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Questions tagged [upper-bounds]

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Upper bound on the number of maximal paths in rooted intransitive DAGs

Let $D(V, A)$ be a DAG. Definition 0: We name a path between two nodes $i$ and $j$ as an $i$-$j$-path. Definition 1: Let $p$ be a path, we call $|p|$ the path length, representing the number of arcs ...
Matheus Diógenes Andrade's user avatar
0 votes
1 answer
124 views

Confusion about lower bounds and upper bounds in learning theory

In computer science, lower bounds and upper bounds are defined as follow: $$m \geq g(n) \implies m = \Omega(g(n))$$ $$m \leq g(n) \implies m = \mathcal{O}(g(n))$$ However, in proving lower bounds and ...
rivana's user avatar
  • 65
3 votes
1 answer
120 views

Is there an established name for this kind of upper bound?

Assume for some algorithmic problem it holds that, for each $\epsilon>0$, there is some algorithm that needs space at most $O(n^\epsilon)$. Is there an established name for this kind of bound? I'd ...
Thomas S's user avatar
  • 1,417
3 votes
0 answers
29 views

Bound on line with minimum zone complexity in a line arrangement

In an arrangement of $n$ (pseudo)lines, the well known Zone Theorem gives a $O(n)$ bound on the complexity of the zone of any given line (for the purpose of this question, the complexity of the zone ...
Tassle's user avatar
  • 881
-2 votes
1 answer
641 views

Solving 3-SAT in O(n^6)?

There's an algorithm (published on GitHub) which is claimed to solve any 3-SAT formulation in polynomial time with a complexity of max O(n^6). I would usually brush claims like this away, but having ...
DanielM's user avatar
  • 73
1 vote
1 answer
105 views

Which is the most efficient of the two following approximation algorithms?

Let $\mathfrak{B}$ be a set, which will be called the set of bins. Suppose we have five maps \begin{align*} \mathrm{Value} &: \mathfrak{B} \to \mathbb{R} \\ \mathrm{Upper} &: \mathfrak{B}...
Bob's user avatar
  • 111
1 vote
0 answers
50 views

What is known about simultaneous protocol set disjointness?

Assume that Alice and Bob have sets $A,B\subseteq[n]$ of size $|A|=|B|=k$. In the simultaneous protocol, they both send a message to Carol (that doesn't observe $A$ and $B$) which needs to determine ...
John's user avatar
  • 11
2 votes
1 answer
154 views

Is the center of a BFS tree a good approximation of the graphs center?

Given a graph $G=(V,E)$, a center is a vertex $v\in V$ with minimal eccentricity (i.e., $v\in\text{argmin}_v\max_u d(u,v)$). Finding the center of the graph can easily be done using all-pairs-shortest-...
Eli's user avatar
  • 21
2 votes
1 answer
72 views

Upper Bound for distance-two chromatic number in terms of maximum degree

Let us consider simple,finite, undirected graphs. A distance-two colouring of a graph $G$ is a function $f:V(G)\to\{1,2,\dots\}$ such that $f(u)\neq f(v)$ whenever $dist_G(u,v)\leq 2$. A distance-two ...
Cyriac Antony's user avatar
7 votes
2 answers
288 views

Is there an Upper Bound on Number of Redundant Clauses in a satisfiable $3-SAT$?

For a non-empty $3-SAT$ with $n\geq3$ variables and $T\geq1$ non-identical non-degenerate clauses $C_i$: $$S=C_1 \wedge \ldots \wedge C_T$$ where a non-degenerate clause is one containing $3$ unique ...
Craig's user avatar
  • 181
5 votes
1 answer
241 views

Upper bound on the expected number of correct bits via a "lossy compression"

Consider the following "compression problem" for a pair $(C,D)$ of algorithms: $C$ receives a uniformly random $x \in \{0,1\}^n$ and outputs a smaller bit string $y \in \{0,1\}^s$. Algorithm ...
Marcel Dall'Agnol's user avatar
0 votes
0 answers
45 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, ....
ghosts_in_the_code's user avatar
4 votes
0 answers
110 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, ...
dohmatob's user avatar
  • 291
6 votes
1 answer
392 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]} = \...
Uthsav Chitra's user avatar
-1 votes
1 answer
90 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 = \...
stats_man97531's user avatar
2 votes
1 answer
120 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 ...
LYH's user avatar
  • 63
1 vote
0 answers
47 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 ...
Sidharth Ghoshal's user avatar
-1 votes
1 answer
121 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 ...
Yhprums's user avatar
  • 111
3 votes
0 answers
223 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 ...
Matt Groff's user avatar
  • 2,100
2 votes
0 answers
121 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 ...
Max Hopkins's user avatar
14 votes
2 answers
530 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 ...
Alexander S. Kulikov's user avatar
1 vote
1 answer
787 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?...
rasul's user avatar
  • 111
6 votes
1 answer
455 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 ...
Christopher King's user avatar
13 votes
2 answers
962 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 ...
William Hoza's user avatar
  • 1,743
2 votes
0 answers
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 ...
Benno's user avatar
  • 121
2 votes
1 answer
124 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 $o$ ...
Luz's user avatar
  • 427
2 votes
1 answer
190 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\...
Vivek Bagaria's user avatar
8 votes
0 answers
365 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 ...
tranisstor's user avatar
2 votes
0 answers
220 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 ...
Veto's user avatar
  • 31
8 votes
1 answer
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 $u$...
Daniel's user avatar
  • 749
6 votes
1 answer
314 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 ...
Alexey Kholodkov's user avatar
3 votes
2 answers
249 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 ("...
vzn's user avatar
  • 11k
7 votes
0 answers
153 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 ...
Matt Groff's user avatar
  • 2,100
3 votes
1 answer
156 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 ...
Danny's user avatar
  • 500
1 vote
2 answers
830 views

Upper bound for number of independent sets

What is the tightest upper bound known for the number of independent sets in a graph?
chtenb's user avatar
  • 155
6 votes
1 answer
191 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} | \...
R B's user avatar
  • 9,458
2 votes
1 answer
115 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 ...
Matteo's user avatar
  • 569
0 votes
1 answer
137 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 ...
user22350's user avatar
4 votes
0 answers
130 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?
ekayaaslan's user avatar
2 votes
0 answers
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 ...
zaloo's user avatar
  • 393
1 vote
0 answers
68 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,...
xmerge's user avatar
  • 13
3 votes
1 answer
760 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 ...
curious's user avatar
  • 173
0 votes
1 answer
140 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). ...
Ran B's user avatar
  • 31
1 vote
0 answers
340 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 ...
Ran B's user avatar
  • 31
10 votes
2 answers
2k 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?
dspyz's user avatar
  • 916
9 votes
2 answers
230 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 = ...
SamiD's user avatar
  • 2,309
1 vote
0 answers
961 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(...
user1742364's user avatar
7 votes
2 answers
456 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 ...
Matt Groff's user avatar
  • 2,100
3 votes
1 answer
188 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 ...
babou's user avatar
  • 1,542
2 votes
0 answers
125 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 ...
Amos's user avatar
  • 201