Questions about finding mappings between problems that allow solving one problem using a solution of another one.

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17 views

How can I convert a list with duplicates into a set for a reduction to the set cover problem?

I'm trying to come up with a reduction for a problem whose description is more or less identical to the first problem given here. Here's a condensed version of the problem: You're given a collection ...
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39 views

Proof that P is closed against switching between polynomially related encodings

Lemma 34.1 Let $Q$ be an abstract decision problem on an instance set $I$, and let $e_1$ and $e_2$ be polynomially related encodings on $I$. Then, $e_1(Q)\in \mathrm{P}$ if and ...
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1answer
50 views

NP Complete Subset GCD Proof

$SubsetGCD$ is described by the following: instance: A set of positive integers $S$ and an integer $k$ question: does there exist a subset $S'$ of $S$ of size $k$ such that $GCD(S') = GCD(S)$ ...
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1answer
30 views

Showing NP-hardness by reducing from a search problem

I'm comfortable with showing NP-completeness of a decision problem: just take some problem that is known to be hard and reduce it to your new problem. This establishes NP-hardness of the new problem. ...
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20 views

Existence of randomized reduction but no deterministic reduction

What is the consequence to complexity theory of having a randomized reduction from an NP-complete problem to problem $\Pi$ while there is no deterministic reduction from an NP-complete problem to ...
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1answer
39 views

Reduction of SUBSET-SUM to SET-PARTITION [duplicate]

There is a similar question that has been asked, but my question addresses particular detail of an answer. I am trying to reduce SUBSET-SUM to SET-PARTITION. I found the following description: ...
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1answer
50 views

How to Prove NP-Completeness of Minimum Crossing Problem?

In graph theory, the crossing number cr(G) of a graph G is the lowest number of edge crossings of a plane drawing of the graph G. (from wikipedia) I know that the problem of counting the ...
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33 views

Is L closed under linear-time reductions?

L is as usual the complexity class DSPACE($\log n$), of languages decidable using a deterministic Turing machine using logarithmic workspace. Is L closed under linear-time reductions? It is ...
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1answer
65 views

Checking acceptance of a word vs finding an accepted word

We know that checking whether some word w is accepted by a turing machine TM is undecidable. But what about the problem of finding one accepting word of a TM? Are these two problems related in some ...
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1answer
45 views

Do problems in P only reduce to NP and coNP problems?

Consider the languages $B,C,D$, such that $B\le_p C$ and $B\le_p D$. Statement: $B\in P, D\in NP, C\in coNP$. Is the statement true for every $B,C,D$? I know that the answer is no and I have ...
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2answers
48 views

$A$ is finite, $B$ is NPC - When there's a polynomial reduction from $A$ to $B$?

$A$ is finite, $B$ is NPC - When there's a polynomial reduction from $A$ to $B$? Basically, I've understood that if $A$ is finite, then there's a reduction for every $B$ which isn't trivial ...
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29 views

mapping reduction for every recursive language [duplicate]

how do I prove that for every 2 languages $A,B\in R$ where $A,B \notin \{ \emptyset , \Sigma^* \}$ I can do a reduction $A \leq_m B$? [EDIT] My try: $A$ is decidable therefore it has a turing ...
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1answer
120 views

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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41 views

Why are decision problems easier than the equivallent optimization problems?

Suppose that we have an optimization problem defined as follows: $OPT$ = Given an input string defining a set of feasible solutions $F$ and an objective function $f$, find $x\in F$ maximizing $f(x)$ ...
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36 views

Are there any known AM-complete problems/is AM-complete well defined?

I'm curious about whether there are any complete problems in the Arthur-Merlin complexity class. Graph Non-Isomorphism (GNI) seems to be the canonical example of a problem in AM, but it's probably not ...
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1answer
49 views

Relation between the “Point-Cover-Interval” problem and the “Interval Scheduling” problem

Point-Cover-Interval Problem: Given a set $\mathcal{I}$ of $n$ intervals $[s_1, f_1], \ldots, [s_n, f_n]$ along a real line, find a minimum number of points $P$ such that each interval contains ...
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1answer
66 views

Prove Vertex-Cover of maximum degree 3 is NPC

This is a homework question. I need to prove that the following language is in NP Complete: 3-VERTEX-COVER = $\{\langle G,k\rangle \mid$ G is an undirected graph, each vertex in $G$ has at most ...
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1answer
64 views

Determine if the language is $R$

Consider the following language: $$L = \{ \langle M \rangle \ |\ M \text { is a TM that decides the halting problem} \}$$ determine whether or not the language is in $R$. Now, from my ...
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3answers
63 views

Reduction and decidability

Consider the following language: $$ L = \{ \langle M \rangle \ |\ M \text { accepts } w \text { whenever it accepts } w^R \}$$ I am trying to understand the following proof that this language $L$ is ...
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1answer
53 views

Cycles in hardness of ST-CON for the class NL

It seems to me that the problem of $s$, $t$ connectivity in a DAG should still be NL-Complete. I am aware that ST-CON without the DAG restriction is complete for NL, so obviously the DAG restriction ...
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50 views

smallest satisfiability-equivalent formulas (generalized Tseitin transform)?

What is known about the following optimization problem for formulas in propositional logic: input: formula $F$ output: formula $G$ in CNF with $\mathrm{Var}(G) \supseteq \mathrm{Var}(F)$ such that ...
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1answer
59 views

If an NP problem reduces to an NPC problem, it is NPC?

Is the following statement true? If a problem P1 is in NP and polynomial time reducible to P2, where P2 is NP-complete, then P1 is also NP-complete. Intuitively I think the answer is No because ...
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1answer
29 views

Reduction of Circuit Satisfiability to CNFSAT [duplicate]

CNFSAT is Karp reducible to Circuit-SAT by replacing all the conjunctions with AND gate, disjunctions with OR gate and negations with NOT gate. However, if we apply the same approach to Karp reduce ...
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1answer
82 views

Reduce our problem to a known np-complete problem

Subgraph isomorphism We have the graphs $G_1=(V_1,E_1), G_2=(V_2,E_2)$. Question: Is the graph G_1 isomorphic with a subgraph of $G_2$ ? (i.e. is there a subset of vertices of $G_2, V \subseteq ...
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38 views

np-complete proof, turing reduction

I have some difficulties with a complexity proof : I work with 3 problems : A, B and C I know : A-> B A-> C C -> B A-> B meaning : if I have a "yes answer " for A , then I have a "yes answer" for ...
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187 views

PSPACE completeness, with different kinds of reductions [closed]

PSPACE-complete$_{FP}$ problems are the PSPACE problems such that every other PSPACE problem can be transformed to it with a polynomial time reduction, i.e. the reduction is an algorithm $\in$ FP. ...
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118 views

Reduction from Vertex Cover to Polygon Cover

Polygon Cover: Input: A set of points $P$, a set of polygons $S$ in a 2D plane, and a positive integer $k \in \mathbb{N}$. Output: True if and only if there exists a subset in $S$ of at most $k$ ...
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1answer
50 views

Proof of Circuit-Sat to Nand-Sat polynomial time many–one reducibility

Given a gate called Nand with the following truth table: A | B | A Nand B ------------------ 0 | 0 | 1 0 | 1 | 1 1 | 0 | 1 1 | 1 | 0 We can ...
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1answer
52 views

Poly-time reduction: D and D Comp [duplicate]

Looking at the Independent Set problem and its complement, I want to show that IS is poly-time reducible to its complement, however I am struggling on coming up with the reduction function. I will ...
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34 views

Reducing a Knapsack-type problem to a known problem

The Quadratic Knapsack problem, introduced by Gallo, is an optimization problem in the following form: $max \sum_{i=1}^n{\sum_{j=1}^n{q_{ij}x_ix_j}}$ $s.t \sum_{i=1}^n{w_ix_i} \leq c$ $x \in \{0, ...
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90 views

P-Completeness and Reducibility

I am taking an algorithm analysis class and am stuck on one of my homework problems and would appreciate it if I could receive some guidance. The problem I'm stuck on is proving that the empty ...
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1answer
198 views

Proving a language is not Turing-recognizable by reduction

I'm having a really hard time understanding some of these concepts. I've read them over from several different sources and still can't reach the a-ha moment. I need to prove a language L is not ...
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3answers
420 views

The relation between 2SAT and 3SAT

Show that proving 2SAT is not NP-Complete would prove that 3SAT is not in P. Or eqivalently - Show that proving 3SAT is in P would prove that 2SAT is NP-Complete. I can see there is an ...
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2answers
74 views

Restricted Integer Programming

The integer feasibility problem is NP-complete: $Ax=b, x \geq 0, x \mbox{ integer}$ $A$ contains elements in $\mathbb{R}$ If we restrict this: $A$ contains only elements in: $\{1,0\}$ ...
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2answers
83 views

If $B$, $\overline{B}\neq \varnothing$ , then for every recursive set $A$, $A \leq_m B$

How to prove if $B$, $\overline{B}\neq \varnothing$ , then for every recursive set $A$, $A \leq_m B$ ? it means every recursive set is mapping reducible to set $B\neq \aleph$. I really have no idea ...
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49 views

Is it true, If $A$ is turing recognizable and $A \leq_m\bar{A}$ then $A$ is recursive?

If $A$ is turing recognizable and $A \leq_m\bar{A}$ then $A$ is recursive? If it is true how to prove it? Update It is my attempt, If $A$ is turing recognizable (r.e.) and $\bar{A}$ is r.e. then ...
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38 views

Methods of turning a decision problem into finding the certificate?

I usually find this in the context of asking about NP-complete problems, but any decision problem works. We start by assuming there's a polynomial time algorithm that gives the yes or no answer. If ...
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114 views

Is the language of Turing Machines that halt on every input recognizable?

I am trying to reduce the complement of the HALTING problem (WLOG, the complement of the HALTING problem is the language of TMs that loop on some string w)to this language in order to show that it is ...
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1answer
86 views

Reduction to $n\log n$ time problem

If a problem $A$ is poly-time reducible to a problem $B$ ($A <_\mathrm{p} B$), and $B$ can be solved in time $O(n\log n)$, can $A$ also be solved in time $O(n\log n)$?
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129 views

Polynominal reduction from unbounded knapsack problem to general integer programming

Given an oracle that can solve in polynominal time: $$a^Tx=b$$ $$x \geq 0$$ So it can solve the feasibility problem with one equality-constraint(a is here a vector and b is a constant, x is required ...
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53 views

Ideas on reducing Traveling Salesman to Metric Traveling Salesman?

I want to show a polynomial reduction from TSP to Metric TSP. I know the rule is: $(G, k) \in TSP \iff (G', k') \in MTSP$ where $G$ is some graph, and $k$ is some bound. It seems like whatever I map ...
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1answer
48 views

Linear and almost-linear perfect hashing

I am trying to understand the sentence from a paper by Patrascu: Unfortunately, we do not have linear perfect hashing. Instead, we use a family of hash functions.... with almost-linearity The ...
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92 views

Let A,B be languages. If A is decidable and B undecidable, then A reducible to B

So I'm learning for an upcoming exam and there's a specific problem which I can't show: Let A be decidable and B undecidable, then $A \le B$ Can someone give me a hint how to solve that? ...
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3answers
268 views

4-color to 3-color polynomial reduction

I know a simple reduction from 3-color to 4-color. But how do you reduce 4-color to 3-color ? I have been searching for the right way to make this reduction for a while now. I would love some ...
3
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1answer
213 views

Variants of the 3-SUM problem

The 3SUM problem has two variants. In one variant, there is a single array $S$ of integers, and we have to find three different elements $a,b,c \in S$ such that $a+b+c=0$. In another variant, there ...
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2answers
72 views

Logical Reduction

Reducing one computable problem to another by providing an algorithm which transforms an instance of one problem to one of the other (and limiting the time or space of that algorithm) is clear to me. ...
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51 views

What is the trick of “adding a huge number” for in the reduction from $\textsf{3-Partition}$?

Problem: To prove the $\textsf{NP-Completeness}$ of the problem of "Packing Squares (with different side length) into A Rectangle", $\textsf{3-Partition}$ is reduced to it, as shown in the following ...
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14 views

Solving $Isomorphism$ using $AUTOM$ in polynomial time

Let $Iso$ be the language of all $<G,H>$ such that $G$ and $H$ are isomorphic, and $AUTOM$ be the language of all $G$'s such that $G$ has a non-trivial automorphism. I'd like to show that, ...
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23 views

$3EQ \leq _P 2EQ$

Let: $2EQ$ - The language of all binary ($\mathbb{Z}_2$) equation sets that have a solution in $\mathbb{Z}_2$, where each multiplication is of at most two $x_i,\, x_j$. Meaning a set of equations of ...
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34 views

What are the fundamental principles/algorithms on the process of equation solving?

I have seen a lot of solvers that are capable of, for example, getting an equation such as x ^ 2 + x = 12 and finding x = [3, -4]. I know some of them are implemented by hardcoding special cases. For ...