# Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

541 questions
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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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### min weight k-closure on DAG

The problem Given a (connected) DAG $G(V,E)$ where each node is assigned an (non-negative) integer weight an integer k where $0\leq k\leq|V|$ Find a induced subgraph $H$ of $G$ consisting of $k$ ...
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### Sparsification Lemma for k-SAT and Exponential Time Hypothesis

According to R. Impagliazzo, R. Paturi and F. Zane, 2001 an instance of $k$-SAT is called sparse if $m = O(n)$ where $m$ denotes the number of clauses and $n$ the number of variables. The ...
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### 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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### Complexity of bounded degree full contraction

This paper defines the problem $\mathrm{B{\scriptsize OUNDED} \ D{\scriptsize EGREE}\ C{\scriptsize ONTRACTION}}$ as follows: Instance: A graph $G$ and two integers $d$ and $k$. Question: Is there a ...
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### Testing whether letters can be scheduled to achieve a word in a regular language

I fix a regular language $L$ on an alphabet $\Sigma$, and I consider the following problem that I call letter scheduling for $L$. Informally, the input gives me $n$ letters and an interval for each ...
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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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### Common insights into hypothetical complexity of graph problems

I came across two examples of hypothetical hardness of some graph problems. Hypothetical hardness means that refuting some conjecture would imply the NP-completeness of the respective graph problem. ...
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### What is the complexity of this graph problem?

Given a simple undirected graph $G$, find a subset $A\neq \emptyset$ of vertices, such that for any vertex $x\in A$ at least half of the neighbors of $x$ are also in $A$, and the size of $A$ 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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### Minimising the root-set of a spanning hyperforest of a hypergraph

I am interested in the complexity of a problem involving spanning hyperforests (a union of hypertrees, which covers all of the vertices) of a $k$-hypergraph. I describe the relevant definitions for ...
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### Is Non-linear Constrained Optimal Exact Cover NP-Hard?

Playing around I ran into a problem which looks like a Exact Set Covering / Partition Problem, but I am unable to find a reduction to categorize the complexity of the problem, despite it looks NP-Hard....
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### Random self reducibility and NP

I was reading the Wikipedia page Random self-reducibility and it states: If an NP-complete problem is non-adaptively random self-reducible the polynomial hierarchy collapses to $\Sigma_3$. I am ...
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### Complexity of the problem of words with fewest distinct letters accepted by a finite automaton

Given a finite (deterministic or nondeterministic, I don't think this has much importance) automaton A and a threshold n, does A accept a word containing at most n distinct letters? (By k different ...
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### Reduction of graph chromatic number to hypergraph 2-colorability

I'm following this paper titled "Coverings and colorings of hypergraphs" by Lovasz 1973, which is referenced in Garey and Johnson's Computers and Intractability, for the Set Splitting Problem. In this ...
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### Matrix Coloring under Vertical and Horizontal Constraints

I'm searching for the correct name of the following NP-complete problem. I would also appreciate answers pointing to problems with similar-looking variations. The input consists of A set of ...
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### 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 ...
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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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### 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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### Tardos Function Counterexample to Blum's $P\neq NP$ Claim

In this thread, Norbet Blum's attempted $P \neq NP$ proof is succinctly disproved by noting that the Tardos function is a counterexample to Theorem 6. Theorem 6: Let $f \in \mathcal{B}_n$ be any ...
### Is Norbert Blum's 2017 proof that $P \ne NP$ correct?
Norbert Blum recently posted a 38-page proof that $P \ne NP$. Is it correct? Also on topic: where else (on the internet) is its correctness being discussed? Note: the focus of this question text has ...