All Questions
19 questions
30
votes
1
answer
3k
views
An edge partitioning problem on cubic graphs
Hello everyone,
I already asked this question on the TCS Stack Exchange, but it has not been resolved yet. Maybe readers of this forum will have other ideas or information, although I suspect that ...
- 1,164
17
votes
0
answers
449
views
Splay trees and Thompson's group $F$
( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but:
Question: Is there a reformulation of the Dynamic ...
- 992
11
votes
1
answer
860
views
Counting colored rook configurations in the cube - when is it even?
Informal Statement
In the $n\times n \times n$ grid, we can places rooks (those from chess) such that no two rooks can attack each other. One way to achieve this is to place a rook in position $(i,j,...
- 1,088
9
votes
0
answers
2k
views
Exactly Counting the Number of Lattice Points in an $n$-Dimensional Sphere
Let $S_n(R)$ denote the number of lattice points in an $n$-dimensional "sphere" with radius $R$. For clarification, I am interested in lattice points found both strictly inside the sphere, and on its ...
- 311
7
votes
1
answer
805
views
Counting Eulerian Orientation in a 4-regular undirected graph
We would like to know how hard it is to count Eulerian orientation in an undirected 4-regular graph. For a given edge orientation to be Eulerian, we mean that every vertex has 2 in-edges and 2 out-...
7
votes
1
answer
357
views
How long are the certificates produced by the Zeilberger and WZ methods for solving combinatorial sums (A=B)?
In the book "A = B" by Petkovesk, Wilf, and Zeilberger, (downloadable here), the authors provide several algorithmic methods for finding closed forms or recurrences for sums involving e.g. binomial ...
- 23k
7
votes
1
answer
304
views
"Separated" version of Sauer's lemma on VC classes
Sauer's lemma, a well-known result in computational complexity theory, learning theory, and combinatorics, states the following:
Let $\Phi$ be a collection of subsets of a set $U$, and assume that ...
- 2,207
5
votes
1
answer
276
views
NP-hardness of a sequence problem
Given $n$ binary sequences $s_i$ ($1\le i\le n$) with common period $T$. Let $s_i^{t_i}$ denote the sequence obtained by cyclically shifting $s_i$ for $t_i$ bits. The $n$ sequences form a good system ...
- 367
5
votes
2
answers
450
views
A simple language and systematic computations
The following somewhat popular simple computer language was enjoyed on sci.math, sci.math.research, pl.sci.matematyka, and perhaps before and after at several places (I wish I knew it's exact history)....
- 7,500
4
votes
1
answer
362
views
Lower bound on the number of solutions of 2SAT
To compute the number of solutions of a 2SAT is a hard problem. Is there some nontrivial lower or upper bound on this number in terms of a “coarse-grained” description of the Boolean formula, for ...
- 1,207
4
votes
0
answers
155
views
Effective "almost enumeration" of monotone boolean functions
Denote by $\mathcal{M}(n)$ the set of all monotone functions $\{0,1\}^n \to \{0,1\}$.
Can $\mathcal{M}(n)$ be represented as $\mathcal{M}(n) = \{ f(t) | t\in \{0,1\}^k \}$ such that:
1) $k = \log |\...
- 2,576
4
votes
0
answers
73
views
Is the $d$-dimensional Arrangement of Trees still $NP$-hard?
The $d$-dimensional Arrangement Problem for general graphs is known to be $NP$-hard since the special case $d=1$ (OLA) already is (Garey et al, [1976]). For Trees however, the one dimensional case can ...
- 41
3
votes
1
answer
96
views
What is known about computing all binary error correcting codes of given parameters?
Define a binary $(n, M, 2e + 1)$ code to be a code $C$ having $M$ code words in $\mathbb{F}_2^n$ whose minimum distance is $2e + 1$.
Are there any sources about using algorithms to find all given ...
- 143
3
votes
1
answer
158
views
Matroids of hypercubes
Let $M_k$ be the (oriented) matroid of the $2^k$ points $B_k = \{-1, 1\}^k$ in $\mathbb R^k$. In other words, the (oriented) circuits of $M_k$ are the minimal (signed) linear dependences among $B_k$.
...
- 459
2
votes
2
answers
249
views
Indexing schemes of binary sequences
I am looking for "low-complexity" indexing methods to enumerate binary sequences of a given length and a given weight.
Formally, let $T_k^n = \{x_1^n \in \{0,1\}^n: \sum_{i=1}^n x_i = k\}$. How to ...
- 1,839
2
votes
0
answers
245
views
Pancake sorting problem – Is computing f(n) NP-hard?
The so-called Pancake flipping problem first discussed by Jacob E. Goodman here yields two entangled problems:
MIN-SBPR (Sorting By Prefix Reversals) - Given a permutation, find the smallest sequence ...
- 51
1
vote
0
answers
157
views
Indecomposability of image transformations (pure algebra). Open questions
W-transformations -- definitions
We will consider a class called finite window transformations $\ T:C^\mathbb Z\rightarrow C^\mathbb Z\ $ defined a paragraph below; $\ \mathbb Z\ $ is the ring of ...
- 7,500
1
vote
0
answers
576
views
Minimizing quadratic form over permutations
Let $Q$ be an $n \times n$ real symmetric matrix and $x$ an $n \times 1$ real vector. Consider the following minimization problem:
$\min_{\pi \in S_n} ~(\pi x)^{\rm T} Q (\pi x)$,
where $S_n$ ...
- 1,839
0
votes
0
answers
59
views
NC0 randomness vs. non-uniformity
In
Ajtai and Ben-Or. A theorem on probabilistic constant depth
Computations. STOC '84, 1984
Ajtai and Ben-Or show a non-uniform derandomization of BPAC0.
Is there a similar relation known for ...
