman problem belong to?

Given a TSP instance $T$, decide whether changing the city coordinates by adding a vector of coordinates $v$ will change the optimal TSP objective by atleast $x$. The city coordinates are integers. ...
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### Is SAT in P if there are exponentially many clauses in the number of variables?

I define a long CNF to contain at least $2^\frac{n}{2}$ clauses, where $n$ is the number of its variables. Let $\text{Long-SAT}=\{\phi: \phi$ is a satisfiable long CNF formula$\}$. I'd like to know ...
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### Is Karp Reduction identical to Levin Reduction

Definition: Karp Reduction A language $A$ is Karp reducible to a language $B$ if there is a polynomial-time computable function $f:\{0,1\}^*\rightarrow\{0,1\}^*$ such that for every $x$, $x\in A$ if ...
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### The $\text{k-key}$ problem

Given an undirected graph, I define a structure called k-key as a path containing $k$ vertices which are connected to a simple cycle which contains $k$ vertices as well. Here's the k-key problem: ...
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### Does a collision oracle for the pigeonhole subset sum problem produce solutions?

I am reading "Efficient Cryptographic Schemes Provably as Secure as Subset Sum" by R. Impagliazzo and M. Naor (paper) and came across the following statement in the proof of Theorem 3.1 (pages 10-11): ...
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### Solve a problem through reduction

I am aware that for a problem to be considered NP-Hard, any problem in NP must be reduceable to your problem (problem which you are trying to prove is NP-Hard). Let's assume that you have proven that ...
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### Turing reducibility implies mapping reducibility

The question is whether the following statement is true or false: $A \leq_T B \implies A \leq_m B$ I know that if $A \leq_T B$ then there is an oracle which can decide A relative to B. I know that ...
I'm studying for my final in theory of computation, and I'm struggling with the proper way of answering whether this statement is true of false. By the definition of $\leq_m$ we can construct the ...