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Questions tagged [polynomial-time-reductions]

Used in questions asking for efficient (polynomial-time) reductions between computational problems.

In computability theory and computational complexity theory, a reduction is an algorithm for transforming one problem into another problem.

If the reduction is further restricted to run in polynomial-time, then it is called a polynomial-time reduction.

173 questions
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NP-hardness for one-dimensional facility location problem with entrance fee for each customer

We have $n$ customers, $(x_1, \dots, x_n)$, sorted on the read line. For convenience, we also use $x_i$ to denote its coordinate on the line. We need to locate $m$ facilities on the real line. We note that these facilities can be located anywhere on…
asdfqwer
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Is there such a notion as "effectively computable reductions" or would this be not useful

Most reductions for NP-hardness proofs I encountered are effective in the sense that given an instance of our hard problem, they give a polynomial time algorithm for our problem under question by using reductions. All reductions in the 21 classic…
StefanH
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vertex cover reduction to subset sum

Subset sum Input: A multi set $S$ of numbers and a natural number $t$ Question: Does $S$ contain a subset $A$ such that $\sum_{x \in A} x = t$? (e.g., $\{1,1,2,3,4,5\}$, by multiset it means duplicates are allowed) Vertex cover Input: An…
sam
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Are these two definitions related to strong NP-hardness equivalent?

Let $P$ be a computational problem whose inputs are integers. Consider the following properties: (a) There exists a polynomial-time reduction from some strongly-NP-hard problem $Q$ to $P$. (b) $P$ is NP-hard even when restricted to instances in…
Erel Segal-Halevi
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Why can't we prove SAT is NP complete just using the Tseytin Transformation?

The Cook Levin theorem proves SAT is NP-Complete, but it is fairly complicated, non-constructive and uses a Turing machine. I am confused as to why just the Tseytin Transformation does not imply/prove CNF-SAT and 3-SAT are NP-Complete. The Tseytin…
G.W.
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Showing the language of all graphs that are both 4-colorable and not 3-colorable is coNP-hard

As the title states, I need to prove that the language of all graphs that are both 4-colorable and not 3-colorable is coNP-hard. I'm not looking for a solution but a clue or something to help me develop the intuition for coNP questions would be very…
OE.omergunr100
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Cardinalities in set coverings

Let $I$ be a set of items; $C \subseteq \mathcal{P}(I)$ be a set of subsets of $I$, where $\mathcal{P}(I)$ stands for the power set of $I$; And $C(i) = \{ c \in C \mid i \in c \}$ be the set of sets, in $C$, containing item $i \in I$. We state the…
Matheus Diógenes Andrade
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Proofs of reduction of any hard problem

Approach:1 To prove any unknown problem $B$ is NPH then take any known NPH problem $A$ (e.g. $3$-sat) which reduces to $R$ in polynomial time. If I take any instance example $I_1$ of $A$, then prepare another instance example $I_2$ of $B$, and if…
S. M.
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Easy proof for $Primes \in NP$

I want to show that $Primes \in NP$ an I've seen multiple proofs that use facts from number theory, like this one. But isn't it much easier to proof $$Composites=\{x\in \mathbb{N}\cup\{0\}:x=1 \vee\exists p,q>1: x=pq\} \in NP$$ first and then doing…
Quotenbanane
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NP completeness of deciding whether a set of examples, consisting of strings and states, has a corresponding DFA?

I'm working on a textbook problem, 7.36 in Sipser 3rd edition. It claims that if we are given an integer $N$ and set of pairs $(s_i, q_i)$, where $s_i$ are binary strings and $q_i$ are states (we are unconcerned here with whether the states are…
xdavidliu
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Complexity of a decision problem: system of linear equations over finite field with restricted solutions

I have a system of linear equations over a finite field $\mathbb F_p \cong \mathbb Z_p$, and I'm interested in the decision problem of whether there exists a solution where all of the variables $x_i$ are in the set $\{0, 1\} \subset \mathbb F$. In…
Peter Kagey
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Prove finding k disjoint paths from n given paths in a directed graph is NP-complete

Problem: Given n paths in a directed graph G(V, E) and an integer k, find out k paths among them such that no two of them pass through a common node. Prove that the given problem is in NP-complete. I was able to prove that the problem is in NP. Hint…
user121450
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Why can KARP reductions be used to define completeness for complexity classes in the polynomial hierachy?

When defining $\Sigma_i^P$ or $\Pi_i^P$ completeness, we want to use a reduction that fulfills the following property: If $L' \leq_p L$ and $L \in \Sigma_i^P$ or $\Pi_i^P$ respectively, then $L'$ is also $\Sigma_i^P$ or $\Pi_i^P$. I can see how…
csstundent
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CLIQUE $\leq_p$ SAT

i'm trying to reduce CLIQUE to SAT: Given: Graph G=(Vertices V, Edges E) and $k \in \mathbb{N}$ Output: Formular F such that if G contains a complete subgraph of size k, the formular is satisfiable (and vice versa). I tried doing it the way…
gxor
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Interpretation of co-NPCompleteness?

Given a Problem $A$ that has an answer $true$ if and only if both conditions $1$ and $2$ are $false$, for some conditions 1 and 2. Whether condition $2$ is $true$ can be tested with certainty in deterministic polynomial time. Thus, we can say that…
J.Doe
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