All Questions

There is no classical algorithm for $n$-bit TQBF with better than $O(2^n)$ complexity. Is that also the best known bound for quantum algorithms / circuits?

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This is somewhat of a meta-cstheory question, and is more historical in nature. What are some good examples of problems for which the literature followed the develpment below: The original algorithms,...

Dependent-linear type theories may be a functional programmer's dream, but is it categorically interesting, i.e. is it the internal language of an "interesting" category? By "...

Are there any dimensionality reduction techniques known for the general $\ell_{p}$-normed spaces for $\ell \geq 1$? In the Euclidean space, there is a classical result: Johnson-Lindenstrass lemma that ...

$\def\dlt{\mathrm{DLOGTIME}}\def\nlt{\mathrm{NLOGTIME}}\def\mr{\mathrm}$During a recent discussion on another question, I mentioned a factoid $\exists\dlt=\nlt$, but then I realized that I may have ...

In a type theory where type checking is decidable we can algorithmically determine whether a given a string defines a valid term of any type. But what about other type theories?

Let's say I have two models trained to do the same thing (EDIT : Clarification here, trained to do the same thing implies they were models trained to solve the same problem (for eg - two different ...

I want to know Scheme language is orthogonal or not, and if yes, then which are the features that make it orthogonal. How can we determine a language is orthogonal or not? and the benefits of ...

I'm building a relevance ranking system for incidents occurrence and prevention. My goal is to use four attributes to establish relevance: tag (About 500 tags), x_coordinate, y_coordinate and time. ...

There's a famous problem these days in the interview prep community (particularly in PRAMP) called the awards budget cut problem. The problem gives you an input of $n$ integers called grants $g_1 ... ...

Given a matrix, What is the computational complexity of the fastest algorithm to compute Jordan canonical form for the matrix?

If a type system allows inductive types (as in e.g. Coq) then we can coin new primitive constants that inhabit types. For example $0:\mathbb{N}$ is constructed when defining $\mathbb{N}$ and does not ...

I'm interested in lower bounds on the complexity in the real-RAM model of solving systems of linear equations which have the sparsity pattern of a three-dimensional cubic mesh. Specifically, consider ...

I'm currently working on graphs classes where the distance between two specific vertices is the same in every connected spanning subgraphs, and I am looking for a name for this class. Given a ...

I'm working on a problem that may be reduced to the following variant of multiple knapsack problem: Each knapsack has its own valuation function; an item brings different profit and weight to a ...

I once read in a paper: "given an algorithm for Majority-SAT, one can solve #SAT with $O(n)$ calls to the algorithm." What is the approach to this?

The definition of a $DLOGTIME$-uniform circuit family is based on a Turing machine that accepts the language $\langle t, a, b \rangle$, where gate $a$ is of type $t$ and has gate $b$ as a child, ...

CodinGame challenges are basically extremely large decision trees, where your program is trying to find the node with the highest possible score. Greedy algorithms don't work because picking a locally ...

It's known that there are different "kinds" of grammar complexity of language $L$ --- nonterminal complexity (minimal possible $|N|$ for grammar $(N, \Sigma, P, S)$ generating $L$), covering ...

tl;dr I'm trying to partition my students into groups with respect to their preferences, i.e. they can declare if they want to be with someone in a group or if they do not want to be with someone in a ...

Maximum independent set and minimum edge cover are a pair of related problems and are cover-packing problems. (Also see p114 of West's Introduction to Graph Theory) Are maximum independent set and ...

I am wondering if it can be shown that graph databases along with the graph database model and cypher queries are computationally equivalent to relational algebra. More specifically can cypher queries ...

Given array if numbers a[1..n]. Pair of numbers (i, j) is interesting, if i < j и a[i] > 2a[j]. How to count number of interesting pairs in O(nlogn)? What is the solution? My solution is not ...

I am sorry if this is not an advanced question. Most computer scientists believed that $NEXP \not \subset P/poly$ but they are not even close to this assumption. The main evidence that they are used ...

$FO^2$, i.e. two-variable first-order logic, has a NEXPTIME-complete satisfiability problem (see Grädel, Kolaitis and Vardi '97). However, the decidability and complexity of this fragment is proved by ...

Let us consider the following optimization problem. $\mathcal{P} =\{P_1,\cdots,P_n\}$, where $P_i\subset\mathbb{R}^d$. Let $m = max_i\lvert P_i\rvert$. The goal is to find a point $c$ such that ...

Given generator $g$ of a multiplicative group mod a prime $p$ the Diffie Hellman problem is to find $$g^{xy}\bmod p$$ from $g^x\bmod p$ and $g^y\bmod p$. The best way to solve this is through discrete ...

First let me describe the metric $k$-median problem. Definition (Metric $k$-Median): Given a set $C$ of clients and a set $L$ of facility locations defined over a distance metric $d$. Open a set $F$ ...

How is orthogonality implemented with Miranda, Haskell, Julia, Scheme, and Erlang languages?

I'm currently trying to find a reduction to this problem: Given a set S of n points (in the plane) in general position, is there a set of at least k triangles (formed using only points in S as ...

Let $G$ be a simple finite undirected graph. Let $\{V_1,V_2\}$ be a partition of its vertex set; that is, $V_1\cup V_2=V(G)$ and $V_1\cap V_2=\emptyset$. The partition $\{V_1,V_2\}$ is said to be an ...

Let ${\cal L}=\{Y_1,..., Y_k, X\}$ be a finite relational language such that $X$ is a unary relation name. Let $\phi(X,\bar{Y})\in{\cal L}$ be a first-order formula (the formula can have the equality ...

I'm trying to formalize the syntax and typing judgments of the Calculus of Constructions in Coq. I'm choosing to use the Pure Type Systems presentation of CoC; however, I've seen mild variations in ...

There are two sets $T$ and $M$. The set $T$ represents a set of tasks and the set $M$ consists of machines. A task $t_i \in T$ has two attributes: 1) a minimum-finish requirement $R_{t_i}$ and 2) an ...

We have a theoretical black box with no obvious inputs and outputs a value. For the sake of imagination, it's a literal black box with a digital counter on one side. We want to figure out what causes ...

Let us define the sketch as mapping $\varphi:R^D\to R^d$, such that for arbitrary $x\in\mathbb{R}^D$, its $\ell^1$ norm is preserved up to $\epsilon$ error, with $1-\delta$ success probability: $$\...

I am doing some research on machine learning algorithms in the context of a seminar, which focuses on Energy-Based Modeling vs Deep Learning specifically in working with images Modeling. Now I know ...

It occurred to me that if a given SAT is unsatisfiable, then a DPLL algorithm must try true/false settings for all literals to determine that there does not exist any satisfying literal combination. ...

The typical definition of dynamic transitive closure (or reachability) uses two types of queries: the first one is an update (edge deletion/insertion) and the second one is a reachability query. Thus, ...

I was wondering if there are any results that say how hard it is to answer the question are there TWO subsets that sum to a fixed value? In other words, the subset sum problem but asking if there are ...

Right now I am trying to understand the concept of simulation in theoretical computer science, focussing on Universal Turing machines. All textbooks that I looked into only explain examples. They ...

Assume that $\mathcal{A} = (Q_A, \Sigma, \Delta_A, q_{i_A}, F_A)$ and $\mathcal{B} = (Q_B, \Sigma, \Delta_B, q_{i_B}, F_B)$ are two NFAs. What is the worst-case time complexity of computing $\mathcal{...

(This question also suits for other similar natural $\mathrm{NP}$-hard problems) I know that there is a $4k^2$ vertex kernel (and $8k^2$ edge kernel) by Thomasse [Thomasse09] for Feedback Vertex Set (...

Underlying motivation for the question: if someone showed that $\text{P}=\text{NP}$ but the algorithm thus produced for, e.g., $3\text{-SAT}$, runs in time $\Omega(n^G)$ where $G$ is Graham's number, ...

I am working on an efficient pattern-matching algorithm (using binary files as input) based on suffix arrays. I would like to ask you If you are familiar with any suffix array construction algorithm (...

After quantum error correction was introduced in mid '90s, in subsequent years many of the classical analogues regarding the structure of code (such as singleton bound, GV bound etc) were obtained in ...

As the title says, I want to know when the treewidth of a bipartite graph is bounded by a constant. What families of graphs are both bipartite and bounded treewidth? More generally, I would like to ...

I'm looking at the nested positivity conditions for inductive types stated in the Coq manual. First off, are there any other references (not necessarily for Coq, but in dependent type theories ...

I have a seemingly simple task that I want to solve using an online algorithm but unfortunately I wasn't able to find relevant resources even though it seems so basic. The inputs are $D=(d_1,\dots,d_m)...