# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...