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Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

4
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1answer
252 views

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 ...
6
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0answers
72 views

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 ...
1
vote
2answers
208 views

Efficiently computable by a “simple” algorithm?

I am interested in the relation between "program complexity" and "computational complexity". In particular, I was wondering What is known about the minimal length a program must have to solve a ...
13
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1answer
683 views

What's the status of Babai's Graph isomorphism result?

It's been over a year since his January 2017 retraction and correction. Is there news? If not is this normal for validation to take this long? I would expect it would get plenty of attention. Has ...
5
votes
2answers
312 views

Complexity of finding a consistent hyperplane

Given $m$ binary labeled points in $\mathbb{R}^d$, it is well-known that in general it's NP-hard to find a hyperplane that minimizes sample error. A brute-force search considers all $O(m^d)$ sample ...
7
votes
4answers
265 views

What are semantic classes that have a syntactic equivalent?

This question is related to Benefits for syntactic and semantic classes. As mentioned there, $\mathsf{PSPACE} = \mathsf{IP}$, which can be interpreted as the semantic class $\mathsf{IP}$ obtaining a ...
2
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0answers
116 views

Hardness of Approximation for minimum path cover in an undirected graph?

Given an undirected graph $G = (V,E)$, a path cover is a set of disjoint paths such that every vertex $v\in V$ belongs to exactly one path. The minimum path cover problem consists of finding a path ...
4
votes
1answer
173 views

Rademacher complexity beyond the agnostic setting

The way I know of to bound generalization error by Rademacher complexity is Theorem 2.4 in this lecture notes, http://ttic.uchicago.edu/~tewari/lectures/lecture9.pdf. Here the quantity on the LHS that ...
22
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2answers
466 views

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 ...
0
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1answer
322 views

What are the problems in EXPSPACE \ EXPTIME?

Based on the diagram below (from Papadimitriou's book) we can see that PSPACE ⊆ EXPTIME ⊆ EXPSPACE. We know that PSPACE ⊊ EXPSPACE, which means that there are problems that can be solved in polynomial ...
0
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1answer
72 views

Reduction from SAT to binary matrix subset problem

I'm trying to understand the answer by @Denis on this question, where he showed a way to reduce from SAT to the binary matrix column subset selection problem. The reason I'm having to post a new ...
1
vote
2answers
190 views

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 ...
17
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1answer
306 views

List of (unsolved) complexity problems arising from PL

What are some major, open computational complexity problems that arise from programming languages, especially program analysis and compilation? I am looking for problems on the lines of "the time ...
0
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0answers
60 views

How hard is nearly Bayes optimal reinforcement learning?

Consider a set of $n$ MDPs (Markov decision processes). An MDP $M$ is selected from this set according to some probability distribution $\xi$ and then interacts with a fixed policy $\pi$ for time $T$ ...
10
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2answers
211 views

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. ...
11
votes
1answer
250 views

Parameterized complexity of inclusion of regular languages

I am interested in the classic problem REGULAR LANGUAGE INCLUSION. Given a regular expression $E$, we denote by $L(E)$ the regular language associated to it. (Regular expressions are on a fixed ...
11
votes
1answer
450 views

Complexity of testing if two sets of $m$ points in $\mathbb{R}^n$ differ only by rotation?

Imagine we have two size $m$ sets of points $X,Y\subset \mathbb{R}^n$. What is (time) complexity of testing if they differ only by rotation?: there exists rotation matrix $OO^T=O^TO=I$ such that $X=OY$...
5
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0answers
92 views

Hard instances in Impagliazzo's Heuristica

In Impagliazzo's imaginary world Heuristica, P ≠ NP but all NP problems are easy on average for any samplable probability distribution. In Impagliazzo's paper, he implies that if you do manage to ...
6
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0answers
213 views

Fischer and Rabin's theorem (1974) for theories of “additive” structures

Fischer and Rabin's Super-Exponential Complexity of Presburger Arithmetic (1974) has the following theorem. (Theorem 12) Let $U$ be any class of additive structures, so if $A = (A, +) \in U$, ...
2
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0answers
73 views

Scaled down and scaled up versions of Impagliazzo-Wigderson Therem

A famous theorem due to Impagliazzo and Wigderson states that if some function in $E=DTIME[2^{O(n)}]$ requires circuits of size $2^{\Omega(n)}$ then P=BPP. When can we change $P$ with some ...
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0answers
95 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 ...
3
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1answer
180 views

How to study the thermodynamics of 2 problems if reduction from $B$ to $A$ exists?

Peter Shor commented on this post: years of experience in theoretical computer science says that the thermodynamic behavior of two NP complete problems are in general not similar. What do we know ...
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0answers
76 views

Reference request for the relationship between approximating degree of Boolean functions and learning algorithms

This paper (http://www.cs.columbia.edu/~rocco/Public/stoc01.pdf) from STOC 2001 is possibly the first paper to show how to convert upperbounds on the $\frac{1}{3}-$approximation degree of a Boolean ...
2
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0answers
93 views

Natural problem in S2P not known to be in any smaller class [duplicate]

Is there a decision problem in $\textsf{S}_2\textsf{P}$ that (presumably) requires the full power of $\textsf{S}_2\textsf{P}$-computations? Specifically, is there a language in $\textsf{S}_2\textsf{P}$...
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0answers
57 views

Logrank allowed extremal matrices?

If we have a $0/1$ real rank $r=O(n^{c})$ size $n^k2^{n^{c'}}\times n^k2^{n^{c'}}$ matrix with all rows and columns distinct with maximum eigenvalue $\leq n^{k'}2^{n^{c'}-\sqrt n}$ for some $1<c'&...
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0answers
53 views

Kolmogorov generic oracle

In Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?, authors defined a new type of generic oracles named Kolmogorov generic oracles. They proved following results relative to $...
4
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0answers
112 views

Reduce $m$-clause 3SAT to PLANAR-3SAT in $O(m^{2-\varepsilon})$

The classic reduction from 3SAT to PLANAR-3SAT requires a removal of $O(m^2)$ crossings from a rectilinear representation of 3SAT with $m$ clauses. However, the crossing number inequality suggests ...
3
votes
1answer
104 views

Connection between algebraic logic and computational complexity of logics?

I'm learning a bit about algebraic logic and I was wondering how knowing the algebraic semantics of a given logic might help the study of the logic itself from a computational point of view. In ...
8
votes
1answer
368 views

Complexity of modal logic IK5

What is the complexity of local satisfiability problem for modal logic $\mathit{IK5}$? Herein we denote by $IK5$ the modal logic over euclidean frames extended with inverse modality. Could you provide ...
12
votes
1answer
1k views

Entropy and computational complexity

There are researcher showing that erasing bit has to consume energy, now is there any research done on the average consumption of energy of algorithm with computational complexity $F(n)$? I guess, ...
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0answers
65 views

Using idea of entropy (maybe Shannon entropy or other continuous entropy) to the topic of functional analysis

I am an electrical engineer without a detailed background in theoretical computer science. I am posting here since I hypothesize that the concept of entropy or other branches of information theory (as ...
5
votes
2answers
221 views

Efficient way to generate random planar cubic bipartite graphs

3-regular bipartite planar graphs appear in a variety of NP- / #P-complete problems. Suppose one wants to test the complexity of these problems via numerical experiments. Is there an efficient way to ...
9
votes
1answer
354 views

Is abelian group isomorphism in $\mathsf{AC^0}$?

An $O(n^2)$ running time algorithm for abelian group isomorphism is easy to see. Later working on this problem in 2003 Vikas improve the result from $O(n^2)$ running time to $O(n \log n)$. In 2007, ...
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0answers
67 views

Computational Complexity of the Helpmate problem in Chess

It is well known that Chess is EXPTIME-Complete. I am interested in the related "Helpmate" problem. Given an $N$ by $N$ chessboard, is there any sequence of at most $k$ moves that leads to Black's ...
3
votes
1answer
202 views

How to compute GCD efficiently?

I want to compute $ A= \langle \text{GCD}(a,j),j=2,3,..,k-1\rangle$ and assume that each value of $j$ is less than $a$. I can compute GCD(a,j), $j=2,3,..,k-1$ and $a \le j$ for single fixed value of $...
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0answers
89 views

Is $\mathrm{DTISP}(n^a,n^b) \subseteq \mathrm{DSPACE}(n^{b/2})$?

The title question arose in the course of discussing a question on MathOverflow. Obviously, from the space hierarchy theorem we know that not only is it false that $\mathrm{DSPACE}(n^b) \subseteq \...
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1answer
122 views

Should GCT focus on $PSPACE\not\subseteq P/poly$?

GCT tries to show $P$ is not $NP$ by showing $NP$ is not in $P/poly$. Could it be useful in showing $\Sigma_{i+1}\not\subseteq P^{\Sigma_i}/Poly$ at every $i>0$? Suppose if it turns out that $\...
11
votes
1answer
195 views

Does the space hierarchy theorem generalize to non-uniform computation?

General Question Does the space hierarchy theorem generalize to non-uniform computation? Here are a few more specific questions: Is $L/poly \subsetneq PSPACE/poly$? For all space ...
4
votes
1answer
200 views

What is the complexity of counting parse trees?

A Counting Problem Given a CFG $G$ and a string $s$, how many distinct parse trees are there for the string $s$? An Example Instance Let's consider an example instance consisting of a CFG $G$ with ...
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0answers
43 views

What is the best known gap between ZPP and Deterministic communication complexity? [duplicate]

I know that $N(f) \times coN(f) \geq D(f)$. This means that $ZPP(f) \geq \sqrt{D(f)}$. Is this separation tight?
10
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1answer
131 views

On sparse complete sets and P vs L

Mahaney's Theorem tells us that if there is a sparse $NP$-complete set under polynomial-time many-one reductions, then $P = NP$. (See "Sparse complete sets for NP: Solution of a conjecture of Berman ...
14
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1answer
181 views

What are the consequences of $P \subseteq L/poly$?

A language is in $L/poly$ if there exists a logspace Turing machine that decides the language with polynomial amount of advice. See here for more info: https://en.wikipedia.org/wiki/L/poly ...
7
votes
1answer
330 views

How to contribute incrementally to Theoretical CS - similar to github

I have been through all the basic undergrad classes in CS. I have finished a masters in CS. I have learned a decent amount about applying Haskell to engineering. I haven't studied abstract math yet. I ...
4
votes
1answer
221 views

Is it known if $CFL \subseteq NSPACE(o(log^2(n)))$?

$CFL$ is the class of context-free languages. Question Is $CFL$ known to be solvable in $o(log^{2}(n))$ non-deterministic space? What about $DCFL$?
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0answers
63 views

Relating classes of computational complexity to finding solution to classes of algebraic equations [closed]

Having related classes of computational complexity to finding solution to classes of algebraic equations, we may relate classes of computational complexity to algebraic geometry or complex geometry,...
5
votes
1answer
162 views

Verifying that a reduction is correct

Alice has a function $f: \{0,1\}^* \to \{0,1\}^*$ which can be computed in polynomial time. She claims that $x \in \mathrm{SAT} \iff f(x) \in \mathrm{CLIQUE}$. Alice sends the circuit computing $f$ on ...
3
votes
1answer
97 views

Complexity of maximising weighted sum of and functions on a set of binary variables

Suppose we have a set of binary variables $a_1, ..., a_n$ that $a_i\in\{0,1\}$. Now we define $m$ and functions over a subset of them: $$j\in\{1,...,m\}: f_j=x_1\land x_2\land...\land x_k$$ in which ...
4
votes
0answers
169 views

Is $NEXP^{NP}$ known to not be contained in $NP/poly$?

To the best of my knowledge, it is known that $NEXP^{NP} \nsubseteq P/poly$, but it's still not known if $NEXP \nsubseteq P/poly$. For more info, see "Superpolynomial circuits, almost sparse oracles ...
12
votes
1answer
670 views

Is sorting $n$ real numbers in time $O(n \sqrt{\log n})$ and linear space possible?

In the recent preprint https://arxiv.org/abs/1801.00776, it is claimed that $n$ real numbers can be sorted in time $$O(n \sqrt{\log n}), $$ and linear space. The paper seems reasonable, though I am ...
7
votes
1answer
125 views

Presburger arithmetic: is it known to be in $EXPSPACE \setminus EXP$?

My understanding of the Presburger arithmetic decision problem is that it requires doubly-exponential time, but only singly-exponential space, meaning that it is in $EXPSPACE \setminus EXP$. ...