**762 reductions questions.**

I'm referring to the decision version of both $TSP$ and $HamCycle$. The first is, given a graph $G=(V,E)$, a weight function $w:E\rightarrow \mathbb R^+$ and an integer $k$, is there a simple cycle ...

What I want to do is turn a math problem I have into a boolean satisfiability problem (SAT) and then solve it using a SAT Solver. I wonder if someone knows a manual, guide or anything that will help ...

Prove that the language $LM =\{\langle M,x\rangle\mid \ M \text{ accepts }x\text{ and rev}(x) \}$, where $\mathrm{rev}(x)$ is the reverse of the string $x$, is undecidable with a reduction from $A_{\...

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^...

I have this HW problem:
Let $F$ be the set of computable total functions, and let $\emptyset\subsetneq S\subseteq F$. Denote
$$L_S=\{ \langle M \rangle | M \text{ is a TM that computes a function ...

I have the following language : $L=\{\langle C_1,C_2\rangle \text{ | } C_1 \text{ and } C_2 \text{ are two circuits that calculate different function}\}$. We can call this language SAT-2C.
Prove that ...

We suppose we have a polynomial algorithm which receives a graph $G$ (any graph) and returns a stable set of $G, SA(G)$ with the following property:
$\alpha(G) − |SA(G)| \leq k$ , for every natural $...

So, I have two problems:
Minesweeper: given an undirected graph where some vertices have a whole number associated with them, check if there is a way to mark some of the non-numbered vertices such ...

It seems that the standard reduction method you see online from 3SAT to 4SAT is that we let $\phi = (a \lor b \lor c)$ be a 3SAT clause, and so there is an assignment that satisfies $\phi$ iff $\phi' =...

So, as is known, ILP's 0-1 decision problem is NP-complete. Showing it's in NP is easy, and the original reduction was from SAT; since then, many other NP-Complete problems have been shown to have ILP ...

Can someone please show me a worked example of a polynomial time reduction of Integer Linear Programming to 3-SAT (in CNF)?
Take a system of inequalities in the form:
$$\mathbf{Ax} \leq \mathbf{b}$...

Disclaimer: This is a homework question.
I would like to reduce vertex cover problem to the following problem:
$$L = \{G \mid G\text{ has a vertex cover of size } |V(G)|/2\}\,.$$
I have divided the ...

A subset $S$ of vertices in an undirected graph $G$ is called almost independent if at most 100 edges in $G$ have both endpoints in $S$. Prove that finding the size of the largest almost-independent ...

We have two $NP$ complete problems $\Pi_1$ and $\Pi_2$. Suppose $\Pi_1\rightarrow\Pi_2$ be a many one parsimonious reduction.
If $\Pi_1$ has an FPTAS then does $\Pi_2$ also have?
If $\Pi_2$ has an ...

I am working on an assignment where one of the sub questions is:
Let $A$ and $B$ be languages. Suppose $A$ is context free and $A ≤_m B$, which means that there is a computable function $f\colon \...

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 ...

In "Computational Complexity" by Arora and Barak they state that the following is $NP$-complete:
$\{ \langle \alpha, x, 1^n , 1^t \rangle : \exists u \in \{0,1\}^n \text{ s.t. } M_{\alpha} \text{ ...

I want to start a wiki post about meta-result of meta-reductions in the theory of $\mathrm{NP}$-completeness.
This can be regarded as a reference request post. Any links are appreciated.
At least, ...

I'm a bit confused about some proof that PATH-SELECTION-PROBLEM is NP-complete (Problem 9, chapter 8 in "Algorithm Design" by Tardos and Kleinberg) that I found in some solution manual here:
https:...

Is this variant of Vertex Cover Cook-complete for $\mathrm{NP}$?
Input: An undirected graph $G(V, E)$ together with a vertex cover $C\subseteq V$
Output: YES if there exists a vertex cover $C'\...

I have a topcoder-like problem that I'm having trouble with:
We are given three strings A, B and C.
What is the length of the longest common subsequence of A and B, which has C as a substring?
...

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. ...

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 ...

The theory of NP-completeness was initially built on Cook (polynomial-time Turing) reductions. Later, Karp introduced polynomial-time many-to-one reductions. A Cook reduction is more powerful than a ...

Are there any decision complexity classes, where the "-Hard" version of the class does not intersect with the original class?

I'm curious about whether there are any complete problems in the Arthur-Merlin complexity class. Graph Non-Isomorphism (GNI) seems to be the canonical example of a problem in AM, but it's probably not ...

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 ...

The most common method of proving that some problem, P, is at least as difficult as some other problem, Q, is to demonstrate that Q can be solved in terms of P (with the usual caveats for the ...

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 ...

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 ...

A widely used example of reductions, is a reduction of $A_{TM}$ to $HALT_{TM}$.
How to show the opposite reduction, meaning of $HALT_{TM}$ to $A_{TM}$, if possible.

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 ...

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 ...

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 ...

I'm asking this, because in every exercise I check if I can relate it to one of the things I know,
like:$A_{TM}$, $\overline{A_{TM}}$, ${HALT_{TM}}$,$\overline{HALT_{TM}}$, $E_{TM}$, $\overline{E_{...

Is there a way to convert a 3SAT formula into a equisatisfiable 2SAT formula? Each method is of interest, even those that grow exponentially.
(So if, for example, my 3SAT formula has 16 variables and ...

If we have two languages $L_{1} \subseteq \Sigma^{\ast}_{1}$ and $L_{2} \subseteq \Sigma^{\ast}_{2}$
I proved that when $L_{2} \in \textbf{P}$ and $L_{1} \leq_{p} L_{2}$ then $L_{1} \in \textbf{P}$
...

Given the following definition of Careful 5COLORING:
A 5-coloring is careful if the colors assigned to adjacent vertices
are not only distinct, but differ by more than 1(mod 5)
how would a ...

Let $A,B$ be two languages, for which we know:
$A \in PSPACE$
$A\le_LB$
Can we conclude from the above that $B \in PSPACE$ ?
I think the answer is no, however I don't know how to ...

The problem I have is as follows:
I have a complete bipartite graph $G=(V \cup C,E)$ as input, where $|V|=1, |C|=n, |E|=n$
The interpretation is that the node of $V$ is a vehicle, the $n$ nodes of C ...

I've read about the reduction from 2-partition for the problem of minimizing weighted completion time with release dates but I'm not very experienced in doing reductions so I want to verify that my ...

Is my logic correct? If so, is this a new reduction and algorithm from 3 SAT to clique? I could only find one SAT to clique reduction; it wasn't this.
Definitions:
A clause group of a SAT instance ...

Lets say Problem A,B are in NP.
Can we reduce Problem A to B? Meaning A $≤_p$ B? or A $≤_t$ B
Is there a difference in "hardness" of a Problem even in NP?
Or must Problem B at least be NP-Complete?

Given:
L1 = {$0^k1^k$|k ∈ N}
L2= {1}
L1 $≤_p$ L2
There must be a function
$f:$$Σ^*$ $\rightarrow$ $Σ^*$
w ∈ L1 $\iff$ f(w) ∈ L2
Lets say a word in L1 is mapped to 1 in L2.
If it is not in L1 ...

When do you use Turing- and when Karp Reduction? What are the advantages and disadvantages?
I've read about Karp Reduction mainly used in the Context of reducing a Language:
e.g. L1 $≤_p$ L2

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