Questions tagged [upper-bounds]
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61
questions
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36 views
Expected value in case of Langevin dynamics
Let $U(\beta) = \exp(-f(\beta))$ where $f$ is $m$ strongly convex and $L$ Lipschitz smooth. I am trying to find an upper bound on $E[\|\beta_t \|^2]$ where $\beta_t \in \mathcal{R}^p$ is defined below
...
0
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0answers
31 views
Is this a proof that diophantine equation solutions can't be bounded by power towers?
From this 2017 paper on upper bounds for solutions to diophantine equations:
Conjecture 1. If a system of equations S ⊆ Bn has exactly one solution
in positive integers x1, . . . , xn , then x1, ....
4
votes
0answers
81 views
Strong data-processing inequality: bound $TV(T_{\#}P_0,T_{\#}P_1)$ if $\|T(x)-x\|_\infty \le \varepsilon;\forall x \in \mathbb R^p$
Disclaimer. I've moved this question from MO hoping that here is the right venue. Also, this is my first post on this channel, so please have some patience.
So, Iet $X = (X,d)$ be a Polish space, ...
6
votes
1answer
234 views
Big-O bounds on the k-th largest element of iid Gaussians
I'm interested in the following problem. Let $X_1, \dots, X_n$ be iid samples with a $N(0,1)$ distribution. Let $X_{[k]}$ be the $k$-th largest element of $\{X_1, \dots, X_n\}$, so e.g. $X_{[1]} = \...
-1
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1answer
60 views
How is additive error handled in this simple algorithm? 'Product of all elements'
Say we have two unit vectors $\hat{u}, \hat{v} \in \mathbb{R}^n$ where $\hat{u} = (u_1,...,u_n)$ and $\hat{v}$ approximates $\hat{u}$. $~\hat{v} = (u_1+\epsilon, ...,u_n+\epsilon)$ where $\epsilon = \...
2
votes
1answer
102 views
References on generalization bounds
I'm looking for references (books, papers, lecture notes etc) on generalization bounds and their proofs. Specifically, I'm looking to fully understand the technique of defining a hypothesis class (or ...
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0answers
43 views
Optimally fair stable matching
There's a nice post by Gil Kalai which outlines the inherent bias in stable matching algorithms quantitatively.
In the traditional loyd shapeley algorithm for $n$ men and $n$ women, given randomly ...
-1
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1answer
109 views
Finding upper and lower bounds of a problem [closed]
We have n balls where 1 is a little heavier than the others and we want to find that heavier ball. We can only put some balls on one side of the scale and some on the other side and see if it leans ...
3
votes
0answers
151 views
Fast way of getting a matrix of sums
We are given an array of variables $A$, along with a matrix $M$. The elements of the matrix $M$ are composed of sums of the variables in $A$. We are allowed to pre-process $A$ in order to find a ...
2
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0answers
80 views
Explicit Formula of Delsarte's Linear Programming Upper Bound for $A_q(n,3)$
The problem of giving an explicit formula for $A_q(n,d)$ is sometimes referred to as "the main problem in coding theory." The value of $A_q(n,d)$ is given by the maximum number of codewords in a q-ary ...
14
votes
2answers
495 views
OR-circuit complexity of a dense linear operator
Consider the following simple monotone circuit model: each gate is just a binary OR. What is the complexity of a function $f(x)=Ax$ where $A$ is a Boolean $n \times n$ matrix with $O(n)$ 0's? Can it ...
1
vote
1answer
646 views
Common terminology used for lower/upper bounds
Suppose you have developed an upper bound on the number of vertices of a particular graph. This bound is the best possible bound that can be found for any given instance. What do you call such a bound?...
6
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1answer
397 views
Do we know a specific $L_{ZFC}$ such that $K(s) \ge L_{ZFC}$ is unprovable in ZFC for all strings $s$?
Chaitin's incompleteness theorem states for any formal system $F$ (which satisfies various criteria), there is a $L$ such that for any $s$ the statement $$K(s) \ge L_F$$ is unprovable in that formal ...
13
votes
2answers
840 views
What is the proof of this nonstandard version of Azuma's inequality?
In Appendix B of Boosting and Differential Privacy by Dwork et al., the authors state the following result without proof and refer to it as Azuma's inequality:
Let $C_1, \dots, C_k$ be real-valued ...
2
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0answers
1k views
Is there a better than brute-force solution to the shortest simple path problem?
Given as input graph which can possibly contain negative weight cycles, we can still ask for the weight of the shortest simple path between two vertices (i.e., a path that does not visit any vertex ...
2
votes
1answer
118 views
Upper bound on the size of a Concept Lattice (Galois Lattice)?
A context is a tuple $(O, A, R)$ where $O$ is the set of objects, $A$ the set of attributes and $R \subseteq O\times A$ is a relation. For $o \in O$ and $a \in A$ we read $oRa$ as the object $o$ ...
2
votes
1answer
161 views
Probability of random variable $X$ less than $max(Y_i)$
For $\lambda > \mu$, consider independent random variables (i) $X \sim Pois(\mu)$ and (ii) $Y_i \sim Pois(\lambda),\;i\in{1,2}$.
Question : Can we upper-bound the following?
$$\mathbb{P}\big(X\...
8
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0answers
345 views
Upper Bound on Number of $n \times n$ Boolean matrices of Boolean rank at most $k$
An $n \times n$ Boolean matrix $B$ has Boolean rank $k$ if there exist matrices $L \in \{0,1\}^{n \times k}$ and $R \in \{0,1\}^{k \times n}$, s.t. $B = L \circ R$. Here $\circ$ denotes the Boolean ...
2
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0answers
195 views
Circuit complexity lower bounds and uniformity
I have troubles to understand how lower bounds w.r.t. circuit complexity and upper bounds w.r.t. uniform machine models can be used to show completeness results.
For example, the word problem for ...
8
votes
1answer
1k views
Probability of two vertices being connected by some path in a random directed graph
Define $G(n, p)$ as a random directed graph ($n$ vertices; we put edge between two vertices with probability $p$).
What are the known results for the following problem:
Fix two vertices $v$ and $u$...
6
votes
1answer
244 views
Best SAT upper bounds based on number of clauses
What's the best upper bounds based on number of clauses? In this question shown fastest algorithms for SAT, but there bounds depends from number of variables ( $O(const^n)$ where n is number of ...
3
votes
2answers
244 views
computing maximal bit density over a FSM
let $L$ be a regular language defined by a FSM over binary symbols $\{0,1\}$. consider a function $f(x)$ on words/ strings that computes "bit density", defined as the number of $1$'s in a word ("...
7
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0answers
146 views
What's the upper bounds for #3-SAT circuits?
We have, from this thread on 3-SAT upper bounds, and this answer on #P that the current best upper bounds for 3-SAT is faster than $O(1.31)^n$, and approximately $O(1.64^n)$ for #3-SAT.
Can we do ...
3
votes
1answer
143 views
Number of different longest common substrings
Given an alphabet $\Sigma$ of size $k$ and two strings $w_1,w_2\in \Sigma^n$ of length $n$. The longest common substring problem asks for a longest string in the set $A(w_1,w_2)$ of all common ...
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2answers
665 views
Upper bound for number of independent sets
What is the tightest upper bound known for the number of independent sets in a graph?
6
votes
1answer
176 views
Bounds on the size of NFA for $r$-skip $k$-distinct language
This question is about an extension of a language discussed in this question.
We define the $r$-skip $k$-distinct language as follows:
$$L_{r,k}=\{\sigma_1\sigma_2\cdots \sigma_{rk}\in\Sigma^{rk} | \...
2
votes
1answer
107 views
Upper bound to number of closed itemsets
Given a set $I$ of $n$ items, and a collection $D$ of $m<2^n$ subsets of $I$, a closed itemset is a subset $A$ of $I$ that is contained in strictly more elements of $D$ than any of its proper ...
0
votes
1answer
122 views
Number of $k$-cuts of grid graphs
Given a $n\times m$ grid, let the bottom-left vertex be $s$ and the top-right vertex be $t$.
Given $k$ non-consecutive edges on the upper horizontal line of the grid, I want to find an upper bound on ...
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0answers
126 views
Deciding transitivity of a directed acyclic graph [duplicate]
Is there any algorithm that decides whether a given directed acyclic graph is transitive or not, in time-complexity asymptotically better than boolean matrix multiplication?
2
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0answers
1k views
Maximum number of triangles in a constrained delaunay triangulation
I'm looking for an upper bound for the number of triangles in a constrained planar delaunay triangulation. I know for d=2 delaunay triangulation, there are at most n+1 triangles where n is the number ...
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0answers
66 views
Bound the number of rounds in the sampling
Suppose we have a sequence $a_1,a_2,\ldots, a_n$, each $a_i$ is sampled uniformly and independently from $[0,1]$.
Define
$$
J_1=1,\\
\text{for}~i>1, ~J_i = 1 \iff a_i < \min \{a_1,a_2,...
2
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1answer
686 views
Subset Sum bounds
Are there any bounds for the subset problem with respect to the number of the terms involved in the sum and the range of the possible values?For example looking for some 2 terms whose sum equals 3 you ...
0
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1answer
128 views
Counting subsets with large sum
Suppose that you have a multiset of positive integers $I$.
$I$ is not given, but it is known that the sum over all elements of $I$ = $k$.
(e.g. if $I$={2,5,7} then k=14 is given, but I is unknown).
...
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0answers
290 views
Bin packing upper bound: total size of items = k, bin size = r
Suppose you have items, whose total size (i.e. sum of sizes) is $k$.
The number of items and their individual sizes are unknown integers.
We need to pack the items into bins of size $r$.
I need to ...
10
votes
2answers
1k views
Is Almost-2-SAT NP-hard?
Is a CNF SAT problem NP hard when the total number (but not the width) of the 3-or-more-term clauses is bounded above by a constant? What about specifically when there's only one such clause?
9
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2answers
226 views
Exact complexity of a problem in $\cap_{m \geq 2}\mathsf{AC}^0[m]$
Let $x_i \in \{-1,0,+1\}$ for $i \in \{1,\ldots,n\}$, with the promise that $x = \sum_{i=1}^n{x_i} \in \{0,1\}$ (where the sum is over $\mathbb{Z}$). Then what is the complexity of determining if $x = ...
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0answers
816 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(...
7
votes
2answers
417 views
What are the current best upper bounds of #P?
#P is the class of counting problems for problems in NP. In other words, a solution to #P returns the number of solutions to a particular problem in NP.
I'm wondering if there have been any studies ...
3
votes
1answer
169 views
Upperbound on cardinality of product of two string sets at pairwise Hamming distance $> 1$
I am considering products $U\times V$ of subsets $U, V\subset \{0, 1\}^p$ with a pairwise Hamming distance greater than 1 : $\forall uv\in U\times V, D(u,v) \geq 2$.
Given $p$, I am looking for a ...
2
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0answers
114 views
What upper bound can we get under 3-wise independence? (comparable edition)
Here is the original question: What bound can we get using $k$-th moment inequality under 3-wise independence? .Yury has given a 3-wise independent example that shows the upper bound is no better than ...
7
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2answers
402 views
What bound can we get using $k$-th moment inequality under 3-wise independence?
Let $X_1,\dots, X_n$ be 0-1 random variables, which are $3$-wise independent. We want to give a upper bound to $\Pr(|\Sigma_iX_i-\mu|\geq t)$. Can we get better bound than $\Theta\left(\frac{1}t\right)...
7
votes
1answer
103 views
How quickly can we perform base extension in a residue number system?
Thinking about residue number systems, one major operation is to extend the set of primes that a given value is modulated by, also known as base extension. For instance, a given number $N$ can be ...
6
votes
1answer
223 views
Best upper bound on rate for q-ary codes
Among the many upper bounds for families of codes in $\mathbb F _2 ^n$, the best known bound is the one by McEliece, Rodemich, Rumsey and Welch (derived through a linear programming relaxation ...
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0answers
370 views
Strongly edge-guarding a 3d triangulation
Let $T$ be a planar triangulation. It is known that one can guard the faces of $T$ using at most $\lfloor n/3 \rfloor$ edge-guards (Worst-case-optimal algorithms for guarding planar graphs and ...
21
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2answers
1k views
Can addition be carried out in less than depth 5?
Using carry look ahead algorithm we can compute addition using a polynomial size depth 5 (or 4?) $AC^0$ circuit family. Is it possible to reduce the depth? Can we compute the addition of two binary ...
20
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0answers
1k views
What's the expressive power of Simply Typed Lambda calculus?
The standard approach to simply typed lambda calculus considers computations over Church numerals.
If input and outputs are Church numerals always typed as $Int$, where $Int = (\tau \rightarrow \tau) ...
14
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0answers
209 views
The best known upper bound for two-way probabilistic finite automata with one-counter
It is known that the class of languages recognized by two-way deterministic finite automata with one-counter (2D1CAs) is a proper subset of $ \mathsf{L} $ (deterministic log-space):
A 2D1CA can run at ...
11
votes
2answers
770 views
Are quasi-polynomial sized circuits for 3-SAT trivial?
Suppose we consider 3-SAT with $v$ variables and $c$ clauses. I am researching a method that appears to take $O(v^{2+\log c})$ time/space to solve any SAT problem fitting this description, to within ...
12
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0answers
609 views
Known upper bounds on the communication complexity of Karchmer-Wigderson games
In 1988 Karchmer and Wigderson established a nice characterization of the circuit
depth $d$ (DeMorgan circuits) of a Boolean function $f \colon \{0,1\}^n\rightarrow\{0,1\}$: $d$ is exactly the number ...
-1
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1answer
615 views
Upper bound of an optimization problem
Please let me know whether there are closed-form optimal results (or upper bound) for the following optimization problem:
$$\max (\prod_{1\leq i\leq n}(x_i)^{y_i}-\prod_{1\leq i\leq n}(x_i-\alpha)^{...
