Questions about the hardest problems in NP, i.e. of those that can be solved in polynomial time by nondeterministic Turing machines.

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1answer
15 views

Hardness and approximation of a problem with a parameter

Let $H$ be a decision problem, where we are given an integer $k$ and some object, say a graph or a formula. We know that $H$ is NP-complete for $k \geq c$, where $c$ is some constant like 3 ($H$ could ...
4
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0answers
56 views

Fixed size set to contain the maximum number of given sets

I asked this question in SO here I have about 1000 sets of size <=5 containing numbers 1 to 100. {1}, {4}, {1,3}, {3,5,6}, {4,5,6,7}, {5,25,42,67,100} ... ...
3
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2answers
109 views

Word tiling, where you must use each tile exactly once

Given words $w_1,\ldots,w_n$ in binary alphabet and another word $w$, decide if $w$ can be written as a product $w = w_{i_1} \cdots w_{i_n}$ (in the monoid $\{0,1\}^\ast$) for some permutation of ...
8
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1answer
149 views

Are all known algorithms for solving NP-complete problems constructive?

Are there any known algorithms that correctly output "yes" to an NP-complete problem without implicitly generating a certificate? I understand that it is straightforward to turn a satisfiability ...
0
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0answers
13 views

When proving a problem is NP-C, how do I select another NP-C problem for the transformation? [duplicate]

I'm taking an algorithms course in which we are discussing proofs that problems are NP-Complete. Our proofs usually take the form: Given a problem $\Pi$, 1. Prove that $\Pi$ is NP. 2. Select an ...
1
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1answer
23 views

How does the Vertex Cover algorithm by Chen et al find its tuples?

I'm still fighting with the aforementioned paper "Improved upper bounds for vertex cover" by Chen, Kanj, Xia (PDF kindly provided by Yuval Filmus). My current problem is that it's specified that the ...
1
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1answer
146 views

Is finding if a graph has k isolated nodes a NP-Complete problem?

I was wondering if finding if a graph has k or more isolated nodes is a NP-Complete problem. I found the following problem: Prove that the following problem is NP-Complete. Given a set of T ...
-1
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1answer
38 views

How to reduce bin-packing problems? [duplicate]

This is my first time with reductions and I can't figure out how to do them. I have read the few standard examples that are given in the standard books. For example, given $n$ numbers $\{ 0 < ...
5
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2answers
204 views

Direct NP-Complete proofs

I'm just starting to learn about NP-completeness. While I understand that reducibility plays a key role in this, I'm astonished how few problems I've been able to find who's proof that they are ...
2
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1answer
76 views

Finding a pair of edge disjoint paths in a graph, such that the weight of each of them is bounded

Given an undirected graph $G=(V,E)$, two distinct vertices $s,t\in V$, a weight function $f:E \to \mathbb{N}$, and a constant $M\in \mathbb{N}$, does there exist a pair of edge disjoint paths ...
1
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2answers
69 views

NP-hardness of an optimization problem with real value

I have an optimization problem, whose answer is a real value, not an integer such as vertex cover and set cover. Therefore, the decision version of my problem is given an input and a real value $r$. ...
7
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1answer
156 views

Is this classic puzzle book game NP-complete?

There is a classic puzzle book game very similar to a crossword puzzle, except a list of words is given and then a $N \times N$ square board made up of unit squares is given, with some squares blacked ...
1
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1answer
67 views

Reduction to Maximum Independent Set

Suppose you had a set $P$ of people. Every person $p_j \in P$ is familiar with atleast one other person $p_i$ (familiarity is symmetric). Is there a subset $S$ of people such that for $|S| \ge k$, no ...
9
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1answer
114 views

Can an NP-hard problem be polynomial on average?

I'm wondering if there are any $NP$-hard problems which are ``polynomial" in the average case. I think there are two ways to interpret this? If $P \neq NP$, can there be an algorithm solving an ...
1
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1answer
44 views

How “coplanar” is a set of points?

Assume that we have 10 points. If all those points are on the same plane, they all are coplanar. But some of them might be at a different place. That disrupts the structure of the plane if we were to ...
1
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1answer
67 views

Set cover problem with constant size subset

Consider a variation of the set cover problem in which the size of the subsets is no larger than a constant $k$. Is this variation still NP-hard?
1
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1answer
39 views

A detail on variant of Mahaney's theorem about reductions of sparse languages vs P/NP

Wikipedia states on sparse languages that There is a Turing reduction (as opposed to the Karp reduction from Mahaney's theorem) from a NP-complete language to a sparse language iff NP $\subseteq$ ...
2
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1answer
50 views

Monotone boolean satisfiability with at most k 1s is NP-Complete

I am to prove that monotone boolean formula satisfiability checking when at most k variables are set to 1 is an NP-Complete problem. Proving that it is in NP is easy, but I'm having difficulty ...
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1answer
98 views

Does the Bible solve an NP-hard problem? [closed]

In the Bible, a census is taken of the 12 tribes of Israel: Simeon: 59,300 Levi: 22,000 Judah: 74,600 Issachar: 54,400 Joseph: 72,700 Benjamin: 35,400 Reuben: 46,500 Gad: 45,650 Asher: 41,500 ...
0
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0answers
56 views

Reduction to the constrained-shortest-path problem (CSP)

How can I reduce the subset sum problem to the CSP? Is this possible ? There are quite a few formulations of the CSP, I am talking about the following: We have an s-t graph, whose edges have costs and ...
3
votes
2answers
105 views

Why does the NP completeness of the Hartree-Fock method not lead to difficulty in practical calculation?

I read Computational Complexity of interacting electrons and fundamental limitations of Density Functional Theory. In appendix, it is claimed that In the following, we show that approximating ...
17
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2answers
216 views

NP-complete problems not “obviously” in NP

It occurred to many that in all the $\textbf{NP}$-completeness proofs I've read (that I can remember), it's always trivial to show that a problem is in $\textbf{NP}$, and showing that it is ...
5
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0answers
135 views

Longest Repeated (Scattered) Subsequence in a String

Informal Problem Statement: Given a string, e.g. $ACCABBAB$, we want to colour some letters red and some letters blue (and some not at all), such that reading only the red letters from left to right ...
0
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1answer
110 views

Concept used in the proof [closed]

In the paper "Resolution for Quantified Boolean Formulas", I am unable to understand the proof of theorem 3.4. Please help me with the basic concept used on page 4: The concept that I am referring ...
3
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2answers
122 views

Need Help Reducing Subset Sum to Show a Problem is NP-Complete

I want to show that the following problem is NP-Complete: For a set of vectors $v_1,\ldots,v_n \in \mathbb{N}^d$ and an integer $k$, does there exist a subset $S \subseteq \{v_1,\ldots,v_n\}$, such ...
0
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1answer
73 views

Is the Berman-Hartmanis Conjecture Solved?

The Berman-Hartmanis conjecture more or less states that if one-way functions exist, there are some problems in $NP$ which cannot be polynomially reduced to $NP$-complete (cf. Ker-I Ko, A Note on ...
3
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1answer
48 views

Why do we assume that a nondeterministic Turing machine decides a language in NP in $n^k-3$ in Sipser's proof

At page 277 of Sipser's Introduction to the Theory of Computation, a proof of the NP-completeness of SAT is given. The following comment is made on the trace of some machine $N$ which can decide a ...
1
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1answer
41 views

Can weighted problem have polynomial complexity if non-weighted problem is NP-complete: hitting set

I am confronted with task to find polynomial time complexity solution for weighted hitting set problem. I have found that usual hitting set problem is NP-complete and therefore the task seems to be ...
3
votes
1answer
60 views

Reduction from PARTITION to MAX-CUT

I am trying to prove the NP-Hardness of the MAX-CUT problem. Other sources seem to reduce from the NAE-3SAT problem, however I have been trying to reduce from PARTITION because PARTITION and MAX-CUT ...
1
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1answer
63 views

Proving DPATH is NP-complete by a reduction from HAMPATH

I have a language DPATH that I'm trying to complete is NP-complete. ...
5
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0answers
87 views

research on OR and AND compression in SAT formulas

this is a new/advanced paper on OR and AND compression of SAT formulas, a newer area of research that seems not covered in any textbooks so far. A simple proof that AND-compression of NP-complete ...
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1answer
39 views

reducing planar 3-colouring from 3-colouring

I was reading this and I'm trying to understand how one would formally describe reducing planar 3 colouring to 3 colouring. The link pretty much describes the process but understanding the ...
1
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3answers
73 views

NP-completeness: Reduce to or reduce from?

Very simple question, but a mistake I make often enough that I'd love to have a standard reference. I'm showing that a problem $P$ is NP-Hard by assuming I have a polynomial time algorithm to solve ...
3
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3answers
333 views

Can't understand why the DP Subset Sum algorithm is not polynomial

I can not understand why the dynamic programming algorithm for the Subset Sum, is not polynomial. Even though the sum to find 'T' is greater than the total sum of the 'n' elements of the set , the ...
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1answer
43 views

NP hardness of Partition

I'm trying to show that PARTITION is NP-hard. I'm not sure if what I have is correct so I'll write what I have. I tried to reduce it from SUBSET_SUM: $$PART= \{S\subset\mathbb{Z}|\exists C \subset S: ...
0
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1answer
200 views

NP Completeness of 3-SAT problem [closed]

I have started reading on algorithmic complexity for my thesis work. Already have studied on Polynomial time reducibility, NP-Complete, NP-Hard. Now trying to prove NP completeness of some of the ...
9
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1answer
1k views

Which NP-Complete problem has the fastest known algorithm?

In terms of worst-case asymptotic runtime, which NP-complete problem has the fastest-known (exact) algorithm and what is the algorithm? Is there something known that is faster than $O(n^2*2^n)$?
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1answer
22 views

How to find partition set of a Partition Problem using its decision problem

I understand Partition Problem is NP-complete. Given we have a magic black box that can answer Yes or No for the partition problem. I was wondering how to come up with a polynomial time algorithm to ...
3
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2answers
433 views

are NP Complete languages closed under any regular operations?

I have tried looking online, but I couldn't find any definitive statements. It would make sense to me that Union and Intersection of two NPC languages would produce a language not necessarily in NPC. ...
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0answers
31 views

Prove that Acyclic Subgraph is NP-Hard by showing Independent Set can be reduced to Acyclic Subgraph

I am trying to prove that the Acyclic Subgraph Problem (AS) is NP-hard by showing that the Independent Set Problem (IS) is polynomially reducible to AS. AS is as follows: Given a directed graph G = ...
1
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1answer
77 views

How is it possible for a problem to be NP-Complete under polylog-time reductions?

I have no source for this, but I've heard people offhandedly mention problems that are NP Complete under polylog reductions (I think SAT was one of them). This confuses me - it seems to me that this ...
5
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1answer
116 views

Is Hamiltonian path NP-hard on graphs of diameter 2?

Let $G$ be a graph of diameter 2 ($\forall u,v\in V: d(u,v)\leq2$). Can we decide if $G$ has Hamiltonian path in poly time? What about digraphs? Perhaps some motivation is in place: the ...
7
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1answer
456 views

How do we know any problem is in NP-complete if we don't know all problems in NP?

A problem is NP-complete if: It is in NP. All problems in NP can reduce to it. It's number 2 that I'm concerned with here. I would be highly surprised if we knew every problem in NP. Based on ...
2
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1answer
181 views

Why is the reduction from Vertex-Cover to Subset-Sum of polynomial time?

In the standard proof why Subset-Sum is (weakly) NP-complete, one reduces Vertex Cover to Subset-Sum by using suitable numbers with O(m+n) bits (where m is the number of edges and n the number of ...
3
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1answer
133 views

Negative numbers in Subset-Sum

If I have a set $A$ with positive and negative numbers, and a number to find C. It is possible to reduce the problem to one with only positive numbers in set $A$? I mean, it is possible to find a ...
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0answers
69 views

A variant of Travelling Salesman: Is it NP-complete if its sub-problems are NP-complete? [closed]

Suppose there is a travelling salesman who wants to travel through N cities in k countries(k <= N). For convenience, he will travel all the cities within a certain country and then move to another. ...
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0answers
22 views

Cyclic definition of NP-completeness [duplicate]

Trying to understand the concept of NP-completeness, I came across this pearl on Wikipedia: From NP-complete: A decision problem L is NP-complete if it is in the set of NP problems and also ...
2
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1answer
27 views

Complexity as it relates to verifiers of languages

So I've been thinking about verifiers and a possible relation between a language's class and it's verifier complexity. From the book, "NP is the class of languages that have polynomial time ...
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1answer
58 views

NP-Complete algorithm defined on a fixed size array [closed]

Given an array, say A, with a finite definite length like N (e.g. 1000) can we define a problem to be NP-Complete without any intentional injection of NP-Completeness by something else : for example ...
1
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2answers
90 views

Is subset sum with a fixed target sum NP-complete?

I've read that subset sum is NP-complete. What happens when I change the decision problem to look for a constant number? So the decision problem would look like this: Input: A collection of ...