# Questions tagged [np-complete]

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### NP-Completeness of the decision problem for the generalized 15-puzzle

I am interested in the natural generalization of the famous 15-puzzle, where you have to slide blocks until you have sorted all given numbers (usally there is a gap of 1 block). Now the ...
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### Is there an NP-complete language that contains precisely half of the n-bit instances?

Is there a (preferably natural) NP-complete language $L\subseteq \{0,1\}^*$, such that for every $n\geq 1$ $$|L\cap \{0,1\}^n|=2^{n-1}$$ holds? In other words, $L$ contains precisely half of all $n$-...
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### How much would a SAT oracle help speeding up polynomial time algorithms?

Access to a $SAT$ oracle would provide a major, super-polynomial speed-up for everything in ${\bf NP}-{\bf P}$ (assuming the set is not empty). It is less clear, however, how much would $\bf P$ ...
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### Is there a natural problem in quasi-polynomial time, but not in polynomial time?

László Babai recently proved that the Graph Isomorphism problem is in quasipolynomial time. See also his talk at University of Chicago, note from the talks by Jeremy Kun GLL post 1, GLL post 2, GLL ...
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### “Almost easy” NP-complete problems

Let us say that a language $L$ is P-density-close if there is a polynomial time algorithm that correctly decides $L$ on almost all inputs. In other words, there is an $A\in$ P, such that $L\Delta A$...
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### Natural candidate against the Isomorphism Conjecture?

The famous Isomorphism Conjecture of Berman and Hartmanis says that all $NP$-complete languages are polynomial time isomorphic (p-isomorphic) to each other. The key significance of the conjecture is ...
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### Are there any heuristic-free NP complete problems?

Are there any NP complete problems with no infinite subset of instances $\Phi$ such that membership in $\Phi$ can be decided in polynomial time, and for all $x \in \Phi$, $x$ can be solved in ...
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### Password hashing using NP complete problems

Commonly used password hashing algorithms work like this today: Salt the password and feed it into a KDF. For example, using PBKDF2-HMAC-SHA1, the password hashing process is ...
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### Poly time superset of NP complete language with infinitely many strings excluded from it

For any arbitrary NP complete language is there always a polytime superset the complement of which is also infinite? A trivial version which does not stipulate the superset to have infinite ...
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### NP-Complete problems that admit an efficient algorithm under the promise of a unique solution

I was recently reading a very nice paper by Valiant and Vazirani which shows that if $\mathbf{NP \neq RP}$, then there can not be an efficient algorithm to solve SAT even under the promise that it is ...
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### Add a matching to a Hamiltonian path to reduce the max distance between given pairs of vertices

What is the complexity of the following problem? Input: $H$ a Hamiltonian path in $K_n$ $R \subseteq [n]^2$ a subset of pairs of vertices a positive integer $k$ Query: is there a matching $M$ such ...
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### Slowest many-one reduction?

When we want to prove that an $L\in \bf NP$ is $\bf NP$-complete, then the standard approach is to exhibit a polynomial time computable many-one reduction of a known $\bf NP$-complete problem to $L$. ...
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### Which complexity class does this number theory problem belong to?

'Given $a,b,c\in\Bbb N$, is there $x,y\in\Bbb N$, $ax^2+by=c$' is $\mathsf{NP}$-complete. Which complexity class does 'Given $a,b,c\in\Bbb N$, is there $x,y\in\Bbb N$, $ax^2+by^2=c$' belong to?
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### Generalization of SAT, where we replace OR with another symmetric function

Let $\sigma(y_1,\dots,y_k)$ denote some boolean symmetric function on $k$ boolean inputs, $\sigma:\{0,1\}^k\to\{0,1\}$. In $k$-SAT, an instance is a conjunction of clauses, where each clause is the ...
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### Does the Cook-Levin theorem relativize?

My only motivation for asking this question is long-standing curiosity, but I am interested in seeing a proof (or disproof) that the Cook-Levin theorem relativizes. If you have a proof that the ...
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### Complexity of finding the largest induced subgraph with all even degrees

What is the complexity of the following problem? Instance: Simple, undirected graph $G$, and a positive integer $k$. Question: Does $G$ have an induced subgraph on at least $k$ vertices, such that ...
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### What's complexity of this set problem which looks like “Linear Programming”?

I came up with a problem below, which looks like a linear programming problem: Given $n$ sets $S_{1}, S_{2},..., S_{n}$, with constraints of :  \forall i=1, 2, 3,...,n\space\space \left | S_{i} ...
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### On the shortest vector problem (is it $NP$-complete?)

Ajtai has shown that shortest vector problem is $NP$-hard by using randomized reduction from subset sum. Has this been derandomized?
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### Magic constant to solve NP-complete problem in polynomial time

Let's suppose that $P\ne NP$. Is that possible to solve all the instances of size $n$ of an NP-complete problem in polynomial time using some "universal magic constant" $C_n$ that has a polynomial ...
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### Analogues of the Berman Hartmanis conjecture and the Creativity Hypothesis

The Berman Hartmanis conjecture which formally states that there is an isomorphism for two $NP$ complete languages $L_{1}$, and $L_{2}$, the isomorphism is a bijective function $f()$ such that $f()$ ...
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### Does p-isomorphism preserve phase transition?

Consider two NP-complete languages that are polynomial-time isomorphic. If we know that one of them exhibits phase transition (with respect to some order parameter), does this imply that the other ...
The PCP theorem can be stated like this : There is a polynomial time reduction from SAT to $Gap\text-MAX\text-3SAT_{c}$ i.e. there is a reduction that maps an instance $\phi$ of SAT to an instance $... 1answer 364 views ### On Zero sum perfect matching Fix$c\geq1$. Input is a$m$vertex complete graph with edges assigned$a_1,\dots,a_{\frac{m(m-1)}2}\in\Bbb Z$in some order. Is it$\mathsf{NP}$-complete to decide if there is a perfect matching of ... 2answers 547 views ### Proof for ACYCLIC PARTITION being NP-complete I'm new to this site, so please pardon me for any mistakes and please feel free to edit the question to help get better answers. I'm interested in reading any proof of ACYCLIC PARTITION (Garey and ... 1answer 111 views ###$NP$completeness of Hamiltonicity of cubic polyhedral plane graphs with bounded face degree? Let$\mathscr{C}_d$be the class of cubic 3-connected simple plane graphs, with face degree bounded by$d$. Is there any$d$such that Hamiltonian cycle is$NP$complete on$\mathscr{C}_d$? If so, ... 1answer 171 views ### Is sparse embedding of a NP-complete problem in a polynomial problem NP-complete? Consider the following problem P: Input is a finite graph G. If the number of vertices in G is 2^2^i for some integer i, then output a minimum vertex cover of G; otherwise output empty set. Can I say ... 2answers 181 views ### Minimising the root-set of a spanning hyperforest of a hypergraph I am interested in the complexity of a problem involving spanning hyperforests (a union of hypertrees, which covers all of the vertices) of a$k$-hypergraph. I describe the relevant definitions for ... 1answer 886 views ### Is approximating Exact Set Cover NP-hard for constant approximation factor? ETH hard? It is known that Exact Set Cover is an NP-hard problem (Reduction from 3-SAT and 3-Coloring). Also, my minor analysis one can realize that this problem is also ETH-hard, i.e. this cannot be solved in ... 0answers 99 views ### Reconstructing a colored grid with vertical and horizontal shifts Consider the following simple problem (puzzle): given a$N \times Nc$-colored grid$G$a$N \times Nc$-colored target grid$G_T$a number$m$represented in unary Can we transform$G$into$...
I'm wondering if the following problem is NP-Complete or has any hardness result. References on related problem are also welcome. Input: integers $n\geq1,k\geq0$ and an invertible matrix \$M\in\mathbb ...