# Questions tagged [reductions]

A reduction is the transformation of one problem into another problem. A example of using a reduction would be to be to show if a problem P is undecidable. This would be achieved by transforming or performing a reduction of a decision problem $P$ into an undecidable problem. If this can be achieved then we have shown that this problem P is undecidable.

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### A partition problem in which some numbers may be cut

In the standard partition problem, we are given some numbers whose sum is $2s$ and have to decide whether they can be partitioned into two subset whose sum is $s$. It is known to be NP-hard. However,...
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### Is Circuit Minimization $P$-hard under logspace reductions?

By Circuit Minimization, I am referring to the following decision problem. Circuit Minimization Input: A bit string $x$ and a number $k$. Question: Does there exist a Boolean Circuit $C$...
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### Can coNLogTime verification be used instead of PTime verification when characterising NP?

I recently read a paper that presented a proof calculus where the verification of whether a given proof is valid was NL-complete. The authors apparently decided that the checking procedure was not ...
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### Could you explain to me the reduction? [closed]

I am looking at the following solved exercise: I haven't really understood at the reduction the part that we construct for each number $a_i$ a package of measurement $(\frac{4}{A}a_i, 5,3)$. Why do ...
1 vote
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### Reduction from independent set in hypergraphs to independent set in graphs

Let me introduce my notations. IS-H : Input : an hypergraph $G=(V,H)$, an integer $k$ Question : is there a (weak) independent set of size $k$, i.e. a set $S \subseteq V$ such that $|S| \ge k$ and ...
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### Maximum weight triangles in dense graphs

There are multiple results (Vassilevska and Williams STOC09, for instance) on computing efficiently minimal-weight triangles (or more generally patterns) in node-weighted graphs. Several of these ...
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### Is ALogTime != PH hard to prove (and unknown)?

Lance Fortnow recently claimed that proving L != NP should be easier than proving P != NP: Separate NP from Logarithmic space. I gave four approaches in a pre-blog 2001 survey on diagonalization ...
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### On reducing the hardness of CNF-SAT to k-Clique

CNF-SAT refers to the following problem: Given a boolean formula $\phi$ in conjunctive normal form, does there exist an assignment to the variables that satisfies $\phi$. There are several ...
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### Verifying a subtlety of Karp's original proof that SAT has a polynomial time reduction to 3SAT

Stated briefly, my question is: is Karp's original proof reducing SAT to 3SAT unnecessarily elaborate? The details are as follows. In his 1972 paper Reducibility Among Combinatorial Problems, Karp ...
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### Analogues of the Berman Hartmanis conjecture and the Creativity Hypothesis

The Berman Hartmanis conjecture which formally states that there is an isomorphism for two $NP$ complete languages $L_{1}$, and $L_{2}$, the isomorphism is a bijective function $f()$ such that $f()$ ...
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### L-reduction From Matrix-Tiling To Minimal Dominating Set in Unit Disk Graph

Recently I read this paper which was published in FOCS2007. In section 4, just before Theorem 4.2, the author mentioned that the gadget construction of Minimum Dominating Set in UDG is similar to ...
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### Is there a FNP problem that's NP-hard but not FNP-hard?

For the reductions, choose a class C such that [it's clear what FC means] and FC is not known to be able to solve the satisfaction search problem, and assume that FC indeed can't solve that search ...
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### Is anything known about Sokoban with only 1 box?

This is intended to be a simpler version of my earlier question here. In this post, 1-Sokoban-search is Sokoban with only 1 box, 1-Sokoban-decision is the corresponding decision problem, and 1-Sokoban ...
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### Improving Cook's generic reduction for Clique to SAT?

I am interested in reducing $k$-Clique to SAT without making the instance much larger. Clique is in NP so it can be reduced to SAT using logarithmic space. The straightforward Garey/Johnson textbook ...
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### Polynomial-time reductions between undecidable languages

The Turing degree $\mathbf{0}'$ is defined as all languages Turing-equivalent to the halting problem. In fact any recursively enumerable language is polynomial-time reducible to the halting problem. ...
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### Slowest many-one reduction?

When we want to prove that an $L\in \bf NP$ is $\bf NP$-complete, then the standard approach is to exhibit a polynomial time computable many-one reduction of a known $\bf NP$-complete problem to $L$. ...
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### Sampling a uniformly random satisfying assignment

Problem: Given $\phi : \{0,1\}^n \to \{0,1\}$ represented by a boolean circuit, generate a uniformly random $x \in \{0,1\}^n$ such that $\phi(x)=1$ (or output $\perp$ if no such $x$ exists). Clearly ...
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