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].

Filter by
Sorted by
Tagged with
0
votes
0answers
9 views

How is this reduction of 3-SAT to Half-SAT not valid?

I am studying algorithms and there is a question in CLRS called the Half-SAT problem We are given a 3-CNF formula with n variables and m clauses where m is even. We wish to determine whether there ...
0
votes
1answer
24 views

Reduction from Clique to IS degree at most 4

This is from the Algorithms textbook by Dasgupta,C. H.Papadimitriou,andU. V. Vazirani question 8.6 (b) that asks: Edit: And missing from the pic as @Nathaniel points out in his answer below: "...
1
vote
0answers
29 views

Polynomial variable of inapproximability after reduction

I proved the inapproximability of a problem that, given a multigraph $G = (V, E)$ and a set of vertices $U \subseteq V$ tries to maximize a score $f(U)$ whose value depends on the edges of the graph, ...
6
votes
2answers
3k views

When problem A reduces to problem B, which problem is more complex?

When discussing complexity classes, when we say that problem $A$ reduces to problem $B$, are we saying that problem $A$ is at least as complex as problem $B$, or the other way around?
2
votes
1answer
69 views

How does strong NP-completeness agree with encoding complexity?

I've recently read about the concepts of weak and strong NP-completeness, but faced a problem in wrapping my head around them. I've understood that problems which have numerical parameters (like ...
0
votes
1answer
128 views

Undecidability of closure under reverse of language accepted by TM

Prove that the following problem is undecidable using a reduction: Given a Turing machine $S$, does $S$ accept a word $w$ iff it accepts its reverse $w^R$? There is a solution here, which I don't ...
2
votes
2answers
251 views

Circuit satisfiability problem : SAT-C to SAT-2C

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 ...
2
votes
1answer
13 views

Padding a 2SAT clause

In http://web.mit.edu/neboat/www/6.046-fa09/rec8.pdf, I see that they pad a 2SAT clause $(x\vee y)$ to make it a 3SAT clause by writing $(x\vee y\vee p) \wedge (x\vee y\vee \neg p)$. Why doesn't $(x\...
1
vote
2answers
25 views

Reduction from Independent Set with fixed vertex to Independent Set

I was looking to solve this reduction, but I dont see how to construct the new graph. It seems very simple but I'm not capable of do it. I give you the complete explanation about this reduction. We ...
5
votes
2answers
5k views

Proving that the shortest simple path problem between two vertices $s$ and $t$ in a graph is NP-complete

How to show that the shortest simple path problem between two vertices $s$ and $t$ (finding a minimum weight path between $s$ and $t$) in a graph is NP-complete? I saw the following proof in a ...
0
votes
1answer
102 views

Mapping reduction from $A_{TM}$ to $INFINITE_{TM}$ same as to $ALL_{TM}$?

I was trying to solve a problem with a mapping reduction from $A_{TM}$ to $INFINITE_{TM}$, and came across a solution that was 100% identical to another solution I saw for $A_{TM} \leq_M ALL_{TM}$. ...
0
votes
0answers
33 views

Show that a set is reducible to another set

Consider an operator $+$ defined on $P(\mathbb{N})$ as follows $$A + B = \{2x:\ x\in A\}\cup\{2x + 1:\ x\in B\}$$ Show that $A$ is $m$-reducible to $A+B$ and $B$ is $m$-reducible to $A+B$ As per the ...
0
votes
1answer
14 views

Show that a set is turing reducible to another set

Consider an operator $+$ defined on $P(N)$ as follows: $A + B = \{2x\mid x \in A\}\cup \{2x + 1\mid x \in B\}$ Show that both $A$ and $B$ are Turing-reducible to $A+B$ I am kind of confused about this ...
1
vote
2answers
41 views

If $f$ reduces $L_1$ to $L$ and also $L_2$ to $L$ is $L_1=L_2$

If the same $f$ reduces $L_1$ to $L$ and also $L_2$ to $L$ does it imply that $L_1=L_2$? My intuition says no, but I couldn't find a counterexample.
1
vote
1answer
49 views

Any problem in P can be reduced to the language of odd integers

Given $A=\left\{n\in \mathbb{N} \mid \text{$n$ is odd}\right\}$, we want to prove that if $S \in P$ then there is a Karp reduction from $S$ to $A$. My attempt: If $S \in P$ we can solve $S$ with a ...
6
votes
2answers
18k views
1
vote
3answers
238 views

How is the Longest Path Problem NP complete?

From the following link: https://www.csie.ntu.edu.tw/~lyuu/complexity/2016/20161129s.pdf So basically, in our iff proof, we have to show two directions: Forward: If Hamiltonian Path has a yes-...
0
votes
1answer
19 views

Find a Cook Reduction from $R_{Clique}$ to its determinist problem

The question is to find Find a Cook Reduction from $R_{k-Clique}$ to its determinist problem. Basically: k-Clique: a group of $k$ nodes in the graph such there is an edge between every two nodes. ...
0
votes
1answer
26 views

reduction of independence problem and cluster problem

independent problem is: there is a simple and undirected graph, we are looking for the maximum vertex in which there is no edge between any two of them. cluster problem is: there is a simple and ...
-1
votes
1answer
28 views

Reduction from $VC$ to $CD$

We define the vertex cover as the problem of finding for a graph $G$, a cover of size $k$. A cover is a set of vertices such that every vertex has an edge to this set. We define CD (cycles destructor),...
0
votes
1answer
90 views

Problem with proving that $RP \subseteq NP$ : a non-deterministic TM for a language $L \in RP$

I'm having a small issue with wikipedia's proof that $RP \subseteq NP$: An alternative characterization of RP that is sometimes easier to use is the set of problems recognizable by nondeterministic ...
-1
votes
0answers
36 views

NP-completeness of disjoint paths with bounded common nodes [duplicate]

Given an undirected graph $G=(V,E)$, $k$ distinct node pairs $(s_1, t_1), ..., (s_k, t_k)$ and an integer $\delta$, determine if there exist $k$ edge-disjoint paths from $s_i$ to $t_i$ $(1\leq i\leq k)...
2
votes
1answer
359 views

Is protein folding really NP-hard and how to show that?

This question has two facets that are related. Is the general problem of protein folding really NP-hard? The hydrophobic-polar protein folding model (Ken Dill et al., 1985) stated the problem on a ...
1
vote
0answers
91 views

Reduction between Parity-SAT and approximate counting

Consider two problems as defined here. Approximate counting: Given a Boolean function $f(x)$, for $x \in \{0, 1\}^{n}$, distinguish between the two cases: The number of satisfying assignments for $f(...
2
votes
1answer
244 views

How to encode reachability in a graph with walls as a SAT problem

Suppose we have a graph that represents a grid of cells. We are given a cell to start in and a cell that's the destination. There are cells that we cannot enter and they are known as walls. Finally we ...
4
votes
2answers
319 views

Polynomial-time linear-reduction from Directed Hamiltonian Path Problem to 3SAT

Is there a polynomial-time reduction from Directed Hamiltonian Path Problem to 3SAT which is linear in the number of vertices? That is, it reduces every directed graph $G$ with $n$ vertices to a ...
0
votes
0answers
19 views

Does having a similar constraint while reducing a problem to similar problem to prove np hard means they are same?

I have been trying to find the computational complexity of my optimization problem and found that it is Np-Hard. To prove it to Np-Hard, I try reducing it Nurse Scheduling Problem. I am quite confused ...
1
vote
1answer
35 views

A question about domains in Karp reductions

A basic question or request for clarification regarding Karp reducibility: Let $\Sigma^*$ be the set of all finite strings of 0's and 1's. Call a subset of $\Sigma^*$ a language. Let $\Pi$ denote ...
1
vote
1answer
18 views

Circuits and Closure Under Reductions

Suppose that $A$ and $B$ are languages such that $A\leq_P B$ (many-to-one Karp reduction), and $B\in \mathbf{P/poly}$. How do we prove that $A\in\mathbf{P/poly}$? Using similar ideas like Cook-Levin (...
1
vote
1answer
402 views

NP Complete Proof - Polynomial Reduction

We know that the INDEPENDENT-SET problem is NP Complete i.e $\langle G',k'\rangle$ means graph $G'$ has an Independent set of size $k'$. I am preparing for the finals an a sample question is to prove ...
3
votes
1answer
174 views

Efficient algorithm to determine if a lambda calculus term is equivalent to one without a given free variable

Consider the following problem: given a lambda calculus term $t$ and free variable $v$ determine whether $\phi(t,v)$, where $\phi(t,v) := \exists t'. t' \equiv t \land v \notin FV(t')$. This problem ...
5
votes
2answers
1k views

Transforming SAT to Quadratic Programming in polynomial time

I would like to show that Quadratic Programming is NP-hard. I am currently reading a couple of papers which state that QP is NP-Hard and prove it by transforming SAT to QP, however I am finding the ...
0
votes
1answer
42 views

Is $L=\{\langle M\rangle\mid L(M)\subseteq HP\}\in coRE$?

My intuition is that $L\notin coRE$, but I haven't managed to prove that $HP \le L$, as previously I only saw reductions from $HP$ or from $\overline{HP}$ with $f$ such that $f((\langle M\rangle,x))=\...
3
votes
0answers
44 views

Sufficient condition for a complexity class's closure under NP-reductions?

Let us say that there exists a $\mathsf{NP}$-reduction from a problem $A$ to another problem $B$ when there exists a non-deterministic, polynomial-time Turing machine $T$ such that for each $a \in A$, ...
1
vote
1answer
113 views

NPC-problem reduction to triangle-free 3-colorability

lately, I have encountered a problem that I struggle to find a satisfactory solution for. I need to prove that triangle-free 3-colorability is NP-complete. Therefore I assume the right way is to find ...
0
votes
1answer
26 views

Condition to prove $f$ is a reduction

A theorem says if $f$ is a computable function and we can prove $x \in A \Leftrightarrow f(x) \in B$, then we can use reduction so $A \leq_m B$. But i'm confused if should I prove if : $(x \in A \...
2
votes
1answer
60 views

If well-founded induction holds, then the relation $\to$ on a reduction system terminates

I am trying to understand a proof from "Term Rewriting and All That" by Baader and Nipkow. Well-founded induction (WFI) is the following statement: $\forall x \in A(\forall y \in A(x \...
0
votes
1answer
57 views

Cook–Levin theorem and reduction as injective function

I saw that the injectivity "derives directly from the theorem", but i can't see how it's happen, any explanation?
0
votes
1answer
27 views

Reducing the Hamiltonian cycle to the travelling salesman problem and self loops

If this is my adjacency matrix for the hamiltonian cycle: $$\begin{pmatrix}0&1&0&1\\ 1&0&1&0\\ 0&1&0&1\\ 1&0&1&0\end{pmatrix}$$ Then a reduction ...
1
vote
3answers
8k views

reduction from SET-COVER to VERTEX-COVER

I've come with an idea for reduction from the set-cover problem to the vertex-cover problem, But I'm not sure if this reduction is correct. I saw in this post's comments that "There cannot be a nice ...
5
votes
1answer
165 views

Weakest reduction for P-completeness

It is common to define $P$-completeness with respect to logspace many-one reductions. I am looking for a complexity class $C$ such that if $C=P$ then all problems in $P$ are $P$-complete under many-...
8
votes
3answers
2k views

Solving the min edge cover using the maximum matching algorithm

To solve an instance of an edge cover, we can use the maximum matching algorithm. Edge Cover: an edge cover of a graph is a set of edges such that every vertex of the graph is incident to at least one ...
0
votes
1answer
34 views

If$A \leq_T B$ is given, can you reduce $\overline{A}$ to $B$ and vice-versa

If you are given two languages $A$, $B$ and $$A \leq_T B.$$ Is it possible to $\overline{A} \leq_T B$ or $A \leq_T \overline{B}$? Here is my shot. Case 1: $\overline{A} \leq_T B$ This is only possible ...
0
votes
2answers
35 views

For every non-trivial language $A$ and every finite strict subset $B \subsetneq A$, it's holds that $A \le_m A \setminus B$

It's claim 1 from Bader Abu Radi's solution to this question. My solution (have no idea how wrong it is): $B$ finite $\Rightarrow$ $B\in R \Rightarrow$ exists TM $\langle M_B\rangle$ that halts $B$. *...
1
vote
1answer
61 views

Can you reduce every decidable language to a regular language?

One of my previous questions on an exam was the following Can you reduce a decidable language to a given regular language? (decidable language $\leq _m$ regular language). If so, does this mean that ...
0
votes
1answer
37 views

Reduction from $A$ to $B$ as execution of Turing machines

As explained in answers to this question, reduction from $A \le B$ can be represented in the following way. But in this example: from here At least as I understand it: The reduction is from $\...
1
vote
2answers
51 views

Reduction as a flowchart

I'm trying to understand the reduction as a flowchart graph. Let's say the boxes $A$ and $B$ are TMs/Functions and $x$ is the input. Is this plot represent reduction from $A$ to $B$ ($A\le B$) or ...
2
votes
2answers
145 views

Proving decidability

Regarding the following languages $L_1$ and $L_2$, I want to prove that $L_1$ is decidable and $L_2$ is undecidable. I want to construct a turing machine which can decide $L_1$ and reduce the halting ...
3
votes
2answers
6k views

Showing that deciding whether a given TM accepts a word of length 5 is undecidable

I'm having trouble grasping this the concept of reductions. I found the solution and it looks like this: Assume that $M_5$ is a Turing Machine that can decide if a given Turing Machine $M$ accepts ...
0
votes
0answers
23 views

Reducing co-NP and NP. (Karp and Turing)

Why is it true that: If every problem in co-NP can be Karp reduced to a problem A, every problem in NP can be Turing reduced to A?

1
2 3 4 5
22