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# 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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### Reducing optimization problem to its decision version

There are two problems that need to be solved. Both problems use a compatibility matrix $C$, where $C[a, b]$ is how compatible students $a$ and $b$ are. (1) Given an $n \times n$ compatibility matrix ...
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### Is the solution to Independent Set or Vertex Cover from 3-SAT optimum?

There are plenty of resources online discussing 3-SAT reductions to Independent Set or Vertex Cover problem. I am unable to find a resource which states that a satisfiable assignment to 3-SAT results ...
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### Is Rectilinear Steiner Tree still NP-complete when points have integral coordinates?

Garey proved that the Rectilinear Steiner Tree problem is (strongly) NP-hard. I wonder if it is still true when we retrict the points to have integral coordinates and lie on a square of side lenght n^...
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### Weakest reduction for 3-$\mathrm{SAT}$

Having read all these posts Constant-depth threshold circuit for $\mathrm{PP}$ Is there any interesting consequence of $\mathrm{DLogTime}$-uniform ${\mathrm{Mod}_6}^0=\mathrm{NP}$ I wonder about ...
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### Looking for a problem provably not in P

My basic position is that everything is in P. Then comes the time hierachy theorem and EXP. That's easy: simulate and then diagonalize. After that comes EXP-completeness; that's difficult to swallow. ...
2answers
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### Is this problem NP-hard? Maximizing selected sets so that their union is less than k?

There is an NP-hard problem called Minimum k-Union where we are given a set system with $n$ sets and are asked to select $k$ sets in order to minimize the size of their union. I'm currently ...
1answer
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### Understanding reductions for NP-completeness

Let's I have to make the following reduction: $$\text{CLIQUE}\le_p \text{VERTEX-COVER}$$ The technique of building the reduction is - Assume you can find a $\text{VERTEX-COVER}$ of size $k$, in ...
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### Path in a vertex-weighted undirected graph

Is it an $NP$-hard problem? You're given an undirected graph $G(V,E)$ with vertex weight $w: V \to \mathbb{N}$ and a function $\mathrm{max}$-$\mathrm{visit}: V \to \mathbb{N}$ and a number $W$. Does ...
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### PCP undecidability

There is a popular proof for the undecidability of the PCP (Post correspondence problem), which is outlined here: https://en.wikipedia.org/wiki/Post_correspondence_problem I'll assume whoever will ...
2answers
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### Solve Hamilton Circuit with Hamilton Path

I want to show the reduction $HC \leq HP$. Let $G=(V,E)$ be my undirected graph. My idea is: For each edge $e=(u,v) \in E$ check whether $(V,E\backslash\{e\})$ has a Hamiltonian Path. If this is true ...
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### If $L=\big\{\langle M_1,M_2\rangle\mid M_1, M_2\text{ are TM and } L(M_1)\cup L(M_1)=\Sigma^* \big\}$ is in $RE$ or $coRE$ or not in $RE\cup coRE$?

I tried to solve it as the following: $$\overline{L}=\big\{\langle M_1,M_2\rangle\mid M_1, M_2\text{ are TM and } L(M_1)\cup L(M_1)\neq\Sigma^* \big\}$$ I'll show that $\overline{L}\not\in RE$ by ...
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### Finding reduction to prove that a language is NP-complete

I need to prove that the following problem is NP-complete: We have $n$ diplomats from $n$ countries and we need to seat them around a round table. We also get a list of diplomats who don't get along ...
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### “Fuzzy” Chinese Remainder Theorem NP-hard?

I have some "fuzzy" congruences like these: \begin{align} \\ x&\equiv a_1 \mod 3 \text{ with } a_1 \in \{0,1\},\\ x&\equiv a_2\mod 5 \text{ with } a_2 \in \{0,3\},\\x&\equiv a_3 \mod 7 \...