Questions tagged [np-complete]

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13
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248 views

Is this problem on unambiguous finite automata NP-complete?

An unambiguous finite automaton (UFA) is a nondeterministic finite automaton (NFA) such that each word has at most one accepting path. In this post, for $n\in \mathbb{N}$, what I call an $n$-UFA (resp....
12
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0answers
474 views

Is satisfying $\sum_{i=1}^{n}{x_i^{y_i}}=r$ NP Complete?

Question I would like to show that satisfying $\sum_{i=1}^{n}{x_i^{y_i}}=r$ is NP-Complete. Consider $L= \{(\bar{y},r):\exists \bar{x} \text{ such that } \sum_{i=1}^{n}{x_i^{y_i}}=r\}$. Where $\...
11
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0answers
370 views

Error in paper “Some NP-complete geometric problems”?

The paper in question: M.R. Garey, R.L. Graham and D.S. Johnson. Some NP-complete geometric problems . This paper proofs the NP-completeness of some well-known problems, such as the Steiner Tree ...
9
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0answers
130 views

Is 4-Coloring restricted to graphs with crossing number 1 NP-complete?

Planar graphs are 4-colorable. Determining if a planar graph is 3-colorable is NP-Complete. A graph with a crossing number 1 (graph such that it can be drawn with $\le 1$ crossing) is 5-colorable. ...
9
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315 views

What would be the consequences if all _infinite_ NP-complete languages are p-isomorphic?

The famous Isomorphism Conjecture of Berman and Hartmanis says that all $NP$-complete languages are polynomial time isomorphic ($p$-isomorphic) to each other. It has been an early attempt (published ...
8
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0answers
145 views

Practical interactive proof schemes for NP-hard problems

Model-checking (in the sense of reachability in a succinct graph) is PSPACE-complete. SAT is NP-complete. Both problems are considered intractable, yet there exist tools capable of solving them on ...
8
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0answers
183 views

Complexity question from mathematical music theory

Fix an positive integer $N$. A row means any linear ordering $R=(n_i)_{0\leq i <N}$ of the additive group ${\Bbb Z}/N{\Bbb Z}$. Call $R$ a (generalized) all-interval row if the elements of the ...
7
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56 views

Complexity of deciding whether subspaces of Z_2^n cover every point 3*x times

When studying the complexity of checking identities in certain finite algebras, I came across the following decision problem: Input: A positive integer $n \in N$ and a set of affine subspaces $H_1,...
7
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196 views

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 ...
6
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397 views

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?
5
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99 views

Reconstructing a colored grid with vertical and horizontal shifts

Consider the following simple problem (puzzle): given a $N \times N$ $c$-colored grid $G$ a $N \times N$ $c$-colored target grid $G_T$ a number $m$ represented in unary Can we transform $G$ into $...
5
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58 views

Hardness result or reference for optimal Gaussian elimination process

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 ...
5
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197 views

When will an NP-complete language remain hard if half of a witness is revealed with the instance?

Let $L$ be an NP-complete language. Let $W(x)$ denote the set of (polynomially length bounded) witnesses that certify $x\in L$. That is, $x\in L$ if and only if there exists a $w$, such that $w\in W(...
5
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189 views

Question about a unary language construction

For any language $L$, let us define another language $Tally(L)$, as follows: $$Tally(L)=\{1^n\;|\;\exists x \;\mbox{with}\; x\in L \;\mbox{and}\; |x|=n\}$$ That is, $Tally(L)$ encodes whether there is ...
4
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71 views

Short $\exists$SO sentences over strings that define an NP-complete problem

[Q1] I'm wondering if there are some "official" SHORT existential second order sentences with ONE binary relation, over strings (over a small alphabet) that define an NP-complete set. (Something ...
4
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0answers
180 views

Is there a language in NSPACE(O(n)) and (very likely) not in DSPACE(O(n))?

Actually I found that the set of context-sensitive Languages, $\mathbf{CSL}$ ($\mathbf{=NSPACE}(O(n))$ $= \mathbf{LBA}$ accepted languages) are not so widely discussed as $\mathbf{REG}$ (regular ...
4
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270 views

Bijection between NP-complete problems

If $A$ and $B$ are NP-complete problems, is there a bijective function $f$ (computable in polynomial time) such that $w\in A$ iff $f(w)\in B$?
4
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140 views

Symmetry of optimal solutions to discrete optimization problems

Given a graph, say one wants to find the clique number, independence number, chromatic number, vertex cover number etc., one knows that a solution exists. However if the solution space has more than ...
3
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248 views

new subset sum approach results

I have been working on a new approach for a subset sum exact solver, and the current state provides an algorithm operating on $O{n/2 \choose n/4}$, demonstrating as well the hardest target value is ...
2
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0answers
67 views

Hardness result or reference for a set partition problem

I'm wondering if the following problem is (or has been proven to be) NP-Complete. Input: integer $n\ge0$, set $S_1,S_2,\ldots,S_{2n}$, set $T_1,T_2,\ldots,T_n$. Accept iff: there exists $\{a_i,...
2
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0answers
95 views

About representability of optimization versions of NP-complete questions as polynomial optimization over the hypercube

Naively, in my very limited awareness, it feels that the Max-CUT is a very "special" NP-Hard problem because for a graph with edge-set $E$, it can be written as the question of trying to maximize a $n$...
2
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0answers
129 views

Space time lower bound with $\mathsf{PSPACE}$ oracle

Does a single tape Turing machine with access to $\mathsf{PSPACE}$ oracle needs more than $\mathsf O(1)$ working tape memory and $\mathsf O(1)$ working time to solve $\mathsf{NP}$-complete problem? ...
2
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0answers
326 views

Which complexity information of Ising model is more important?

In 1982, Barahona proved that finding the ground state of an Ising model is NP-hard. Later, in 2000, Istrail proved that it is NP-complete. When I look up the citations of these two papers using ...
1
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38 views

Efficient algorithm for finding segregators in a directed acyclic graph

Given a directed acyclic graph $G=(V,E)$, we define a $(\alpha,\beta)$-segregator of $G$ to be a subset $S$ of $V$ of size $\alpha$ such that no vertex in $G\setminus S$ has more than $\beta$ ...
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45 views

Minimum rank graph cut

Consider the following problem: Input: A graph $G=(V,E)$ and a matroid $M$ on $E$, given by an independence oracle. Task: Find a cut $C\subseteq E$ in the graph, such that the rank of $C$ in the ...
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128 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 ...
1
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0answers
77 views

Prove that finding set of $k$ vertices $S$, such that $G{\setminus}S$ is claw-free is NP-Complete

The claw in a graph $G(V,E)$ consists of a vertex $v\in V$, and it's three neighbours - $\{x_1,x_2,x_3\}\in V\setminus \{v\}$, if $\{x_1,x_2,x_3\}$ form an independent set in $G$. The problem asks us ...
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0answers
76 views

Distributing bags of apples equally

Assume you have $N$ bags of apples that you want to equally distribute to $K$ people. Each bag contains $n_i$ apples and you are not allowed to open and divide the bags; you must distribute the bags ...
1
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0answers
281 views

Finding an equivalent NP-complete instance for this game-theory problem

I apologize if this question is not a good fit for CSTheory. I'm a PhD student who has just started out and I'm working on a game-theory problem in one of my classes. Although my professor hasn't ...
1
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0answers
960 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 ...
1
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0answers
125 views

Properties of “second-order” NP (complete) languages

Reading the question Natural NP-Complete Problems with Large Witnesses, I was interested in this language: $L = \{ \varphi ~~:~~ \varphi \text{ is SAT formula with more than } |\varphi|^2 \text{ ...
1
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0answers
841 views

Are there upper bounds on the worst case complexity of NP-complete problems?

I have proven some problem to be (weakly) NP-complete and try to find out some algorithm to solve it exactly. Except for some pseudo-polynomial stuff, I would be happy with an algorithm running in $O(...
0
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0answers
108 views

Complexity of multi-objective optimization problems

How can we define and prove the worst-case complexity of multi-objective optimization problems (MOOP)? It is easy to see that, if one of the objectives is an NP-Hard optimization problem, then the ...
0
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0answers
68 views

Complexity of variant polynomial factorization

Few years back a question on whether a variant of integer factorization was $\mathsf{NP}$ complete was asked, which according to state of mathematics today does not have a conclusive answer. Is ...
-1
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1answer
161 views

Will a non-linear lower bound on some NP complete problem prove non-linear lower bound on 3SAT?

A problem $\Pi$ is $\mathsf{NP}$ complete if there is a polynomial time reduction from an $\mathsf{NP}$ complete problem $\Pi^\circ$ to $\Pi$ with polynomial blow up on number of variables and ...
-2
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0answers
38 views

How hard is this special 3-partition problem?

The definition of the original 3-partition problem is that: given a set $A$ of elements $A=\{a_1,a_2,...,a_{3m}\}$ and a positive integer $B$, and a positive size $s(a_i)$ for each $a_i \in A$ where $...
-4
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0answers
59 views

3-partition problem is NP-complete. How about “N-partition problem”?

It is well known that 3-partition problem is strongly NP-complete. (https://en.wikipedia.org/wiki/3-partition_problem) Then, My question is, how about "N-partition problem"? Is it NP-...
-5
votes
1answer
26 views

Polynomial-time reducibility of Primality and 3-SAT

Is 3-SAT $\leq_{p}$ Primality? And/or is Primality $\leq_{p}$ 3-SAT? I think the answer is no and yes, respectively, but I'm not sure. Any help would be appreciated. Thank you.