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Questions tagged [polynomial-time-reductions]

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

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8 votes
2 answers
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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 ...
asdfqwer's user avatar
4 votes
0 answers
38 views

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, ...
Matheus Diógenes Andrade's user avatar
3 votes
1 answer
205 views

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 ...
xdavidliu's user avatar
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3 votes
0 answers
596 views

Is there any polytime reduction from feedback vertex set to vertex cover?

I know that feedback vertex set (FVS) problem is $\mathrm{NP}$-complete since there is a simple and nice polytime reduction from vertex cover (VC) problem to FVS. Specifically, given a undirected ...
Blanco's user avatar
  • 623
3 votes
0 answers
87 views

Multipoint evaluation of a given polynomial

You are given a polynomial of degree n. We have to find the value of the polynomial at n different points in O(n(log(n))^2). The answer should be modulo 998244353. I have read various blogs on it and ...
Abhijeet Narang's user avatar
2 votes
0 answers
80 views

Limited number of calling for a decision blackbox to compute all the solutions

I am trying to reduce between a solution problem and a decision version of the same problem. The problem is the orthogonality problem. Given $2$ sets $L$ and $R$, whose size each is $n$ vectors over $\...
Dan D-man's user avatar
  • 534
2 votes
0 answers
114 views

A polynomial time reduction and the size of problem (exact cover)

An exact cover problem is one of the NP-complete problems. Given a family $\mathbb{I}$ of subsets of a set $[n]=\{1,\dotsc,n\}$, whether there exists a subfamily $\mathbb{I}'\subseteq \mathbb{I}$ ...
Sanghack Lee's user avatar
2 votes
0 answers
262 views

NP-completeness of Induced disjoint paths between a set of sources and a set of sinks

In a given undirected graph $G(V,E)$, a set of $k$ paths is said to be induced if: They are vertex-disjoint. Each one is itself an induced path. No edge connects two vertices of two different paths. ...
Thinh D. Nguyen's user avatar
1 vote
0 answers
35 views

Reduction from Hamiltonian path to Tripartite decision problem

I teach a fairly advanced algorithms class to high schoolers and I accidentally presented them with a bunk reduction from Hamiltonian path to the Tripartite graph decision problem. My attempt involved ...
bbg07's user avatar
  • 11
1 vote
0 answers
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Scheduling jobs with the same release time and different due dates on a single machine

Consider the problem of scheduling jobs with different lengths on a single machine while the jobs have the same release times and different due dates. The goal is to schedule the maximum number of ...
Soroush Vahidi's user avatar
1 vote
0 answers
134 views

Does there exist an FPTAS for bin packing problem?

We know that there does not exist an FPTAS for the bin packing problem because it is a strongly NP-HARD problem, as the 3-partitioning problem which is strongly np-hard, can be reduced to the bin ...
Soroush Vahidi's user avatar
1 vote
1 answer
196 views

Show problem is NP-hard

I'm preparing for my exam and I got stuck on the following problem: The gardening problem: We have access to a set of different types of seeds and a number of plant pots.For each plant pot, there is ...
kim120's user avatar
  • 21
1 vote
0 answers
32 views

example of an NL-completeness reduction?

I'm looking for simple examples of nondeterministic log-space completeness reductions. In particular I seem unable to construct any nontrivial widget using 2-SAT clauses, which is known to be NL-...
Albert Hendriks's user avatar
1 vote
1 answer
857 views

Show that a subproblem of Sparse Subgraph is $\mathcal {NP}$-Complete

I want to show that a subproblem of the known, $\mathcal {NP}$-Complete, Sparse Subgraph problem is also $\mathcal {NP}$-Complete. Sparse Subgraph problem: Input: Undirected graph $G(V,E)$, two ...
entechnic's user avatar
  • 143
1 vote
0 answers
51 views

Relationship between complexity classes W[1]-hard and NP-hard?

If i have a parameterized reduction from multicolored independent set (W[$1$]-hard) to some problem $A$, which take polynomial time. Can i say that problem $A$ is NP-hard? in other words, Is ...
Yuhang Bai's user avatar
1 vote
0 answers
94 views

Covering Salesman Problem (CSP) polynomial reduction to the TSP

I am facing one problem that consists in polynomially reducing the Coverging Salesmen Problem (CSP) to the Traveling Salesman Problem (TSP). So, let me first define the CSP. The CSP, I am working on, ...
Matheus Diógenes Andrade's user avatar
1 vote
0 answers
23 views

Solve the K_ELEMENT problem using INTERSECT

Suppose you have a machine that takes inputs a set of sets, $\{S_1,S_2,\dots S_n\}$, and an integer $k$. The machine then returns True if $S_1$ intersects every ...
Addem's user avatar
  • 367
1 vote
0 answers
42 views

Show that if vertex cover is reducible to a mod-inverse than P=NP

Let MOD-INVERSE consist of all pairs $\langle N,c \rangle$ such that $c$ has an inverse modulo $N$. Let VERTEX-COVER consist of all pairs $\langle G,k \rangle$ such that $G$ is an undirected graph ...
pensee's user avatar
  • 131
1 vote
0 answers
141 views

Reduction between Parity-SAT and approximate counting

Consider two problems as defined here. Approximate counting: Given a Boolean function $f(x)$, for $x \in \{0, 1\}^{n}$, distinguish between the two cases: The number of satisfying assignments for $f(...
Sid Meier's user avatar
  • 249
1 vote
0 answers
24 views

Time-Sensitive Reductions for Undecidable Problems

I'm studying Comparability and Complexity, and through the course, a number of problems (namely, the halting problem for Turing Machines, etc.) have been proven undecidable through elementary proofs ...
matan129's user avatar
  • 111
1 vote
0 answers
60 views

Finding $l$ subsets such that their intersection has less or equal than $k$ elements NP-complete or in P?

I have a set $M$, subsets $L_1,...,L_m$ and natural numbers $k,l\leq m$. The problem is: Are there $l$ unique indices $1\leq i_1,...,i_l\leq m$, such that $\hspace{5cm}\left|\bigcap_{j=1}^{l} L_{i_{j}}...
Christian Singer's user avatar
1 vote
0 answers
103 views

What are the requirements for a superset of P to be closed under karp reductions?

So today in our exercise session on complexity theory we discussed that P, NP, and BPP are closed under karp reduction. We also figured that the proofs could likely be expanded to straight ...
SEJPM's user avatar
  • 325
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0 answers
21 views

If A ∈ coNP, B ∈ NP and $NP \neq coNP$, is it possible to Karp reduce A to B?

If A $\geq_p$B and $B\in NP$, $A\in coNP$, then we can build a Turing machine $M_A$ using $M_B$ machine of B. Input: w We make a new word with a reduction function $f(w)$. Then we run $M_B$ on $f(w)$ ...
Naneless's user avatar
0 votes
0 answers
54 views

Why does this approach not work on the SubSet Sum Problem?

I was reading this post, and in it I learned how to make difficult instances of the SubSet Sum Problem. There the guy who responded to the post says that it is necessary to have density 1.0 and all ...
Edu's user avatar
  • 1
0 votes
0 answers
51 views

HAMILTONIAN PATH AND SUBSETSUM

If we were to discover a deterministic algorithm capable of deciding, in polynomial time, whether a given graph contains a Hamiltonian path, would that imply that the problem SUBSETSUM belongs to P? ...
Drat's user avatar
  • 1
0 votes
0 answers
44 views

Reduction from novel problem to Set Cover

i would like to perform a reduction for my novel problem to preferably the set cover problem, but i am a bit lost.. My problem can be described as follows: Suppose you have given an binary word as ...
Sven Fiergolla's user avatar
0 votes
0 answers
39 views

How to prove Set-Cover problem is NP hard via reduction from Clique problem?

Since I know that the reduction Clique $\leq_{p}$ Vertex-Cover and Vertex-Cover $\leq_{p}$ Set-Cover are possible, I know how to show this using transitivity property of reductions. However is there ...
Yavuz Bozkurt's user avatar
0 votes
0 answers
34 views

Reduction between problems where problem A solves problem B with probability $\frac{2}{3}$

Suppose we have a problem, $A$, and a machine $T_A$ that solves $A$. Now, let's say we have a problem $B$ that is solvable with a polynomial number of calls to $T_A$, and we call $T_B$ the machine ...
Loic Stoic's user avatar
0 votes
0 answers
53 views

Is this minimizing problem NP-hard?

We have $c$ arrays of positive integers, named $j[1],j[2],j[3]..,j[c]$. We know that the length of $j[i]$ is $c-i+1$. In addition, the sum of the numbers of $j[i]$ is greater than or equal to the sum ...
Soroush Vahidi's user avatar
0 votes
0 answers
53 views

Question about unary languages

Let $A$ be any unary language. Prove that if $A$ is NP-complete, then P=NP. Approach 1: We just have to prove that we can decide $A$ in polynomial time. Consider a Turing machine that on input $w$, ...
Keio203's user avatar
  • 257
0 votes
0 answers
52 views

I would not be able to get my simulation in my life time

I am running a simulation on my computer. I tried to multiply two polynomials $g(x), h(x)\in GF(2)[x]$, with $degree(g(x))= 8165$, and $degree(h(x))=25$. This multiplication took almost $20$ minutes ...
Robin Kurtz's user avatar
0 votes
0 answers
224 views

Polynomial Reduction from 3SAT

Given an undirected graph $G=(V,E)$ where $V$ is a set of vertices, and $E$ is a set of edges and given a set $D$ where $D \subseteq V $ and $ \forall v \in V \setminus D \: \mid \: \exists w \in D : ...
Emily's user avatar
  • 1
0 votes
0 answers
259 views

Horn Satisfiability is NP Complete, isn't it?

To show that any formal language is NP Complete first it must be showed that this formal language is both in NP and NP Hard. So to show that Horn Satisfiability is NP Complete first it must be showed ...
Farewell Stack Exchange's user avatar
0 votes
0 answers
201 views

Prove minimum vertex bisection problem is reducible from bisection width, thereby proving NP-Completeness

The bisection width problem gives you a yes or a no -> if there exists a bisection of size at most $K$ for $G=V,E$. Minimum Vertex Bisection problem gives you a bisection of the smallest size. ...
Prboetic's user avatar
0 votes
0 answers
43 views

Difficulty in finding a counter example for a polynomial reduction

I am doing some exercises on proving NP Complete problems and the first problem was directed bipartite graphs with a Hamiltonian cycle and the second one was undirected bipartite graphs with a ...
sari98's user avatar
  • 1
0 votes
0 answers
55 views

In-place Acceptance Problem

In-place Acceptance Problem (InAP) Instance: A deterministic Turing Machine M and a w input for it. Question: Does M accept the input w without going through cell (|w|+1)? Show that InAP is PSPACE-...
user146767's user avatar
0 votes
0 answers
68 views

Reduction techniques in complexity

I am learning computational complexity and parameter complexity. In order to proof that a problem is np-hard, we should reduction one which is np-hard to the problem. However, I don't have any idea ...
Yuhang Bai's user avatar
0 votes
0 answers
70 views

Reduction rules to lower bound minimum degree of a graph

I'm trying to come up with a list of rules that return an equivalent instance to the following problem, while eliminating all vertices of degree 2 or less from the graph: Given a graph $G=(V,E)$, the ...
giorgioh's user avatar
  • 317
0 votes
0 answers
36 views

If a problem A has a poly-time reduction to a problem B in co-NP, is A in co-NP as well?

i.e. $A\leq_pB\:\wedge\:B\in\text{co-NP}\rightarrow A\in\text{co-NP}$ ? I feel like it's the case but I can't think of a straightforward proof. Clarification: I am talking about polynomial-time Turing ...
nc404's user avatar
  • 111
0 votes
0 answers
42 views

Reducing the Hamiltonian Cycle problem to the problem of finding a $m$-length Hamiltonian Cycle

I was working through some exercises regarding NP complexity, when I came across this one: Let $mH$ be the problem of finding a Hamiltonian cycle of length $m$ ($m$ fixed) in a graph $G$. A cycle of ...
modak32's user avatar
0 votes
0 answers
34 views

How to prove the complexity of this modified version of the minimum dominating set problem?

I have an optimization problem and I want to show its complexity. The optimization problem is the same as the minimum dominating set problem, but with an additional constraint. The constraint is easy. ...
samie's user avatar
  • 21
0 votes
0 answers
41 views

$A \leq_p {\overline{A}} \Leftrightarrow {\overline{A}} \leq_p A$

I want to prove that $$A \leq_p {\overline{A}} \Leftrightarrow {\overline{A}} \leq_p A$$. Does anyone have a Idea how to solve this ?
Frank's user avatar
  • 147
0 votes
0 answers
31 views

planar 1-in-3 sat described as a planar graph for independent set

Given a planar 1-in-3 sat formula, can someone reduce that formula into a graph that asks the question when ever there is an independent set for it, that's also planar?
user3700810's user avatar
0 votes
0 answers
77 views

Find decidable sets such that $A$ reduces to $B$ but not vice versa

I am stuck in this problem, so any help is appreciated. The problem asks to show that there exists decidable sets $A$ and $B$ such that $A \leq_{m}^{p} B$ but $B \not \leq_{m}^{p} A$, and that $A$, $B$...
Link L's user avatar
  • 481
0 votes
0 answers
105 views

If any problem in NP is not in P then NP C ∩ P = ∅

If any problem in NP is not in P then NPC ∩ P = ∅ The proof is: We have $X ∈ NP$ and $X \not\in P$. Assume $Y ∈ NP C ∩ P$. As $X ≤_P Y$ we have $X ∈ P$, which is a contradiction. I have not clear ...
Gianni Spear's user avatar
-1 votes
1 answer
34 views

If two languages are polytime reducable, does that imply they are also turing reducable

Is it possible for a pair of languages where A ≤T B but not A ≤p B? I am not sure if this could be the case since a turning reduction would imply we can use a decider for one language to decide ...
user145121's user avatar
-1 votes
1 answer
51 views

Are the following assertions true if P != NP?

We consider the NP-complete $CLIQUE$ problem. Let furthermore $MST^*$ be the minimum spanning tree problem. Assume that $P \ne NP$ and explain whether the following assertions hold: $MST^* \le_{P} ...
pekusbill's user avatar
-1 votes
1 answer
220 views

Close To Cook Reduction given NP != coNP

I am struggling to answer these two questions: Prove or wrong: Both are given the assumption that NP != coNP. For any 2 decision problems S, S', if there is a Cook reduction from S' to S then there ...
Neo182's user avatar
  • 29
-5 votes
1 answer
129 views

Proving correctness of Polynomial reduction

Given a problem A is NP-Hard and A ≤𝑝 B, is there a way to prove that B is also NP-Hard?
pog's user avatar
  • 1