Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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NP-hardness of a winner determination auction

We would need some suggestions for the proof of NP-hardness of an optimization problem. The problem $$ \max_{x_{a,s}} \sum_s \sum_a x_{a,s} q_a \lambda_s \prod_{s' < s} \prod_{a' \neq a} (1 + ...
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Reduction from OR-SAT to Exact CNF-SAT, keeping the number of variables polynomially bounded?

Let me define both the problems first: $OR$-$SAT$: $m$ Boolean formulae are given in $CNF$, $\phi_1$,$\phi_2$, $\ldots$, $\phi_m$, each on variable set, $x_1, x_2, \ldots, x_n$. ($m$ $<$ $2^n$, ...
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70 views

Small area containing large amount of patterns

The problem: I come across a theoretical problem when designing characters for electron-Beam lithography. Abstractly, given an integer $m$, let $\mathcal{M}$ be the set of $(0,1)$-matrix $A_{p\times ...
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254 views

Is this variation of the “sequencing with release times and deadlines” problem NP-complete?

The following problem is known to be NP-complete. It can be found in pages 236 and 70 of Garey & Johnson. In this book this problem is known either as ...
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252 views

Help with an np-completeness proof [closed]

I know that graph contractability is NP-complete: given $G=(V_1,E_1)$ and $H=(V_2,E_2)$, can a graph isomorphic to $H$ be obtained from $G$ by a sequence of edge contractions? Consider the following ...
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391 views

Post Correspondence Problem “binary” variant

Bounded Post Correspondence Problem is defined as follows: given list of pairs of words $ (x_1,y_1), \ldots, (x_n, y_n) $ and $K$ find sequence of indexes $i_1, \ldots, i_k$, $k \leq K$ so that $x_{...
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200 views

Is there a list of NPC or harder problems for specific real world problem domains? [closed]

The domains of interest to me are: 1. Robotics 2. Search 3. NLP 4. Image feature extraction 5. Network optimization 6. Network security
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211 views

If only pathological cases of NP-hard problems are difficult to solve, then why isn't NP-hard defined to only include those pathological cases?

NP-hard problems are not used in cryptography, because they are believed to be computationally-intractable in the worst case but are not computationally-intractable in the average case. Is there a ...
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241 views

Two Decision Problems About Graphs — Original Results?

I have a couple of short but pleasing results. I was wondering (a) if they're original (b) if so whom should I tell? I don't have easy access to any standard texts that would help me out here. Nor ...
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1k views

Removing all but a few cycles in a graph

Let problem $S$ be defined as Given undirected graph $G$ and a set of cycles $C_1,C_2, \ldots, C_n$ in G, find minimum number of vertices that need to be deleted to remove all cycles in the ...
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301 views

Huffman Tree Depth, Is there any theory?

I'd like to as a variation on this question regarding Huffman tree building. Is there any theory or rule of thumb to calculate the depth of a Huffman tree from the input (or frequency), without ...
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2k views

Bin packing approximation with different bin sizes

Is there any greedy solution with an approximation bound for the bin-packing problem when we have bins of different size? More formally, there are $n$ bins of size $b_i$ for $i=1,\dotsc,n$, and $m$ ...
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1answer
200 views

What is the computational complexity of this SAT Variant

Given a 3SAT problem. The question being: 'This Problem has exactly K Solutions'? Now, lets say K=1 (without loss of generality). If the problem has a exactly 1 solution and the answer is True. So, ...
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2answers
207 views

Is Asymptotic PTAS $\subseteq$ APX?

The definition of asymptotic polynomial-time approximation scheme (Asymptotic PTAS) is defined as follows: A minimization problem $\Pi$ is Asymptotic PTAS if for all $\epsilon$ there exists an ...
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1k views

Is longest common subsequence with bounded occurrences NP-complete?

The general longest common subsequence problem (LCS) over a binary alphabet is NP-complete. Does the problem remain NP-complete if each input string has m zeros and n ones, where m and n are constants?...
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1answer
206 views

Node-weighted steiner problem with few terminals

Consider the node-weighted steiner problem: Input: a graph $G=(V,E)$, a set $T\subseteq V$ of terminals, a weight function $w: V\setminus T \to \mathbb{R}_+$. Output: a minimum weight subset ...
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1answer
154 views

What languages can be reduced to a NP-complete problem in polynomial time

NP-complete: Language is NP-complete, when it is in NP and every problem in NP is reducible to it in polynomial time. But what languages are reducible to a NP-complete problem (for example SAT) in ...
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1answer
96 views

3 dimensional matching shortest solution NP-hard?

We have array of arbitrary number of elements - 3d vectors with positive integers components - for example ...
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1answer
87 views

Minimum number of columns making each row different

I'm curious whether this problem is NP-hard: suppose you are given an arbitrary $m\times n$ 0-1 matrix (each element is either 0 or 1, for the simplicity of the problem), and any pair of rows (i.e. a ...
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1answer
206 views

Does a weighted graph have a path with weight zero?

Given a weighted digraph $G=(V,E)$, where each edge is associated with a weight (could be positive, negative, or zero). We define the weight of a path to be the sum of the weights along this path. ...
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1answer
258 views

Another edge partitioning problem on cubic graphs

This question was motivated by a closely related problem An edge partitioning problem on cubic graphs Input: at most cubic graph ( maximum node degree is 3) $G=(V,E)$, a natural number $k$ Question:...
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1answer
528 views

Clique cover problem

Consider the following graph problem. We are given a graph $\mathcal{G} = (\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ is the set of vertices and $\mathcal{E}$ is the set of edges. For each vertex $...
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1answer
201 views

Max-weight connected & co-connected subgraph problem

The max-weight connected subgraph problem (MWCS) may be described as follows: given a simple graph $G=(V,E)$ and a weight function $w:V\to\mathbb{R}$, one seeks for a subset $L\subseteq V$ for which ...
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1answer
669 views

Positive 1-in-3 SAT FPT or Fixed Parameter Intractable

There are a number of satisfiability problems that are difficult to solve even in the fixed parameter sense. For example, Weighted q-CNF Satisfiability is W[1]-complete when parameterized by the ...
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334 views

Is there a FNP problem that's NP-hard but not FNP-hard?

For the reductions, choose a class C such that [it's clear what FC means] and FC is not known to be able to solve the satisfaction search problem, and assume that FC indeed can't solve that search ...
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338 views

Hardness of XSAT

The standard NP-hard SAT problem is the problem of Boolean satisfiability of conjunctions of clauses, where clauses are disjunctions of literals. I am interested in the problem of the Boolean ...
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1answer
1k views

Vehicle routing problem over Manhattan distances

I am looking for references to the variant of the vehicle routing problem over Manhattan distance metric where the aim is to optimize the number of tours starting at the depot. Is the following ...
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1answer
268 views

Is the following optimization problem (another variant to a previous problem) NP-hard?

This problem is a following up question on this one. The only difference is the addition of $3^{rd}$ constraint "$\sum_{i} x_{ij} \le M$", where M is a constant number. This constraint essentially ...
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1answer
119 views

A dominate vector subset sum problem

Let $k$ be some constants (e.g. one can take $k=2$ for simplexity), for any $u,v\in \mathbb{R}$, we say $u$ dominate $v$ if $\forall 1\le i\le k,~ u[i]\ge v[i]$, write it as $u\succ v$. Consider the ...
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2answers
177 views

Check the match of the maximum of each subset

Given a number of vectors with $n$ elements, i.e., $S=(a_1, \cdots, a_n)$, $T_j=(b_1^j, \cdots, b_n^j)$ for $j=1,\cdots, m$ where each $a_i$ or $b^i_j$ is a natural number. Question: determine ...
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1answer
204 views

Max weight travel on a graph with deadline

Given a deadline $D>0$ and a complete graph $K_n$ (with loops) in which each edge $e_{ij}$ has a weight $w(e_{ij}) \ge 0$ and a travel time $l(e_{ij}) > 0$. Starting from one of the nodes, we ...
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219 views

Packing sets to maximize overlap

For a set of sets $A$, let $\cup A := \cup_{S \in A} S$. Consider the following problem: Input: a list of $m$ weights $w = (w_1, \ldots, w_m)$, a list of $n$ distinct subsets $T = (S_1, \ldots, ...
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1answer
230 views

NP-Hardness for an optimization problem

I want to prove that the following optimization problem is NP-Hard. max $\prod_{i = 1}^{N} \frac{\left[\sum_{j =1}^M x_j \mathcal{R}_{ij}\right]^2}{ \sum_{j=1}^M x_j}$ subject to $x_j \in \{0,1\}\...
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1answer
260 views

Complexity of an edit distance problem

Given an array $A[1...n]$ of non-negative integers, we want to transform $A$ into $A'$ such that $|A[I] - A[I + 1]| \leq 1$ in the minimum number of operations. One operation consist of picking ...
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1answer
203 views

Efficient verification of Kemeny-optimal rankings

The problem of finding a Kemeny optimal aggregation is as follows: Given items $1 \ldots n$ and a list of partial rankings $q_i$ (i.e. permutations of subsets) of these items, find a ranking $r$ of $1\...
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31 views

Scheduling with Separation Constraint

There are $N$ types of jobs. For each $i$, we have to schedule $T/D_i$ jobs of type $i$ in $T$ timeslots. We know that $\sum_{i=1}^N 1/(D_i+1) = 1$. For each type $i$, the distance between two ...
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81 views

Minimum cut with nonlinear objective function

Let $G$ be an undirected graph. The classic minimum (cardinality) cut problem asks for a cut $C\subseteq E(G)$, such that $|C|$ is minimum. Let us generalize it the following way: let $f$ be a ...
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57 views

Reasoning about NP hardness of optimization problems with closed form functions as input

(This may not be a research level question per se. I can delete this question if the community thinks this way too) I am trying to understand how to reason about hardness of optimization problems ...
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Directed NP Hard Problem on DAG

There are problems that are NP-Hard on undirected graphs(maximum weight independent set and graph coloring) but are polynomial time solvable on trees. Tree decomposition is a good tool to talk about ...
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Best way to represent NP-Hardness result for decision problem with two decision parameters

What is the best way to represent a NP-Hardness result for a decision problem with two decision parameters? Suppose we have a problem $P$ which asks to minimize two parameters $x$ and $y$ and we show ...
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Is NP-complete the existence of paths of a given length in a directed graph? [closed]

Given a directed graph G= (V,E), a pair of vertices s and t, a natural number K encoded in binary, whether the problem to decide there exists a path (not necessarily simple) from s to t of length K is ...
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111 views

Generalized path cover problem in DAG

Let $G=(V,E)$ be a directed acyclic graph. Two vertices is transitive if there is a directed path between them. A Path Cover for a Set of Transitive Pairs (PCSTP) is a set of directed paths such that ...
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149 views

Are there any NP-complete for continuous mathematics? [closed]

Looking at this wiki page, it seems most NP-complete problems are based on discrete structures, such as graphs. What are some problems that involve real or complex analysis instead of discrete ...
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1answer
155 views

Is pooling-aware bin packing NP-Hard?

I am unable to prove whether the following problem is NP-Hard. It seems like a bin-packing or a partition problem, without being close enough to either of them (at least I do not see the reduction to ...
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31 views

Sparse coding and matching pursuit algorithms

Is it true that all known sparse coding algorithms which work efficiently in practice don't have convergence proofs and always use an intermediate step of a matching/subspace pursuit algorithm on the ...
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27 views

How does one know what is not in a certain class of pseudo-distributions?

We consider working in the function space $\mathbb{R}^{\{ -1,1\}^n}$ where the expectation inner-product makes the juntas form a $2^n$ dimensional orthonormal basis. Now say one has found a degree $...
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182 views

At what parameters is following $NP$-hard?

Problem Instances at given $\alpha>0$. $(1)$ Given $a_1,\dots,a_{n^\alpha}\in\Bbb Z$ with $|a_i|\in(2^{n-1},2^n-1)$ is there a subset of that sums to $0$? $(2)$ Given $a_1,\dots,a_{n}\in\Bbb Z$ ...
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53 views

About increasing the objective values of certificates for Max-Clique SDP

Say you write a round-k Lasserre (or any other hierarchy!) SDP relaxation of the Max-Clique problem. Lets say one now finds (or knows how to sample with high probability) a graph $G_1$ of size $n$ and ...
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169 views

Is “Binary Interval Tree” NP-hard? [closed]

The input is set of (disjoint) intervals $I$. The output should be the following rooted binary tree. Each leaf node corresponds to an interval from $I$. Each interior node contains an interval which ...
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142 views

Quantum computer versus Random 3-SAT?

It seems to be commonly believed that quantum computer cannot efficiently solve NP-hard problems. What about the challenging problems in average-case, such as Planted Clique and Random 3-SAT?