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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Is L decidable or not

Let $L = \{\lt M\gt | M$ is a $TM, L(M) = \{1^n0^n | n\ge0\}\}$. Create a reduction from $A_{TM}$ (acceptance problem) to $L$. Is $L$ not decidable? But isn't $L$ decidable since it is possible to ...
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Is the following fuction computable?

I'm trying to show that $K_1 \le_1 K$ where $K$ is the diagonal halting set $\{x : \varphi_x(x) \downarrow\}$ and $K_1=\{x: \exists y \,\, \varphi_x(y) \downarrow\}$, then I defined the function $$\...
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Is the number of NP-complete problems finite?

It should be straight forward to show that there are infinitely many NP-hard problems: Proof: Take the problem Remove 1 Vertex 3-COL ($R1V3COL$) which takes a graph $G=(V,E)$ as an instance and ...
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if there is no reduction from A to B

I'm facing the following question : If there is no $𝐴\leq_𝑚𝐵$ reduction, does this necessarily mean that A is not decidable? for any choice of B. thanks in advance.
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Is the Clique Problem polynomial time reducible to the graph-Homomorphism Problem and if so what does the reduction look like?

Is the k-Clique Problem (given a Graph G and a natural number k does G kontain a Clique of size k) polynomial time reduzible to the graph-Homomorphism Problem (given two graphs, G and H, is there a ...
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What is the result of a reduction operation called?

Is there a commonly used term for the result of a reduction operation? "reductant" does not quite sound right...
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Undecidability of two Turing machines acting the same way on an input

So I need to find a reduction to the (undecidable) problem of deciding if two Turing machines $M_1$ and $M_2$ behave the same way on an input $x$. "Behaving the same way" is defined like this: $M_1$ ...
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In the reduction from HALT to ALLHALT, why does the constructed Turing machine loop indefinitely when the inputted Turing machine rejects?

Let HALT be the language $\{\langle M, w\rangle : M\text{ is a TM that halts on }w \}$. Let ALLHALT be the language $\{\langle M\rangle : M\text{ is a TM that halts on all inputs}\}$. Use a reduction ...
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Polynomial reduction proprietries: reflexive, transitive, but not symmetric

From theory I know the polynomial reduction is: reflexive $A ≤ A$ (A Language) transitive $A ≤ B$ and $ B ≤ C$, then $A ≤ C$ (A, B, and C Languages). What about symmetric? There is an example with ...
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Complexity problem reduction?

Let say A and B are two decesion problems where A $\le$ B polinomial reduction is true. Is this : A̅ $\le$ B̅ also true? If so, can you show an exemple, if not why?
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Can polynomial many-to-one reduction be done to a specific problem instance?

Let's say I reduce the problem $A \in L$ to $B \in K$ , with a function $f: \Sigma^{*} \rightarrow \Gamma^{*}$ such that $w \in L \Leftrightarrow f(w) \in K$ . I know if I want to solve $A$, given ...
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NP-completeness for integer linear program

This is a homework problem, so I don't want the solution. I need a hint which problem to reduce to the following and/or how to start on it. We were thinking of TSP or independent set but couldn't come ...
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PSPACE-hardness of the unbounded puzzle

Given a board of size $n \times 1$ where $n=\infty$ (basically a tape, one way infinite), and a set of colors $C$, some starting color $c_{start} \in C$, a set of templates $T$ in a form $(c_k, i, c_i,...
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Prove the Droid Trader Problem is NP-complete

This question is from chapter 8, exercise 35, of Algorithm Design by Kleinberg and Tardos. A player in the game controls a spaceship and is trying to make money buying and selling droids on different ...
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Proving NP-Complete problem by reduction of subset-sum

My assignment question as "given a multiset of symbols (letters) L from an alphabet Σ (thus, the same letter may appear in L multiple times), and a set of words W ⊆ Σ' , UseAllLetters asks if it is ...
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Show that SQUARED-SUM-PARTITION is NP-complete

Consider the following problem SQUARED-SUM-PARTITION. You are given positive integers $x_1, \dots, x_n$, and numbers $k$ and $B$. You want to know whether it is possible to partition the numbers $\{ ...
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How do I reduce subset sum to another problem in NP?

I'm trying to solve the following problem about arranging pens on rows. The problem goes as the following. Given $n$ integers $l_1, \dots l_n$, the lengths of the pens, r rows and a goal G. Is it ...
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Hamiltonian cycle, verifying and finding

If we have an algorithm that in polynomial time says if a graph G has an hamiltonian cycle, can we have an algorithm that in polynomial time find an hamiltonian cycle? My attempt is to delete an edge ...
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Decide if Turing machine's language contains either a or b string

during school exercises we worked on decidability problems and there was one I don't really understand. We were provided with solution and explanation regarding this exercise but still I need more ...
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Reduction from Vertex Cover to Dominating Set

I am trying to reduce the vertex cover (decision) problem to the dominating set (decision) problem in order to prove that the latter is NP-hard. After some research online, I found that many articles ...
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Are there any problems that reduce to the halting problem?

I'm reading through sipser and there is a lot of computability problems that the halting problem reduces to, i.e. if $A_{TM} = \{\langle M,w\rangle : M$ accepts input $w\}$ then $A_{TM} \leq P$ where ...
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Example of two undecidable languages that cannot be mapping reduced to each other

I want to find two undecidable languages $A$ and $B$ that $A$ cannot mapping (i.e. many-one) reduce to $B$, $B$ cannot reduce to $A$. One of my thought is to let $A$ be the halting problem, let $B$ be ...
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Given a set, partition it into ordered triples

I have a set $S$ of $3m$ positive numbers $\{a_1,a_2,\ldots,a_{3m}\}$. The question is: can you select $m$ disjoint triples $(a_i,a_j,a_k)$ from $S$ such that $a_i-a_j-a_k\geq1$? I was trying to ...
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How do we construct reductions for NP-Completeness

I'm wondering in what direction we construct reductions to prove that a problem is NP-complete. Say the question is asking to prove that the vertex cover problem is NP-complete given that the ...
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Why does such reductions work [duplicate]

In class we saw examples of reductions like from Independent Set (IS) to Longest common subsequence (arbitrary number of sequences) (LCS) $V = \{v_1,\ldots,v_n\} E =\{ e_1,\ldots, e_m \}$ The ...
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Is finding the minimum feedback arc set on graph with two outgoing arcs for each node np-complete?

I have a graph with at most two outgoing arcs for each node and I need to extract a DAG by removing the least number of arcs. I know that the general problem is np-complete but i can't reduce it to ...
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Reductions from non decision problems

I want to show a minimization problem $Y$ has no approximation factor of 1.36. To be more specific the problem $Y$ is the exemplar distance problem between two genomes. Could I reduce from the min ...
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Partition into pairs with minimum absolute difference, NP-hard?

I have a set $S$ of an even number of positive elements $2m$ and $m$ values $t_1,t_2,\ldots,t_m$ where each $t_i\leq1$ for all $i$. The question is: can you select $m$ disjoint pairs $(a_i,b_i)$ from ...
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Can current quantum computers decide languages that Turing Machines cannot?

I am currently learning Computing Theory at university, and we were on the topic of Turing-Decidability, Recognizability, etc. Showing that a problem is undecidable with Turing machines due to ...
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How do I go about creating a mapping reduction?

So I understand what a mapping reduction is and how to create them for simpler problems such as a reduction from the set of even numbers to set of odd numbers however, seemingly more complicated ...
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Reduce duplicate subset sum problem to distinct subset sum problem?

In duplicate subset sum problem (DuSSP), we are given a multiset $\{a_1,a_2,\ldots,a_n\}$ where some of the $a_i$ are duplicates. We can assume that $a_1\leq a_2\leq \cdots\leq a_m.$ We are also given ...
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Clique-or-almost reduction to clique

I saw the posted question here about a direct reduction from near-clique to clique. Clique-or-almost is like near-clique but with the option for a complete clique of size $k$, I mean that perhaps an ...
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What are the requirements for a superset of P to be closed under karp reductions?

So today in our exercise session on complexity theory we discussed that P, NP, and BPP are closed under karp reduction. We also figured that the proofs could likely be expanded to straight ...
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If $Q$ reduces to $L$ then $\overline{Q}$ reduces to $\overline{L}$

The following exercise is taken from Chapter 17 of Languages and Machines by Thomas Sudkamp: Let $Q$ be a language reducible to a language $L$ in polynomial time. Prove that $\overline{Q}$ is ...
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LogSpace reductions vs. PTime reductons for defining PSpace-completeness [closed]

Continuing Is every PSPACE-complete problem complete with respect to logspace reductions? : earlier, PSPACE-completeness was defined via logspace reductions (e.g., cf. http://www.cs.cornell.edu/~kozen/...
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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 ...
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Deciding whether $f(x) = f(y)$ is beyond RE and coRE

I would like to prove that the following subset is outside both RE and coRE: $$A = \{ (p, (d_1, d_2,\dots, d_k)) \mid \text{for each } 1 \le i,j \le k, \; [p]d_i = [p]d_j \}, $$ where $p$ is a ...
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Why log-space reduction is used for NL-completeness while PSPACE reduction isn't used for PSPACE completeness?

NL-Complete languages are defined by Log-space reduction, while PSPACE complete languages are defined by poly-time many-to-one reduction. According to these posts : Why not polynomial-space reductions ...
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Concurrent Element-wise Reduction Algorithm (multi-threaded) (C++)

I'd like to implement a high-performance implementation of a multi-threaded reduction, element-wise, on x86 CPUs. Without loss of generality, assume the reduction operation is a sum of integers (so, ...
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Is Finding A Hitting Set of Size n/2 NP-Hard?

In Hitting Set problem we are given a collection E of subsets of V and we want to find smallest subset H of V which intersects (hits) every set in E. In decision version of the problem, we are asked ...
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Concrete example of Vertex Cover to Subset Sum reduction

In Computational Intractability, we often come across a need to reduce Vertex Cover (VC) problem to a Subset Sum problem, mostly to prove Subset Sum is NP-Complete. I also see a reduction in the line ...
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I want to know where there is the flaw in my argument

I came across following problem to finding whether the following language is decidable or semi-decidable or not even a semi-decidable. $L: \{\langle M\rangle: M\space is\space a\space TM\space and\...
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How to prove NP-hardness from scratch?

I am working on a problem of whose complexity is unknown. By the nature of the problem, I cannot use long edges as I please, so 3SAT and variants are almost impossible to use. Finally, I have decided ...
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Proof by reduction and Turing machines [closed]

This is a practice question I have, but I can't wrap my head around it. ............. Let L = {M | M is a TM that halts with exactly two words on its tape in the form Bw1Bw2B}. B = Blank Position the ...
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Can't find a mistake in reduction from RE language to a non-RE language

In the book Introduction to Automata Theory there is a question 9.3.4 that asks if a question "whether a language L(M) is infinite" is RE or non-RE? I've seen the answer, that its non-RE, however I ...
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Change the structure from 3SAT to 1in3 3SAT

There is a variable set V = {x1,x2,x3} and clause set C1={x1,x2,-x3} C2={x1,-x1,-x2} C3={-x1,-x2,x3} C4={x2,x3,-x3}. For this structure, no matter each variable is positive or negative, the clause can ...
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Variant of TSP: allow each vertex to be visited at most twice

We are given a finite set $V$ and a set of distance $d : V\times V \rightarrow R\ge 0$ and we wish to compute a tour. Suppose we allow each vertex to be visited at most twice in the tour. How can we ...
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Is finding a minimal set of seed variables for a complete deduction of a system of equations NP-complete?

Suppose we have a set of variables $V$. We also have a set of equations $E$, which are sets of at least two variables. We don't know anything about these equations, except if we know all but one of ...
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Halting problem in EXP-complete

I have some troubles understanding why the halting problem is in EXP. In Wikipedia the following is written: It is in EXPTIME because a trivial simulation requires O(k) time, and the input k is ...
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A doubt on converting NOT gate to CNF formula

For a NOT gate if $x_1$ is input and $x_2$ is the corresponding output, I see the equivalent CNF (conjunctive normal form) is $(x_1 \lor x_2) \land (\overline x_1 \lor \overline x_2)$. My ...

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