# 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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### Is it possible for two languages in NOT ( R E ∪ c o − R E ) to not have a reduction between them?

In computability theory, the class NOT(RE∪co−RE) refers to languages that are neither recursively enumerable (RE) nor co-recursively enumerable (co-RE). These languages are considered to be highly ...
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### Is PP non-adaptively random self reducible?

It is well known that $\mathsf{\#P}$ is non-adaptively random self-reducible, with the common proof given via the permanent. Feigenbaum and Fortnow showed that this implies $\mathsf{PP}$ is adaptively ...
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### Reduction from any succinct language to tiling

Given a set of colors $T = \{1, \ldots, k\}$, a set of horizontal or vertical rules $H, V \subseteq T \times T$ of ordered pairs, and a chosen color $b = 1$, the tiling language is defined as follows: ...
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### How to prove that a problem is not smoothed-polynomial?

Many research works use Smoothed analysis to prove that some NP-hard problems can actually be solved efficiently in typical cases. A different notion with a similar goal is Generic-case complexity. ...
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### Is a problem, that is $L$-complete under non-uniform $AC^0$ reductions, necessarily outside of (non-uniform or uniform) $NC^1$?

I don't have much intuition about non-uniformity so the question may be quite naive.
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### Natural reduction from partially observable Markov decision process planning to reconfiguration

Suppose I want to prove the PSPACE-completeness of reconfiguration by reducing partially observable Markov decision process (POMDP) planning to it. Is there a known natural / succinct reduction from ...
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### Is this kind of "multi-reduction" interesting?

Given a decision problem $P$, the usual way to show that it is NP-hard is to find a known NP-hard problem $Q$, and show a polynomial-time algorithm that transforms every instance $I_Q$ of $Q$ to an ...
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### Can you find a counterexample / disprove my P=NP solution?

I've posted the full article here. The source code is available here. Basically, in the Linear Programming (LP) task, we solve a system of inequalities: one inequality per BSAT clause. In each ...
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### This is a variant of the unsolved problem of $a^6+b^6+c^6≠d^6+e^6+f^6$, for distinct primes. Does it have any significance for Exact Three Cover?

Suppose, I'm solving Exact 3 Cover, given a list with no duplicates $S$ of $3m$ whole numbers and a collection $C$ of subsets of $S$, each containing exactly three elements. The goal is to decide if ...
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### Why is the model-checking problem for MSO $\textsf{PSPACE}$-complete?

I am currently reading "Parameterized Complexity Theory" by J. Flum and M. Grohe. In Chapter 10.3 they state in the first paragraph: Let us remind the reader that the model-checking problem ...
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### Reductions That Acts on Witnesses

We say that a language $X$ is polynomial time reducible to $Y$, intuitively, if given an algorithm for solving $Y$, there's an algorithm for solving $X$. I know this can be formalized using Karp ...
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### strong NP-completeness of multi-dimensional Equal-Subset-Sum

I want to show that the multi-dimensional Equal-Subset-Sum is NP-complete in the strong sense: Given a set of $d$-dimensional vectors of non-negative integers, does there exist two distinct nonempty ...
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### converting K-SAT clause to a p-in-L-SAT equation

Given a generic K-SAT instance $S$ with $n$ boolean variables. Is it possible to convert a clause of this instance into an equivalent p-in-L SAT system of equations such that the number of new clauses ...
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### Is Optimal Swap Sorting NP-Hard?

Given an array of integers with duplicates, find the minimum number of swaps to sort the array. According to this question, the problem is NP-Complete but the reference given proves NP-Completeness ...
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### Proving #P-hardness for the number of subsets of a set of positive integers with a sum of at most T?

Consider the given problem: you have a set S of positive integers, and you want to find how many subsets have a sum of at most T. I highly suspect that the problem is hard since a polynomial time ...
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### Set Cover with Multiple covers

I am interested in whether a set cover instance that covers all elements $q$ times may have the property that every sufficiently small subset of this set cover will not cover the elements even once. ...
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### Complexity of Identifying SAT Problems with a Unique Solution from Satisfiable Instances

I am curious about the computational complexity involved in identifying SAT problems that have only one solution from a set of satisfiable SAT instances. input and output: input: A satisfiable cnf ...
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### Nonlinear GAP similar to Min-GAP but with minimum quantities and without capacity

I have $m$ items and $n$ bins where each item $i$ and bin $j$ has a value $v_{i,j}$. Each bin $j$ has a value $V_j$. I want to pack the items into the bins such that (1) I minimize the ratio of the ...
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### Variation of (derandomized) Valiant-Vazirani

I am interested in the following "improvement" of the Valiant-Vazirani reduction. As pointed out here, under the right derandomization assumptions one can obtain a deterministic polynomial-...
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### Are there complexity teaching resources that do not treat NP-hardness gadgets as Voodoo magic?

I am teaching a mini-complexity course to high achieving high-school students from my country this fall, and they have all expressed strong interest in learning more about what $P, NP$, reductions, ...
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### Is universal hashing fully black-box reducible to error correcting code?

Fully black-box reduction is defined as in Notions of reducibility between crytpographic primitives, O. Reingold et al. Error-correcting code is used in the black-box abstract way in the sense that ...
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### Does GHC use graph reduction?

I have read somewhere that GHC does not use graph reduction for compiling/evaluating expressions. Is this right? If yes, what does it use as an alternative?
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### On the Reductions of Functional complexity Classes

In Chapter 10 of Computational Complexity by Christos Papadimitriou, it is noted that reduction between problems of functional complexity classes are defined as follows: Function problem A reduces to ...
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### Statements equivalent to strongly polynomial time linear programming

Say a problem is SPT iff it admits an SPT algorithm. What statements of interest are known to be equivalent to "LP is SPT"? Examples: "linear feasibility solving is SPT" (due to ...
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### Reduction from unweighted graphs to weighted graphs?

Are there any examples where you take a problem for the unweighted case and reduce it to the weighted case? (Not the other way around). My objective is to do something similar to the following: If the ...
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### Existence of W[1]-Hard construction from multiple hard problems

I am a research scholar working on parameterized complexity. Currently, I am exploring ways to prove the hardness of a problem by providing W[1]-Hard constructions. A problem is known to be fixed-...
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### Unclear proof step in Feder and Greene's 1988 paper showing NP-Hardness of approximating k-center problem within a factor of 1.82

I was reading the paper "Optimal Algorithms for Approximate Clustering", Feder and Greene [1988] (https://dl.acm.org/doi/10.1145/62212.62255). Specifically, I was trying to look at the $1.82$...
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### What is region construction in timed automata?

I've recently started self-learning timed automata. There's this theorem in there that a timed automaton can be converted to a DFA using a "region" construction. I've looked up references on ...
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### A reduction from the maximum $k$-closure problem to the clique problem

Fix a partially ordered set $(P, \le)$ with $N$ elements and real weights $w(p)$ for each $p \in P$. A subset $S \subset P$ is called closed if for any $x, y$ with $y \in S$ and $x \le y$ we also ...
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### Can this NP-hardness proof for Super Mario Brothers (and other games) be simplified?

In "Classic Nintendo Games are (Computationally) Hard", a generalized framework based on reducibility of 3-SAT for proving NP-hardness of classic Nintendo games is presented, and several ...
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### complexity class of a function - linear combinations and reductions (Fermionant, immanant, $GL_n$ representations)

The fermionant is a matrix function from physics, which is indexed by a positive integer $k$: \begin{align} \operatorname{Ferm}_k(A) = \sum_{\lambda} d_{\lambda}^{(k)} \operatorname{Imm}_{\lambda^T}(A)...
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### Some examples of tools to demonstrate problem is in $NC$ [closed]

Unlike the class $P$ or $NP$ the class $NC$ does not have any complete problems. To show a problem is in $NC$ one needs to marshal efforts to directly show the problem is in $NC$ since there are no ...
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In the standard 3-partition problem, there are $3 m$ integers, their sum is $m T$, and they have to be partitioned into $m$ subsets of sum $T$ and size $3$. Consider the variant without the ...