4
votes
2answers
78 views

Theoretical computer science text for mathematician

I am a high school student, I know some basic programming in java,python and visual basic. I love combinatorics and I have encountered various cases in which I have found some problems are really ...
2
votes
0answers
31 views

How do you find a minimum of a function with these tools?

Let's say I can define a group $G$ acting on a set of combinatorial objects $X$ and I have a function $f: X \to \Bbb{N}$ that I want to find a minimum of in $X$. Is there a polynomial time ...
0
votes
0answers
36 views

Application of Combinatorics, Logic and computability theory in physical science: Tiling of Wang Tile with proportionality

The original problem of Domino Tiling and Wang Tile has great theoretical interest on computability theory... However, the great emerging problem on application of Wang Tile in material science and ...
1
vote
1answer
74 views

The complexity of counting solutions to $x_1 + \dots + x_m = N$ in non-negative integers under constraints

Consider the equation $$x_1 + \dots + x_m = N$$ where $x_1,\dots,x_m \ge 0$ and under the additional constraints $x_k \le a_k$ for $k=1,2,\dots,m$. I'm interested in knowing whether the number of ...
6
votes
1answer
689 views

Why is Dantzig's solution to the knapsack problem only approximate

For a bunch of items with values $v_i$ and weights $w_i$, and with a total weight $W$ that our bag can carry, how do we achieve maximum total value without breaking the bag? Dantzig proposed that we ...
0
votes
3answers
63 views

How to find a set of ascending natural numbers which when added to another set of ascending natural numbers sums to a certain number

Given: $$ X = \left\{ x_1, x_2, \ldots , x_n \right\}\text{ with }x_i \in \mathbb N\text{ and }1 \le x_i \le x_{i+1} $$ $$ z \in \mathbb N $$ Wanted result: $$ Y = \left\{ y_1, y_2, \ldots , y_n ...
2
votes
2answers
121 views

number of strictly increasing sequences of length $K$ with elements from $\{1, 2,\cdots,N\}$?

What is the number of strictly incremental sequences of length $K$ with elements from $\{1, 2,\cdots,N\}$ ? Is there any exact value? How about approximations?
0
votes
0answers
18 views

Reducing the complexity of a Combinatoric Equation

Given the equation: $$ P = \sum\limits_{n=1}^{\lfloor {\frac{q}{2}} \rfloor} {\dbinom{2n-1}{\frac{W}{2t}+n-1}\frac{1}{2^{2n-1}}} $$ Are there any algebraic tricks (or any others for that matter) ...
1
vote
1answer
98 views

Cover the n-sphere with sub-hemispherical caps

Original Question (answered): Define a cap (x,Phi) to be the set of all points of the sphere that are within an angle Phi of the point x. $ 0 \le \phi < \frac{\pi}{2} $. (define the angle ...
2
votes
3answers
54 views

Best Sum of Three Elements in a Sequence

I encountered the following problem: Given an integer sequence $\left(s_1,s_2,\dots,s_n\right)$ and an integer $l$, find $$\min\left|s_i+s_j+s_k-l\right|,$$ where $i\neq j\neq k\neq i$, and return ...
1
vote
0answers
17 views

The $k$-th term in the graded lexicographical order is recursive

I recently constructed a proof that a computable universal function exists for the class of polynomials of $n$-variables. To this end, I adopted the graded lexicographical monomial order. However, I ...
0
votes
0answers
21 views

Time complexity of combinatorial number? [duplicate]

What is the time big-oh complexity of the following combinatorial number? $$\binom{h+m-1}{m-1}.$$ where $h \gg m$. I guess that it is $O((h+m)^{m-1})$. Thank you very much.
2
votes
0answers
70 views

Number of orderings of subset sums

In short: In how many ways can all $2^n$ subset sums of $n$ real numbers $a_1,\ldots, a_n$ be ordered? I am not concerned about the case in which different subsets sum to the same number; you may ...
1
vote
3answers
306 views

Project Euler's Problem Number 88

I am tackling Project Euler's problem number 88, which in a nutshell reads: Let $S_n$ be the set of sequences of natural numbers $(s_1,s_2,...,s_n)$ where $s_1\leqslant s_2\leqslant\cdots\leqslant ...
8
votes
1answer
113 views

Finding the smallest set on which a group acts faithfully

Given a finite group $G$, how efficient can one make an algorithm to find the size of the smallest set $S$ such that $G$ is isomorphic to a group of permutations of the members of $S$? And does the ...
2
votes
1answer
128 views

Finding the shortest/“most negative” closed directed trail in a weighted digraph with negative weights

I'm using the following definition of a "closed directed trail": a closed directed trail is a directed cycle in a digraph where all edges are distinct. Note that vertices may be repeated, so long as ...
4
votes
4answers
314 views

Computing partition numbers

Today a friend and myself came up with the question of computing partitions of numbers, i.e.: given a number $n$, what is the number $p(n)$ of was of different ways writing $n$ as a sum of non-zero ...
1
vote
0answers
36 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 ...
1
vote
1answer
214 views

How to deduce the psition mapping of entries of a matrix?

I would be thankful if any peer shed light on me. Assume that the mapping of a set is unknown. By knowing n number of E element sets and the transformed sets with positioned elements, How can I ...
6
votes
2answers
1k views

Minimum distance of a binary linear code

I need to find parameters $n$, $k$ and $d$ of a binary linear code from its Generator Matrix. How can I find parameter $d$ efficiently? I know the method that compute all the codewords and take ...
3
votes
0answers
139 views

$\sum _{i=1}^{n} \sum _{j=1}^{n} \sum _{k=1}^{n}\sum _{l=1}^{n} A(i,j)A(i,k)A(i,l)A(j,k)A(j,l)A(k,l) $

I want to find an efficient algorithm for calculating a sum of products with entangled indices. For example, $\sum _{i=1}^{n} \sum _{j=1}^{n} \sum_{k=1}^{n} A(i,j)A(j,k)A(k,i)$, where A(i,j) is the a ...
2
votes
1answer
275 views

Combinatorics and bin packing

http://www2.informatik.hu-berlin.de/alkox/lehre/lvws1011/coalg/bin_packing.pdf Bin Packing: We are given a set $I=\{1,\ldots \, n\}$ of items, where items $i \in I$ has size $s_i \in (0,1]$ and set ...
0
votes
1answer
856 views

Proof that computing composition of permutations is in P

Consider the following problem: A permutation on the set ${1,…,k}$ is a one-to-one, onto function on this set. When $p$ is a permutation, $p^t$ means the composition of $p$ with itself $t$ times. ...
2
votes
1answer
283 views

approximation of binomial coefficient sum

I would like to find some approximation or upper & lower bounds on the next simple expression: \begin{align} \sum_{i = 0}^{k} \binom{h}{i} \qquad h \geq k \end{align} But I need this ...
1
vote
2answers
747 views

Is there a winning strategy for Scrabble?

I am sure many of us are addicted to the popular Facebook app: Words with Friends, which is basically an online version of Scrabble. In Playing Games with Algorithms:Algorithmic Combinatorial Game ...
0
votes
1answer
81 views

What is the time complexity of determining coefficients of generating functions?

My question is inspired by the following problem: Given $k$ coins with denominations $\{c_1, ..., c_k\}$, how many ways are there to generate $n$ cents? This can be solved in $\Theta(nk)$ time using ...
6
votes
1answer
2k views

Complexity of counting the number of triangles of a graph

The trivial approach of counting the number of triangles in a simple graph $G$ of order $n$ is to check for every triple $(x,y,z) \in {V(G)\choose 3}$ if $x,y,z$ forms a triangle. This procedure ...