All Questions

Filter by
Sorted by
Tagged with
2
votes
1answer
134 views

Maximum Positive Negative Set Cover Problem

I am considering the following problem. Input: Given two disjoint subsets $A$ and $B$ and a collection $C$ of $k$ sets $S_1,S_2,\ldots,S_k$ where $S_i \subseteq A \cup B$ for all $i=1\ldots k$. ...
4
votes
0answers
99 views

Expected vs actual amount of information leaked by an $l$-bits message

Say we have a random variable $X$ that contains $k$ bits of information, and a message $M = f(X)$ ($M$ is deterministic given $X$) that is $l$ bits long, where $l<k$. This implies $H(X) = k$ and $...
2
votes
1answer
121 views

Lower bound on the worst-case unbiased coin flips to sample a distribution?

Say that we have a distribution $\mathcal{D}$ such that all probabitilities associated with it are $p$-bit fixed precision numbers, so: $$ \Pr_{X\sim \mathcal{D}}[X = k] =\sum_{i = 1}^p \frac{k_i}{2^i}...
3
votes
0answers
43 views

Extending the sequential calculus (logic over words) to allow a hierarchy of languages like the arithmetical hierarchy

Let $\Sigma$ be some finite alphabet. Then consider the logical language $\mathcal L = \{ R_a : a \in \Sigma \} \cup \{ <,= \}$ and first order formulas. For a given first order formula $\varphi$ a ...
-1
votes
1answer
46 views

Min Cut with Vertices

I have an undirected graph G with a set of vertices and edges. Each vertex has a weight w. Let's assume we have all vertices connected with some paths. I'm looking ...
0
votes
0answers
89 views

How much computer science / software experience is needed for a math major to do work in AI?

As a math major who is planning on getting their PhD in mathematics, how much computer science or software engineering experience/education do I need to comfortably do research in AI, i.e., would a ...
1
vote
1answer
58 views

QPIP minimal client quantum capabilities

It is conjectured that classical (BPP) client blind quantum computing is implausible according to Aaronson et al: https://www.researchgate.net/publication/...
3
votes
0answers
46 views

Hardness of Approximation of Set Cover with Growing Size Bound

I'm considering the minimum set cover problem with the constraint that each set contains at most $k$ elements. Here, $k$ depends on the size of the universe. For example, $k$ may equal $\log n,\sqrt ...
2
votes
1answer
97 views

Verified type checkers

Most of the work on programming language metatheory mechanization focus on the declarative properties of the languages (e.g., type soundness), but fail to address the algorithmic side, i.e. the type ...
1
vote
1answer
68 views

Embedding a n-tree into a b-dimensional space

Given a (directed) n-tree $T=(N,E,r)$ rooted in $r\in N$, I want to represent each node $n\in N$ at most as a $m$-dimensional vector $v_n\in \mathbb{R}^m$ (From the current Yuri's reply, m cannot be $...
7
votes
1answer
72 views

Simple proof that splay trees have the dynamic finger property?

Splay trees are conjectured to be dynamically optimal, and they're known to have a number of nice properties, including the dynamic finger property, which says that the amortized cost of an access in ...
1
vote
0answers
73 views

Agnostic query learning of decision trees

Gopalan, Kalai, Klivans gave an algorithm https://dl.acm.org/citation.cfm?id=1374376.1374451 for agnostically learning decision trees $h:\{0,1\}^n\to\{0,1\}$ under the uniform distribution given ...
1
vote
0answers
61 views

Straight line programs with $\sqrt{~}$

A straight line program with division is very powerful https://rjlipton.wordpress.com/2012/10/16/one-mans-floor-is-another-mans-ceiling/. Is it possible to reduce a straight line program with $\sqrt{~...
7
votes
1answer
291 views

Planar Exact Cover by even-size sets

Major edit on June 6, 2019: Replaced the target problem with a simpler (but equivalent) one. Is the following problem NP-complete? Planar Exact Cover by even-size sets Input: A set $U$, a ...
5
votes
0answers
68 views

Complexity of a specific class of definite integrals

INTRODUCTION: From the answer to this question I learned that deciding whether a definite integral is $0$ or not can be NP-complete, as the following integral representation of the Number Partition ...
2
votes
0answers
86 views

A question about UE

Much has been written about the class UP see related (even more in literature) example question here. Much is understood about the class UP, and its place in collapsing the PH too. UP has a played ...
2
votes
1answer
105 views

Difference between statically and dynamically typed languages

When writing a course on computer science where students get an introduction to both Python and OCaml, I was on the verge of saying that Python is dynamically typed and OCaml is statically typed. I ...
6
votes
2answers
224 views

Razborov-Smolensky polynomial argument on $\textrm{ACC}[q]$ where $q$ is a prime power

It seems to be a folklore that we can "handle" $\textrm{ACC}[q]$ circuits not only for prime $q$ but prime power $q$. For example, authors of this paper say that ... any constant depth circuit ...
3
votes
1answer
127 views

Reductions in Descriptive Complexity

Reducing one problem to another are well known in various settings, such as many-one, randomized, truth-table, logspace or a whole slew of other reductions. Descriptive complexity can alternately ...
1
vote
0answers
56 views

Graph automorphism with prescribed values

Consider a graph $G$ with vertices labeled $1,...,n$ and edge weights $w_{ij}$. Recall an automorphism of G is a permutation $\sigma$ of the vertex labels such that $w_{\sigma(i),\sigma(j)}=w_{ij}$ ...
10
votes
0answers
115 views

Diameter of “almost” always connected Erdős-Renyi graphs

Let $G=(V,E)$ be a random Erdős-Renyi Graph, i.e., $G\in\mathcal{G}(n,p)$. It is well known that if $p=(\log n +c +o(1))/n$ with $c\in\Re$ then $$ P(G \text{ is connected})=e^{-e^{-c}}\ . $$ However, ...
1
vote
0answers
158 views

On planar $4$ regular graphs

It is $NP$-hard to decide if a $4$-regular planar graph can be $3$-colored. Is an exact algorithm possible that under uniform distribution is in average polynomial time?
4
votes
2answers
798 views

Attempted proofs of P vs NP

What are the most recent (say in the last 3 years) attempts at disproving $P = NP$, and where can I find the papers?
1
vote
1answer
94 views

How many samples are needed to reconstruct a path?

Consider an input set of vertices $V$ and vertices $s,t\in V$. The goal is to learn some unknown shortest path from $s$ to $t$; the set of edges of the graph is hidden at first and there may be ...
3
votes
1answer
72 views

Agnostic query learning for DFAs

Angluin's membership+equivalence query algorithm allows to efficiently and exactly learn a target $n$-state DFA. But what if the target DFA is huge, or the target concept is not even a regular ...
2
votes
0answers
31 views

A Context-Sensitive Grammar which cannot be recognised by a Parsing Expression Grammar

It is (currently) an open question of whether every context-free grammar can be recognised by some parsing expression grammar. [1] However, has it been proven that there exists an example of a ...
1
vote
0answers
166 views

Can a hash preimage be used to amplify BPP probabilities?

Suppose we are given a (univariate) polynomial $P$ of degree $d$, and we wish to determine if $P$ is identically $0$. A standard way to do this is to use a classical PRG to randomly sample a number $...
2
votes
1answer
144 views

Is the unbounded fan-in model realistic?

Does the unbounded fan-in circuit model apply in "practical" settings? In other words, are there real-world realisable computers with unbounded fan-in gates? As I understand, standard silicon ASICs ...
5
votes
0answers
50 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\...
4
votes
0answers
89 views

Counting matchings on 3-regular bipartite graphs

What I call a graph here allows parallel edges. Is the following problem #P-hard: INPUT: a 3-regular bipartite graph $G$ OUTPUT: the number of matchings of $G$. It is known that counting matchings ...
0
votes
0answers
7 views

Extension complexity of convex hull of vertex intersection of nicely behaved polytopes?

Take two convex bounded polytopes $P_1$ and $P_2$ where a. $P_2\subseteq P_1$ b. $\mathcal{V}(P_1)\cap\mathcal{V}(P_2)\neq\emptyset$ where $\mathcal{V}(P_i)$ is vertex set of $P_i$ at $i\in\{1,2\}$. ...
-1
votes
1answer
85 views

Comprehensive list of functions used in Big-$O$ notation

We all know that exponential functions grow faster than polynomials. Let us consider the following function: $f(n)=n^{a_1}⋅(\log n)^{a_2}⋅(\log\log n)^{a_3}⋅(\log\log\log n)^{a_4}⋯ $ where the leading ...
19
votes
3answers
2k views

Why colon to denote that a value belongs to a type?

Pierce (2002) introduces the typing relation on page 92 by writing: The typing relation for arithmetic expressions, written "t : T", is defined by a set of inference rules assigning types to ...
1
vote
1answer
64 views

How to play the following game? (placing balls into bins)

Let $n,\ell\in\mathbb N$ for some $n\gg \ell\gg 1$. The goal is to pick two sequences of numbers, $x_1,\ldots,x_\ell$ and $y_1,\ldots,y_\ell$ such that $$\Sigma_{i=1}^\ell x_i = n\quad{}\mbox{and}\...
2
votes
2answers
198 views

Under what models do we know linear time sorting?

The best we know for general case sorting is $O(n\log n)$ (which is also $\theta(n\log n)$ is decision tree model) and the problem of $O(n)$ sorting is open for turing machine models. Under what ...
0
votes
0answers
33 views

2D-Interval partition problem

The classical interval partition Problem ascs for a minimal colouring of an interval graph: Let [a_i, b_i] be a collection of (closed) intervals (for i in {1,2,...,n} ). Find a partition of {1,2,...,n}...
2
votes
0answers
58 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,...
0
votes
0answers
22 views

Does simplex algorithm work in this case?

Essentially I want to know if we have two polytopes $P_1$ and $P_2$ in $\mathbb R^n$ and we maximize on $P_1\cap P_2$ where a. $P_2\subseteq P_1$ b. $\mathcal{V}(P_2)\cap\mathcal{V}(P_1)\neq\...
1
vote
0answers
48 views

Mapping of entire balls using Locality Sensitive Hashing (LSH)

LSH functions are useful for approximate nearest neighbor search. They are usually defined, for distance metric $d$ and $c>1$ as follows: A family of hash functions is $(r, cr, p_1, p_2)$-LSH ...
1
vote
0answers
86 views

Does simplex algorithm run in polynomial on Bipartite Perfect matching polytope?

It is well known that simplex algorithm runs in exponential time in worst case. However are there situations (necessary and sufficient conditions) where simplex algorithm runs in polynomial time? In ...
20
votes
1answer
416 views

Is prime-counting function #P-complete?

Recall $\pi(n)$ the number of primes $\le n$ is the prime-counting function. By "PRIMES in P", computing $\pi(n)$ is in #P. Is the problem #P-complete? Or, perhaps, there is a complexity reason to ...
2
votes
0answers
58 views

$XP_{\text{uniform}}=FPT$ and update to $EPTAS$ section in complexity zoo?

Complexity zoo in https://complexityzoo.uwaterloo.ca/Complexity_Zoo:E#eptas has the following: $FPT = XPuniform\implies EPTAS = PTAS$. Fundamentals of Parametrized complexity on page $534$ has ...
7
votes
1answer
147 views

Sampling monotone Boolean functions

I'm interested in sampling monotone increasing Boolean functions on $n$ input bits uniformly at random. I understand that this is equivalent to approximating the Dedekind numbers ($D_n = $ the number ...
0
votes
0answers
19 views

The set of weight functions for which the assignment problem has non-trivial solutions

The standard assignment problem is specified with a square matrix ${\bf W}$ of weights (values, costs): $$ V_{\cal P} = \sum_i w(i, b(i)) = \sum_{(i, j) \in {\cal P}} w_{ij}, $$ where $\cal P$ is a ...
2
votes
0answers
133 views

Is the following problem in $coNP$?

Given an $n\times n$ matrix $M$ with $\mathbb Z$ entries is 'does an $\frac n2\times\frac n2$ minor of $M$ vanish?' in $\bf{coNP}$? At least one $\frac n2\times\frac n2$ minor non-vanish implies rank ...
0
votes
0answers
46 views

What is the complexity of Parametric Mixed Integer Linear Programming?

We know $$\forall\bf y\in\mathbb Z^n:K\bf y\leq b$$ $$\exists\bf x\in\mathbb Z^m:A\bf x + B\bf y\leq c$$ is in $\bf P$ if $n,m$ are fixed from Kannan's result (refer page $1$ in reference). What is ...
1
vote
0answers
35 views

TSP variant in which edge costs depend on the already visited vertices

Does a TSP variant exist in which edge costs depend on the vertices already visited? For instance, if you already visited vertices A, B, and then C, in that order, then now the cost to traverse CD = 5,...
0
votes
1answer
33 views

Graph path problem [duplicate]

I am trying to solve one graph traversing problem which might be classical to guys who are familiar with the topic. However, I am not. I have directed graph where nodes are cities and plane can fly ...
5
votes
1answer
65 views

NP-intermediate approximation regimes for natural problems within the MAX-k-CSP family

I would like to know whether there are any examples of natural problems within the MAX-$k$-CSP family for which (under standard/reasonable conjectures) we believe the following: There is a value $\...
2
votes
1answer
193 views

How far has computer science moved past Knuth's TAoCP, if at all? [closed]

The seminal book The Art of Computer Programming got its start in 1968. I have been finding references to it in many literature reviews, apparently there are many problems for which a review by Knuth ...

15 30 50 per page