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Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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914 views

Is there a reduction from a 0-1 knapsack problem to the unbounded problem?

As we know, an unbounded knapsack problem could be described as: $\max \sum_{i=1}^nc_1x_i$ s.t. $\sum_{i=1}^na_ix_i\le b$ $x_i\ge0,x_i\in\mathbb Z,i=1,\cdots,n$ And for an 0-1 knapsack problem, we ...
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0answers
45 views

Bounding the cost of an approximation algorithm when subtraction involve [closed]

Given an algorithm with approximation ratio $\alpha$, and another algorithm with approximation ratio $\beta=n^\epsilon$, and a solution to a problem with cost $c$. What is the standard way to bound $\...
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0answers
57 views

Help in NP-Hardness proof of a certain type of Class Cover problem

Class Cover Problem is nothing but finding an optimal cover of certain class (Point Set) with a particular shape only i.e. finding minimum number of a certain shaped polygon (for example, rectangle) ...
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0answers
273 views

Partially filled jigsaw puzzle with six types of tiles

This is a slight variation of the question Are 'zero-one' jigsaw puzzles NP-complete? asked on cs.stackexchange.com. What is the complexity of the following problem? Input: an $n\times n$ Jigsaw ...
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0answers
241 views

What are the consequences of a ${\bf O}$(m) algorithm for SAT

We are given a Boolean formula $F$ in conjunctive normal form with $n$ variables and $m$ clauses and we would like to know if there exists at least one assignment to the $n$ variables that makes $F$ ...
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0answers
297 views

NP-Completeness of Certain Bounded Degree Graphs [closed]

I was studying time complexity when it comes to bounded degree graph problems and I was wondering if I can get help with the following two problems. 1) Is the set of all (G, k) where G is a graph ...
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0answers
367 views

Hardness of min-max problems

Consider the following min-max problem Given a graph $G=(V,E)$ and an integer $k \geq 0$, delete at most $k$ nodes in $G$ to maximize the size of the minimum dominating set in the residual graph. ...
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0answers
1k views

A variation of the subset sum problem [closed]

The subset sum problem is NP Complete. I was wondering if the following variation can be proved to be NP Complete : The cardinality of set of integers is $m$. And each element of the set is between $...
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2answers
908 views

Where is the error in the following P != NP pseudo-proof ?

Here is a necessarily wrong proof of $P \neq NP$ as it relativizes, but I can't find the error : Le $U$ be a universal Turing machine, whose inputs are restricted to Turing machines accepting ...
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1answer
491 views

Formalized configuration of Subset sum problem Worst-case ? [closed]

Is there a formal proof of the worst-case configuration of the subset sum problem? In other words - is there a set proven to be the hardest to find a subset equals to 0 from? thanks
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2answers
499 views

Is DNF-Equivalence Problem $\mathsf{NP\mbox{-Hard}}$?

I have the following Equivalent DNF problem: Input:Two DNF formulas, $F_1$ and $F_2$,with variables $a_1,a_2,...a_n.$ Output: $1$ if $F_1$ and $F_2$ are equivalent, $0$ otherwise. $F_1$ and $F_2$ ...
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1answer
765 views

graph coloring with 3 colors

I'm searching for an algorithm that can calculate a suboptimal solution for: color a graph with 3 colors some vertices already have a color and can't be changed the edges have values and the ...
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1answer
143 views

Subset Sum Problem and hard looking instances that are not really hard

I have been working in a subset sum solver (some new approach) and while working on the time complexity analysis I found what I describe below. Maybe this could explain why some "hard looking" ...
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1answer
139 views

Find the shortest s-t trail(edge disjoint path) in a graph with negative weight edges

A walk in a graph is a finite or infinite sequence of edges which joins a sequence of vertices. A trail is a walk in which all edges are distinct. Note that a trial may visit a vertex multiple times ...
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2answers
214 views

NP-hard problems on the class of caterpillars

My question is whether there exist an NP-hard problem that has only a caterpillar as input. By saying only caterpillar as input, I wanted to emphasize that no function (eg: weights on vertices or ...
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2answers
153 views

Geometric max cover

Consider $n$ points and a distance function $d$ that satisfies the triangle inequality. We are also given a number $r$. Each point $p$ defines a set $B_p$ (or a ball) that covers all other points ...
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1answer
267 views

NP-complete problems with optimal approximation in poly-time

I'm looking for examples of hard optimization problems, for which we have an optimal approximation (not that this is not the same as $PTAS$, as we require a completely tight approximation, and not $1+\...
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1answer
363 views

Hardness of an extended maximum set packing problem

(Edited) The maximum set packing problem when the sets are all of equal size, say $k$, is known to be NP-hard for $k \ge 3$. The requirement in this problem is that the sets in the solution will be ...
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3answers
555 views

Comparing graphs

I am looking for a kick in the right direction. What i am trying to do is to compute "similarity" between two graphs, where I define "similarity" as the number of shared paths. example: ...
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1answer
75 views

Constrained Topological Sorting with bounded number of chains

In general, constrained topological sorting is NP-hard. Now we add another constraint to it, such that take any k+1 nodes and there will be at least one pair ...
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1answer
245 views

PTAS (polynomial time approximatin scheme) for euclidean TSP/Minimum-Cost k-Connected subgraph problem

Problem 1 I have read "On Approximation of the Minimum-Cost k-Connected Spanning Subgraph Problem" (by A. Czumaj, A. Lingas), and even in the abstract are 2 statements "We present a polynomial time ...
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1answer
61 views

About using smoothness of the Hessian for getting to approximate criticality of a non-convex objective

Is there any algorithm which shows that under the assumption of Lipschitz smoothness of the Hessian of a non-convex function one can get to its critical point faster?
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1answer
225 views

Is there any reference on the hardness of approximation for 2-partition problem?

I tried to look for some references but could not find any. I knew it is proved to be NP-complete via a transformation from Knapsack or 3DM problem. But I couldn't find a way to apply PCP theorem to ...
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1answer
17 views

What Is the Complexity of This Two-to-One Matching Problem?

Given a graph $G=(V,E)$ and a function $c:V\mapsto\{1,2\}$. The function $c(\cdot)$ divides the vertices into two disjoint sets $V_1$ and $V_2$, where for all $v_1\in V_1$, we have $c(v_1)=1$ and for ...
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0answers
101 views

Network Reliability Problem

Network reliability, in which we are given an undirected graph $G$ with a failure probability $p_e$ for each edge and we are asked to calculate the probability that the network becomes disconnected ...
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0answers
59 views

Generalization of k-Coloring: maximizing the number of vertices with no neighbours of same color

One can consider the following generalization of the $k$-Coloring problem: Let be given a graph $G$ and an two integers $k$ and $p$. A vertex $v$ of $G$ is properly colored if $v$ has no neighbour ...
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0answers
60 views

About complexity of recovering or learning Bayesian networks

Are there complexity theoretic results about recoverability or learnability of the marginals (of the source vertices) and the conditionals (along each of the edges) of a Bayesian network from having ...
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0answers
33 views

On approx-preserving P- and A-reducibilities

Let $X$ and $Y$ be two NPO problems. Let $(f,g)$ be a reduction between $X$ and $Y$, in particular, assume that $(f,g)$ is both P-reduction and A-reduction, i.e., there exist two poly-time ...
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0answers
154 views

NP-hardness of minimizing sum of complicated objective function

In our research, we faced the following problem optimization problem: Input: a list of $k$ pairs of positive integers $(n_1,d_1), \ldots, (n_k,d_k)$; an integer $m$. Output: $P$, a partition of the ...
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0answers
432 views

Hardness of Minimizing Submodular Functions with Cardinality Constraints

I am new to submodular functions and I am reading the introductions to submodular functions and applications ( https://www.ima.umn.edu/optimization/seminar/queyranne.pdf ). In this introduction, it ...
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0answers
70 views

Finding exact value with a quotients of products of random values

Sorry for the haphazard title: really not sure what this should be called Suppose we have a set of $z$ random values $S = r_1, \dots, r_z$ drawn from $\mathbb{Z}_N$ (where $N$ is some large prime). ...
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111 views

Minimum Weight Ordering of nodes on a directed graph

I'm bumping my head against the wall trying to prove this problem is NP-complete (it might not be) Let $G = (V,E)$ be a directed graph with weights $w:E \to \mathbb{R_{\geq 0}}$ on the edges. The ...
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0answers
751 views

Algorithm to maximize profit: ways to solve/approach? (Advanced NP-Complete)

This one's hard, so all help really appreciated! I know it is NP-Complete and thus cannot be solved in polynomial time, but looking for help in analysis, i.e. what type of NP-Complete problem it ...
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0answers
347 views

Maximize Covering Minimizing the Overlap

I have this problem: Given a collection of sets $S:\{S_{1},...,S_{k}\}$ where each set $S_{j}$ is a subset of $U:\{e_{1},...,e_{n}\}$ universe of elements. I would find-out a subset $C \subseteq S$ ...
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0answers
225 views

Does coNP-hardness imply NP-hardness [duplicate]

Possible Duplicate: Do many-one reductions and Turing reductions define the same class NPC Hi, Is the following true: If L is coNP-hard, then L is NP-hard. I have found statements of this, ...
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0answers
164 views

np completeness [duplicate]

Possible Duplicate: Reduction Algorithms Hi, I am thinking of researching in this area and am doing some analysis now. Need your help understanding the NP Completeness. I went through the ...
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2answers
760 views

What progress has been made to prove whether or not p=np? [closed]

I know that it is still one of the biggest mysteries of computer science whether non-deterministically polynomial problems can be solved in polynomial time. I am curious to know what makes this ...
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1answer
53 views

NP-Complete graph problems where a special vertex is given as input?

I am currently working on a graph theory problem where the instance includes a graph and a special vertex in the graph. I am trying to prove the NP-completeness of the problem as well as explore ...
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1answer
137 views

Given oracle for Max-3SAT compute clauses that cannot be satisfied

We know that Max-3SAT is NP-hard to compute exactly (and also hard to approximate to a particular constant multiplicative factor). However, suppose you are given an oracle for Max-3SAT that tells you ...
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1answer
75 views

Restricted Universe Exact Cover

Apologies for a simple question - I am a beginning graduate student in TCS. Consider the following $\mathrm{ExactCover}$ problem: Given a collection $\mathcal{S}$ of subsets of a universe set $U$ and ...
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1answer
131 views

A conceptual question regarding hardness proofs by reduction [closed]

If we restrict the input domain of a known NP-hard problem P so that this restricted domain is equal to the input domain of another problem S, then show that we can reduce a solution to P given input ...
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1answer
61 views

How to make any graph 2-degenerate?

I have to show a PPT(polynomial time reduction) from 'Colorful graph Motif' to '2-Degenerate Steiner Tree'. As input graph should be 2-degenerate, but here is normal graph G (that is, basically an ...
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1answer
304 views

What is the simplest known solver for a np-complete problem?

Lets define the simpler of two terms as the one with shortest description length on the untyped λ-calculus. Trying to find the simplest solver for a np-complete problem, I've got this: ...
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1answer
138 views

NP Hardness of Metric Steiner Tree

It is known that the metric steiner tree problem is NP hard (Garry and Johnson [1977]). I wanted to know if there is a simpler way of proving the same. Specifically, I am trying to find a polynomial ...
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1answer
79 views

Example of decidable NP-hard problem that is not NP-complete [closed]

I am looking for an example of a decision problem which fulfills the following conditions: 1. It is decidable 2. It is NP-hard 3. It is not NP-complete All my search attempts yielded examples that ...
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1answer
116 views

Difference between 'Reductions' in algebraic problems vs “Reductions” in Computational Intractability [closed]

When I read NP-completeness for the first time, I really wondered why is the concept of Reductions given such high emphasis, after all we have been looking at concepts such as reductions and 'special ...
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1answer
312 views

What is meant by “if there exists a $\rho$-approximation algorithm with $\rho < 2$, then P = NP”?

For example, for the $k$-center problem we want to prove that a 2-approximation algorithm is optimal. A proof is presented on page 39 (Theorem 2.4) in Williamson and Shmoys, The Design of ...
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1answer
174 views

Can we map this problem to subset-sum?

Let there be $n$ set of ordered pairs $s_1=\{(c_1,f_1),(c_1,f_2) ...(c_1,f_m)\}$, $s_2=\{(c_2,f_1),(c_2,f_2) ...(c_2,f_m)\}$, $s_3=\{(c_3,f_1),(c_3,f_2) ...(c_3,f_m)\}$, .... $s_n=\{(c_n,f_1)(c_n,f_2) ...
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1answer
498 views

Prove that the problem of rectilinear picture compression is np-complete

I need a demonstration that the rectilinear picture compression is NP-hard, I know that this fact was proven using 3SAT by Masek in 1978 but I can't find the paper.
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1answer
61 views

Proving NP-complete problem

Suppose the following problem: Given an undirected graph G=(V,E), is it possible to choose a subset V' of vertex set V, such that deleting it removes all triangles (cycles of length 3), where |V'| is ...