Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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5
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2answers
251 views

Can we recover integers $a_i$ from the sum $a_0 + a_1e+a_2e^2+\cdots+a_ne^n$?

Since $e$ is transcendental, the function $f:\mathbb Z_{\geq 0}^{n+1}\to \mathbb R$ is injective, $$ f(\underset{\text{Integers}\ \geq\ 0}{\underbrace{a_0,a_1,\ldots, a_n}}) = a_0 + a_1e+a_2e^2+\cdots ...
1
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0answers
46 views

Living in Minicrypt, but sampling hard instances without the solution

In Impagliazzo's worlds, Minicrypt is the one, where one way functions exist. In other words, we can sample hard-on-average instances of NP complete problems. Question: Is living in Minicrypt, where ...
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1answer
120 views

Complexity of a satisfiability problem

I would like the know the complexity of a specific satisfiability problem. I have a feeling it could be solved in polynomial time, but I am not sure about it. The problem is described below. Given $n$ ...
2
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1answer
125 views

Complexity of MAX-ONEs Monotone 2-SAT with $n^{3/2}$ or $C n^2$ clauses?

Let $\phi$ be negative monotone 2-CNF on $n$ variables and $n^{3/2}$ clauses. What is the complexity of finding satisfying assignment with maximum number of ones $k$? Alternatively let $G$ be a graph ...
3
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0answers
138 views

If $\sf{E} = \sf{NE}$. Then $\sf{NP}-{P}$ contains no sparse sets [closed]

I am reading "The Complexity Companion" by Hemaspaandra & Ogihara, I have a question about lemma 1.21. In its proof, they suppose $L$ is some sparse language in $\sf{NP}$ ($||L^{=n}||&...
28
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3answers
2k views

Is Descriptive Complexity dead?

I recently started reading about Descriptive Complexity, the branch of Complexity Theory studying the logic languages needed to express complexity classes. The main milestone in the area seems to be ...
6
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1answer
238 views

Are all computational models of quantum computing equivalent?

So the question was inspired by a seminar which presented the following models of quantum computing: Quantum Computing with Photons Quantum Computing with Rydberg atoms Quantum Computing with trapped ...
0
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0answers
83 views

Is the traveling salesman problem still NP-hard if all edges need to be covered as well?

If we formulate the travelling salesman problem with an added edge-covering constraint as follows, is it still NP-hard? Given a graph G with non-negative edge weights, is there a circular walk in G ...
0
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1answer
61 views

How hard is deciding the existence of a polygonization with prescribed perimeter?

Polygonization problem of a set of points in the Euclidean plane (2D lattice) is to find a simple polygon that passes through all points. Deciding the existence of a polygonization with minimum (or ...
8
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1answer
330 views

On $\text{ETH}$ with $m$ as parameter: consequences of algorithm running in time $2^{\delta m}$ where $\delta \to 0$ as $k \to \infty$

It has been shown in [1] that $k\text{-SAT}$ has a $2^{o(n)}$ algorithm if and only if it has a $2^{o(m)}$ algorithm, $n$ being the number of variables and $m$ being the number of clauses. Being $s_k=\...
6
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1answer
227 views

Why does Dinur's proof of the PCP theorem fail to work for unique games?

What is the critical step where things go wrong if one attempts to use Dinur's proof the PCP theorem to prove the unique games conjecture by starting from a unique label cover instance and doing gap ...
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0answers
60 views

Some examples of tools to demonstrate problem is in $NC$ [closed]

Unlike the class $P$ or $NP$ the class $NC$ does not have any complete problems. To show a problem is in $NC$ one needs to marshal efforts to directly show the problem is in $NC$ since there are no ...
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0answers
50 views

How and How fast can we infer a logical formula that expresses a given graph in C$^2$( logic with 2 vars and counting quantifiers)?

In the following paper the author's claim that almost all graphs can be expressed in first order logic with counting quantifiers and two variables. I would like to know, is there any algorithm that ...
14
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1answer
838 views

Are there languages decidable in linear time by RAM machines that have superlinear time complexity lower bounds for Multitape Turing machines?

Question: Are there languages decidable in linear time by RAM machines that have superlinear time complexity lower bounds for Multitape Turing machines? Background: I recently stumbled upon the ...
8
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1answer
218 views

Is there any online lecture series that covers "Computational Complexity: A Modern Approach"?

I am trying to cover the Computational Complexity :A Modern approach by Boaz Borak and Sanjeev Arora. Are there any nice lecture series that cover this material ?
6
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0answers
364 views

On an unsubstantiated number in a proof of a weaker PCP theorem

I am a mathematician studying Arora and Barak's textbook Computational Complexity, and I am having trouble following one detail in their proof of this weaker PCP theorem: Theorem 11.19. We have NP $\...
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0answers
168 views

What does it mean by the statement: "a problem is hard to approximate "?

In most of the research papers that I have read so far, I often come across the statement of the following form: "the problem is hard to approximate within any factor smaller than some constant&...
-2
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1answer
158 views

How to calculate complexity in a high dimensional space?

Edit: 'Fitness landscape analysis' was mentioned as a relevant measure. If you're going to downvote the post, at least leave a comment what is wrong. For a specific f(), I'm defining a term '...
10
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1answer
274 views

On the usage of Arora and Barak's main lemma in their proof of the PCP theorem

I am a mathematician working toward understanding a proof the the PCP theorem using Arora and Barak's textbook Computational Complexity. I believe I found a few (fixable) errors in Section 22.2, in ...
7
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0answers
150 views

Reference for computing the rank of a matrix in polynomial time

In a recent paper, I need to use the fact that computing the rank of a matrix over the integers has polynomial complexity. Given the context, I don't particularly care about the exact asymptotics, as ...
19
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1answer
1k views

Has parameterized complexity led to better algorithms?

I know that for the vertex cover problem, if we know that the parameter $k$ (which is the number of vertices in the solution) is small, then we can expect to solve it feasibly in practice. So far, ...
9
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1answer
374 views

Complexity Theory Consequences of $\mathsf{NP} = \mathsf{QP}$

I have a certain impossibility result that holds unless $\mathsf{NP} = \mathsf{QP}$. It seems quite likely that one could strengthen this to hold unless $\mathsf{NP} = \mathsf{P}$, which I would not ...
2
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1answer
195 views

Is it possible to reduce an NP language to a NEXP language with exponentially smaller input length?

Suppose we have an NP-complete language $L_1$ and a NEXP-complete language $L_2$. For any deterministic exptime machine $M_1$ with oracle access $M_1^{L_1}$, is it possible to find a deterministic ...
4
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0answers
84 views

Complexity of Encoding a Matroid Flow Problem in a Matrix

Context: Take a directed graph $G$ with a specified subset of source vertices $S$ and target vertices $T$. We say a subset $I\subseteq T$ of size $r$ is independent if there exist $r$ distinct ...
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0answers
165 views

Complexity of Edge Coloring Regular Graphs With Large Degrees

There is an interesting series of papers on edge colorability / $1$-factorization of regular graphs with large degrees, which over the years have shown better and better lower bounds for the degree $\...
4
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1answer
256 views

Proof of $LP$ is in $coNP$ without showing it is in $P$?

Is there a proof that linear programming is in $coNP$ without showing it is in $P$? If so what is the strategy?
1
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1answer
102 views

Common solutions to 3SAT and 2SAT models comprised of the same variables

I have a problem which is a combination of 3SAT & 2SAT instances. Consider $L$ is a set of variables $(x_1 ... x_n)$. $S_3(L)$ is a 3-SAT instance and $S_2(L)$ is a 2SAT instance, both made of ...
7
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1answer
190 views

Isomorphism of ‘ordered’ DAGs / acyclic semiautomata

I am wondering what is known about the isomorphism problem on ordered DAGs, in particular how to find a canonical representative modulo isomorphism. By ordered I mean that each vertex has a list of ...
5
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0answers
107 views

Separating QMA and QCMA

A separation between $QMA$ and $QCMA$ remains a notoriously difficult open problem. Even an oracle separation remains elusive. Have there been any recent efforts towards establishing such a separation,...
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0answers
56 views

Complement of Multi-colored Clique with an extra condition

In the problem Multi-colored clique, we ask for a $k$-clique of the input graph $G$ where $G$ is guaranteed to be $k$-colorable. In the complement problem Independent Set given Clique Partition, we ...
2
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1answer
147 views

Complexity of Multi-colored Clique when every color pair induce biclique+isolated vertices

I am interested in the MulitColoredClique problem with an additional restriction. (Def.: A $k$-coloring $V_1,V_2,\dots,V_k$ of a graph $G$ is a partition of the vertex set of $G$ into $k$ independent ...
6
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0answers
118 views

Computational complexity of finding paths with specified product in a (group-labeled) directed graph

This question came up in the analysis of the puzzle game Swish. One way of representing the solvability problem is this: given a directed graph $G$ where each edge of the graph is labeled with an ...
12
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1answer
301 views

Why is the reduction from 3-SAT to 3-dimensional Matching Parsimonious?

In this talk at the Simons Institute, Holger Dell notes that there is a parsimonious reduction from 3-SAT to the 3-dimensional Matching (3-DM) problem. In other words, there is a reduction between ...
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0answers
68 views

Can the theory of Bidimensionality be applied to weighted instances of a problem?

So my understanding of bidimensionality is you are assured the problem solution is about O(k^2) so you can pay O(k) purely to reduce the instance to one of bounded treewidth. As far as I know, this ...
1
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1answer
97 views

Consequences of turning $\oplus \text{SAT}$ into few satisfying assignments

Suppose there is a reduction which, given a $\oplus \text{SAT}$ instance $\phi$, returns another $\oplus \text{SAT}$ instance $\psi$ having all the following properties: The size of $\psi$ is ...
-3
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1answer
70 views

tables of reductions in literature [closed]

I'm interested in tables of which problems are reducible to which other problems. Particularly for graph problems, but any such tables/graphs would be neat, just so I know how to look for them. ...
1
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0answers
223 views

Disproving $\oplus$ETH by reducing $\oplus k$-SAT with $n$ variables and $m$ clauses to planar graph with $o(m^2)$ vertices?

In this question and its answer, they discuss about reducing CNF-SAT with $n$ variables and $m$ clauses to a (problem on) planar graph $G=(V,E)$ with $|V|$ as small as possible. It is said that the ...
6
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0answers
105 views

Counting on grid graphs

Are there problems defined on graphs, such as counting 2-factors, Hamiltonian cycles, connected spanning subgraphs etc., that are in $\#P$ and remain hard for grid graphs? Since there seem to be ...
22
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3answers
6k views

Implications of proving NP=RP on complexity theory

Edit: As indicated below by Mahdi Cheraghchi and in the comments, the paper has been withdrawn. Thanks for the multiple excellent answers on the implications of this claim. I, and hopefully others, ...
2
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0answers
52 views

The Edge Cover Equilibrium Problem

Let the Edge Cover Equilibrium Problem be the following: INPUT: a simple undirected graph $G$. OUTPUT: YES, if the number of edge covers of $G$ having odd cardinality is equal to the number of edge ...
4
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0answers
84 views

Complexity of finding the mean of the subset with smallest variance

Let $x_1,\ldots, x_n \in R^d$, and $\alpha \in (0, 1)$. Suppose that $\alpha n$ is an integer. Let's consider the following problem $\min_{\mu \in R^d} \frac{1}{n} \sum_{i=1}^n F\left(\frac{\pi(i)}{n}\...
6
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2answers
478 views

Implication of solving 3SUM problem of a certain size on the Exponential Time Hypothesis

In the recent question 3SUM Complexity—A special(?) Case I asked about why the set size $O(n^3)$ was an interesting value for the 3SUM Problem and got a nice answer. My reference was the paper “...
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1answer
169 views

What evidences are there that $PP$ is in $BQP$ and $PP$ is not in $BQP$?

Unlike hierarchy collapse arguments for classical complexity we have that quantum complexity is different. What evidences are there that $PP$ is in $BQP$? What evidences are there that $PP$ is not ...
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0answers
46 views

Non-rigid isomorphic structures

In many of the problems trying to solve hidden shift over some objects like graphs mainly the rigid classes are considered. For eg. in this and this isomorphism problem restricted over rigid graphs is ...
6
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2answers
341 views

3SUM Complexity—A special(?) Case

In the paper “Consequences of Faster Alignment of Sequences” by Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann which appeared in ICALP 2014 and is available here the following version of ...
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0answers
163 views

Why is it difficult for $GCT$ to prove super quadratic lower bound?

We have a quadratic lower bound for the Permanent versus Determinant problem. Why is it difficult for $GCT$ to improve it?
-4
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1answer
146 views

Is mathematical proof itself NP-hard?

At the 8:00 mark of this video, he claims that proving things is itself an NP problem. I'm looking for more insight into this. Could someone help explain this concept to me and also provide a link to ...
0
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0answers
112 views

Can you diagnolize without mentioning simulation?

Are there any known diagonalization proofs, of a language not being in some complexity class, which do not explicitly mention simulation? The standard diagnolization argument goes: here is a list of ...
3
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0answers
91 views

Are there enumerations of machines for all languages in 𝑃 such that there exists a simulator that can efficiently run all the machines enumerated?

From Kozen INDEXINGS OF SUBRECURSIVE CLASSES: "the class of polynomial time computable functions is often indexed by Turing machines with polynomial time counters.... The collection of all (...
12
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2answers
447 views

Complexity relative to the graph of the Busy-Beaver function

This question is inspired by the comments made on this other question that I asked, and by an attempt to provide an explicit example of a complexity question beyond the Turing degree $\mathbf{0}$. (...

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