# 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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### What are common techniques for reducing problems to each other?

In computability and complexity theory (and maybe other fields), reductions are ubiquitous. There are many kinds, but the principle remains the same: show that one problem $L_1$ is at least as hard as ...
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### How do I construct reductions between problems to prove a problem is NP-complete?

I am taking a complexity course and I am having trouble with coming up with reductions between NPC problems. How can I find reductions between problems? Is there a general trick that I can use? How ...
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### Graph problem known to be $NP$-complete only under Cook reduction

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 ...
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### Teaching NP-completeness - Turing reductions vs Karp reductions

I'm interested in the question of how best to teach NP-completeness to computer science majors. In particular, should we teach it using Karp reductions or using Turing reductions? I feel that the ...
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### Sorting as a linear program

A surprising number of problems have fairly natural reductions to linear programming (LP). See Chapter 7 of  for examples such as network flows, bipartite matching, zero-sum games, shortest paths, ...
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### If I can solve Sudoku, can I solve the Travelling Salesman Problem (TSP)? If so, how?

Let us say there is a program such that if you give a partially filled Sudoku of any size it gives you corresponding completed Sudoku. Can you treat this program as a black box and use this to solve ...
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### Converting (math) problems to SAT instances

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 ...
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### Is the “subset product” problem NP-complete?

The subset-sum problem is a classic NP-complete problem: Given a list of numbers $L$ and a target $k$, is there a subset of numbers from $L$ that sums to $k$? A student asked me if this variant of ...
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### Can one show NP-hardness by Turing reductions?

In the paper Complexity of the Frobenius Problem by Ramírez-Alfonsín, a problem was proved to be NP-complete using Turing reductions. Is that possible? How exactly? I thought this was only possible by ...
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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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### Could min cut be easier than network flow?

Thanks to the max-flow min-cut theorem, we know that we can use any algorithm to compute a maximum flow in a network graph to compute a $(s,t)$-min-cut. Therefore, the complexity of computing a ...
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### Reducing the integer factorization problem to an NP-Complete problem

I'm struggling to understand the relationship between NP-Intermediate and NP-Complete. I know that if P != NP based on Ladner's Theorem there exists a class of languages in NP but not in P or in NP-...
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### If P = NP, why wouldn't $\emptyset$ and $\Sigma^*$ be NP-complete?

Apparently, if ${\sf P}={\sf NP}$, all languages in ${\sf P}$ except for $\emptyset$ and $\Sigma^*$ would be ${\sf NP}$-complete. Why these two languages in particular? Can't we reduce any other ...
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### Minimal size of contracting a DAG into a new DAG

We have a DAG. We have a function on the nodes $F\colon V\to \mathbb N$ (loosely speaking, we number the nodes). We would like to create a new directed graph with these rules: Only nodes with the ...
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### Is Hidoku NP complete?

A Hidoku is a $n \times n$ grid with some pre-filled integers from 1 to $n^2$. The goal is to find a path of successive integers (from 1 to $n^2$) in the grid. More concrete, each cell of the grid ...
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### Types of reductions and associated definitions of hardness

Let A be reducible to B, i.e., $A \leq B$. Hence, the Turing machine accepting $A$ has access to an oracle for $B$. Let the Turing machine accepting $A$ be $M_{A}$ and the oracle for $B$ be $O_{B}$. ...
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### How to prove the existence of a number which cannot be written by any algorithm?

I have the problem: Show that there exists a real number for which no program exists that runs infinitely long and writes that number's decimal digits. I suppose it can be solved by reducing ...
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### Poly-time reduction from ILP to SAT?

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 ...
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### Is There a Complete Problem for the Class of Turing Decidable Problems?

Languages such as $\text{HALT}_{TM}$ are $\textsf{RE-complete}$ under many-one reductions. It is trivial to see that $\text{co-RE}$ has complete problems, too. S. Schmitz  considers some classes ...
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### Planarity conditions for Planar 1-in-3 SAT

Planar 3SAT is NP-complete. A planar 3SAT instance is a 3SAT instance for which the graph built using the following rules is planar: add a vertex for every $x_i$ and $\bar{x_i}$ add a vertex for ...
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### Does every NP problem have a poly-sized ILP formulation?

Since Integer Linear Programming is NP-complete, there is a Karp reduction from any problem in NP to it. I thought this implied that there is always a polynomial-sized ILP formulation for any problem ...
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### Reductions among Undecidable Problems

Im sorry if this question has some trivial answer which I am missing. Whenever I study some problem which has been proven undecidable, I observe that the proof relies on a reduction to another problem ...
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### If A is mapping reducible to B then the complement of A is mapping reducible to the complement of B

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 ...
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### Showing that minimal vertex deletion to a bipartite graph is NP-complete

Consider the following problem whose input instance is a simple graph $G$ and a natural integer $k$. Is there a set $S \subseteq V(G)$ such that $G - S$ is bipartite and $|S| \leq k$? I would like ...
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### Can we construct a Karp reduction from a Cook reduction between NP problems?

We have had several questions about the relation of Cook and Karp reductions. It's clear that Cook reductions (polynomial-time Turing reductions) do not define the same notion of NP-completeness as ...
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### What do complexity classes look like, if we use Turing reductions?

For reasoning about things like NP-completeness, we typically use many-one reductions (i.e., Karp reductions). This leads to pictures like this: (under standard conjectures). I'm sure we're all ...
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### Infinite alphabet Turing Machine

Is a Turing Machine that is allowed to read and write symbols from an infinite alphabet more powerful than a regular TM (that is the only difference, the machine still has a finite number of states)? ...
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### NP-complete proof from Dasgupta problem on Kite

I am trying to understand this problem from Algorithms. by S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani, chapter8, Pg281. Problem 8.19 A kite is a graph on an even number of vertices, say $2n$, ...
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### Hardness and directions of reductions

Let us say we know that problem A is hard, then we reduce A to the unknown problem B to prove B is also hard. As an example: we know 3-coloring is hard. Then we reduce 3-coloring to 4-coloring. By ...
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### Reducing max flow to bipartite matching?

There's a famous and elegant reduction from the maximum bipartite matching problem to the max-flow problem: we create a network with a source node $s$, a terminal node $t$, and one node for each item ...
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### For any language $A$, there is $B$ such that $A \le _T B$ but $B \nleq _T A$

I am trying to come up with a proof for the following: For any language $A$, there exists a language $B$ such that $A \le_{\mathrm{T}} B$ but B $\nleq_{\mathrm{T}} A$. I was thinking of letting $B$...
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### Simple proof for NP-completeness of Edge Dominating Set

In a graph, an edge dominating set is a subset D of the edges such that any edge in the graph is either in D, or shares an endpoint with an edge in D. The Minimum Edge Dominating Set problem is to ...
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### Graph 3-colorability is self-reducible

I am interested in self-reducibility of Graph 3-Coloralibity problem. Definition of Graph 3-Coloralibity problem. Given an undirected graph $G$ does there exists a way to color the nodes red, green, ...
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### What NP decision problems are not self-reducible?

So we just learned about self-reducibility in class. My professor and our textbook would not commit to saying that all problems in NP are self-reducible, but there didn't seem to be any examples of ...
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### NP-hardness of finding a subset of vertices in a vertex-weighted graph

This is task from the German IT contest ("Bundeswettbewerb Informatik"), but since the deadline is past, asking this question is no cheating. Given a vertex-weighted, directed graph $G=(V, E)$ and ...
### Mapping Reductions to Complement of A$_{TM}$
I have a general question about mapping reductions. I have seen several examples of reducing functions to $A_{TM}$ where \$A_{TM} = \{\langle M, w \rangle : \text{ For } M \text{ is a turing machine ...