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6
votes
1answer
143 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
68 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
66 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 ...
4
votes
0answers
118 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
votes
0answers
124 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 ...
1
vote
0answers
59 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 ...
2
votes
1answer
171 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
votes
1answer
116 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). ...
1
vote
0answers
112 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
231 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
votes
2answers
190 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 = ...
0
votes
0answers
222 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 ...
7
votes
2answers
342 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
144 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
votes
0answers
103 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
votes
2answers
322 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 ...
7
votes
1answer
82 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
153 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 ...
5
votes
0answers
322 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
votes
2answers
930 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 ...
8
votes
0answers
445 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
votes
0answers
182 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 ...
10
votes
2answers
616 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
votes
0answers
383 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
votes
1answer
280 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 ...
1
vote
0answers
299 views

Succinct representation of boolean functions

Let $f$ be a boolean function over $n$ variables $f: \{ 0, 1 \}^n \rightarrow \{ 0, 1 \}$. We are looking now for a representation of $f$ s.t. when given that representation and values $x_1, \ldots, ...
5
votes
2answers
297 views

Number of edges in $K_4$-free graphs

What can be the upper bound on the number of edges in a graph of $n$ vertices such that the graph does not have $K_4$ as a minor? Is there some relevant paper/book that I can look into it or it would ...
13
votes
1answer
705 views

What are the limits of computation in this universe?

I understand that Turing completeness requires unbounded memory and unbounded time. However there is a finite amount of atoms in this service thus making memory bounded. For example even though ...
1
vote
0answers
181 views

Upper bound for set cover with respect to m that is better than trivial when $n \ge 3m$

Does anyone know of an upper bound for Set Cover $(\mathcal{U}, \mathcal{S}, k)$ with respect to $m=|\mathcal{S}|$ that is better than trivial when $n =|\mathcal{U}|$ is at least $3m$? (Set cover). ...
5
votes
1answer
454 views

What is the running time of taking a limit?

I'm interested in finding the running time(s) for determining mathematical limits. For instance, $\lim_{x \to 2} \frac{1}{x} = \frac{1}{2}$. I'd like to know more about algorithms for determining ...
36
votes
3answers
1k views

A fixed-depth characterization of $TC^0$? $NC^1$?

This is a question about circuit complexity. (Definitions are at the bottom.) Yao and Beigel-Tarui showed that every $ACC^0$ circuit family of size $s$ has an equivalent circuit family of size ...
5
votes
1answer
308 views

What are the current best known upper and lower bounds on the (un)satisfiability threshold for random k-sat and/or 3-sat?

I would like to know the current state of the phase transition for random k-sat, given n variables and m clauses, what is the best known c=m/n for upper and lower bounds.
29
votes
7answers
3k views

Best Upper Bounds on SAT

In another thread, Joe Fitzsimons asked about "the best current lower bounds on 3SAT." I'd like to go the other way: What's the best current upper bounds on 3SAT? In other words, what is the time ...
6
votes
4answers
674 views

What are the best known upper bounds and lower bounds for computing O(log n)-Clique?

Input: a graph with n nodes, Output: A clique of size $O(\log n)$, Providing links to references would be great
14
votes
2answers
516 views

How large a treewidth can a tree plus half the edges have?

Let G be a tree on 2n vertices. The treewidth of G, tw(G) = 1. Now suppose we add n edges to G to get a graph H. An easy upper bound on tw(H) is n + 1. Is this essentially the best possible? It ...