Questions tagged [combinatorics]
Questions related to combinatorics and discrete mathematical structures
1
vote
1answer
53 views
Counting models satisfying a boolean formula
I'm trying to implement the #2-SAT algorithm from the paper "Counting Satisfying Assignments in 2-SAT and 3-SAT" (Dahllöf, Jonsson and Wahlström, Theor. Comput. Sci. 332(1–3):265–...
8
votes
2answers
134 views
Find an optimal ordering
I came across this problem and am struggling to find a way to approach it. Any thoughts would be greatly appreciated!
Suppose we are given a matrix $\{-1, 0, 1\}^{n\ \times\ k} $, for example,
...
0
votes
1answer
29 views
Number of possible min heaps
The number of possible min-heaps containing each value from {1, 2, 3,
4, 5, 6, 7} exactly once is --------------
According to me, the answer should be 48. The first element 1 is fixed as root. The ...
1
vote
1answer
20 views
Minimum number of tree operations to normalize a labeled tree
Given a binary tree with labels on the leaves, like $(bc)(ad)$ or $((af)e)(c(db))$, which we can interpret as a product of terms with respect to a commutative associative operation, how many ...
3
votes
0answers
72 views
Adjacent Gray code
Gray code is permutation of $\{0,1,2,\dots,2^n-1\}$ such that each of consecutive number is differs only one bit in binary representation.
Example for $n = 3$
$000\\
001\\
011\\
010\\
110\\
111\\
...
3
votes
1answer
28 views
Sampling numbers from a weighted set that sum to constant value
So I have a multi-set of positive integers $S = \{n_1, n_2, \dots\}$ with associated weights $W = \{w_1, w_2, \dots\}$. I want to sample some numbers, without replacement, from $S$ according to ...
2
votes
2answers
44 views
Efficient n-choose-k random sampling
Is there an efficient method of sampling an n-choose-k combination at random (with uniform probability, for example)?
I have read this question but it asks for generations of all combinations, not ...
1
vote
1answer
62 views
Placing items into compatible bucket types to find an optimal total value
Suppose we have a list of buckets, each with a unique type and a maximum capaciy. We also have a list of items, each with a value and a list of compatible bucket types. An item is compatible with a ...
2
votes
1answer
31 views
Generate a random combination in O(k) time and space?
How to generate a random combination of $k$ numbers from $n$ choices in $O(k)$ time and space, if we can generate a random number between 1 and $O(n)$ in $O(1)$ time?
I know only 3 algorithms: with $...
0
votes
0answers
17 views
Combinatorial Optimisation of a College Timetable
I'm a relative newcomer to the world of combinatorics, and would like a suggestion for how to tackle this problem.
We have an event that 750 students will be attending. The event is split into 4 ...
0
votes
1answer
73 views
What is the fastest algorithm of generating all possible permutations (within a given set of constraints) of a multidimensional array?
There is D-dimensional array A. The number D and the size Sd of every dimension d=1..D is input from keyboard. There is also 1-dimensional array E of size N. It consists of unique integer numbers 0..N-...
2
votes
1answer
49 views
Subset on boolean cube with largest sum of biases
On the boolean cube $\mathcal{B}=\{0,1\}^n$, we assign each vertex a value by $p:\mathcal{B}\rightarrow[0,1]$. Let
$$\tilde{p}_i=\sum_{x\in\mathcal{B}}(-1)^{x_i}p(x).$$
What is the value of $\max_p\...
8
votes
1answer
65 views
Given a constant k, find the biggest possible rooted tree, if for every path from root to leaf, the sum of the arity of its nodes equals k?
As an example, here are all possible trees for the case $k=3$:
On each node written is its arity (= the number of children).
While this should be solvable by dynamic programming, I think there was a ...
1
vote
1answer
12 views
Bounds on “well dispersed” sparse matrices
Suppose we have an $n\times n$ zero/one matrix $M$, with $k$ ones. Let us say that the extent of $M$ is the maximum of $i+j$ over all ones at positions $(i,j)$ of the matrix, and the quality $q(M)$ is ...
2
votes
1answer
38 views
Why 2 different edge min-cuts in an undirected multigraph must be completely disjoint?
For the proof of a maximum of (n 2) min-cuts in any n-vertex undirected multigraph using the random contraction algorithm, we need to know that no min-cut shares an edge with another different one.
...
1
vote
0answers
40 views
Find a partition of multiset of binomial coefficients with constriants
Given the multiset $S$ where the elements are defined by the binomial coefficient ${n \choose k}$ where $n \in \mathbb{N}$ and $
0\leq k \leq n$ find the partition $P$ of $S$ such that the sum of ...
0
votes
0answers
38 views
O(1) algorithm to get approximate number that is larger as a binomial coefficient
Ideally I need a to calculate the binomial coefficient ${p \choose n}$. But since the fastest algorithm to do this is an $\mathcal{O}(n)$ algorithm I would look to do something different. I don't ...
0
votes
1answer
30 views
Generation of all combinations of a set in max-differing/random order
I'm looking for an algorithm that generates all k-combinations of a set, such that each successive combination generated differs as much as possible (or in practice, a lot) from all previous ...
0
votes
0answers
45 views
Counting chords intersections in a circle
The problem is:
Given 2n distinct endpoints of n chords on the unit circle, count the number of intersections between chords (if k chords intersect at one point,
that point counts as $\binom{n}{2}$ ...
1
vote
1answer
67 views
Given a bitstring generate all bitstring with n flipped bits
For an algorithm I need to be able to iterate over all bit strings where $k$ bits are flipped given a bit string with length $n$ and $n \geq k$. For instance let's say I have the bit string $1001$ and ...
4
votes
3answers
272 views
Hard connected instances for Weisfeiler-Lehman test of isomorphism
There are instances when WL algorithm fails. For example graphs G1 and G2 below have the same coloring after WL-1 algorithm.
However, one of these graphs is disconnected. So what are the instances ...
1
vote
0answers
15 views
Relation between deficiency and color class parity of graphs
Let $G$ be a graph with total vertices $|V(G)|$. Let the maximum degree of the graph be $\Delta$. Let us assume the graph is total colourable( no adjacent vertices, adjacent edges and an edge and its ...
0
votes
1answer
36 views
invite 12 person from 24 that we have 6 men and 6 womens [closed]
i had a question and its
"A man has 5 female and 7 male friends and his wife has 7 female and 5 male friends. In how many ways can they invite 6 males and 6 females if husband and wife are to invite ...
2
votes
1answer
47 views
Variant of interval scheduling (multiple machines with given availability)
I am looking for an algorithm to solve the following variant of interval scheduling : schedule some tasks on multiple machines, which are only available during a given interval of time. Two tasks ...
1
vote
1answer
31 views
Linear ordering of all subsets of size k
I was wondering if there is an obvious way to 'name' the ${n \choose k}$ subsets of size $k$ of the integers from $1$ to $n$. So I am looking for a bijection from the subsets of $\{1,\ldots, n\}$ into ...
2
votes
1answer
36 views
Alternate proof of the Caro-Wei theorem for lower bounding the independence number
Let $G$ be a graph on $n$ vertices whose degree sequence is $d_1,d_2,...,d_n$. Let $\alpha(G)$ denote the size of maximum independent set of $G$, i.e., the size of a maximum subset of vertices of $G$ ...
0
votes
1answer
41 views
Sum of zero nim sum series
The problem is proposed here and related to this question. Given $n$ and $k$, I would like to know how to compute$$\sum_{\substack{x_0 ⊕x_1⊕\cdots⊕x_k=0\\x_i≥0,\ 0≤i≤k\\\sum\limits_{i=0}^kx_i≤n-2k}}\...
2
votes
0answers
67 views
Balls in Bins with Pairwise Distance
Given $n$ bins in a row (numbering from $1$ to $n$) and $2k$ balls ($n \ge 2k$), one may put all balls into bins with each bin having at most one ball (there are $\binom{n}{2k}$ configurations). ...
1
vote
1answer
34 views
Number of ways to choose same number of elements from two different sets
Given two sets of elements S and R, with p elements and q elements respectively. 1 <= p,q <= n. Now, the number of ways to choose same number of elements from set S and R is $$\sum_{i=0}^{\min(p,...
0
votes
1answer
85 views
Hiring problem from CLRS
Hiring problem is discussed in section 5.1 and 5.2 of the CLRS and I'm referring this for exercise solutions.
However, for Exercise question 5.2-2 my solution deviates from the one given in the ...
4
votes
2answers
138 views
A problem on constrained combinatorics
Not sure if this is a proper place, but I really don't know where else to ask. I'm craving for an algorithm generating certain sequences of numbers (the problem comes from physics).
I'm looking for ...
1
vote
1answer
67 views
Algorithm to organize a tournament where the team componentes change each round
So, I was tasked with creating an app that generates the schedule of a doubles tennis tournament (i.e., teams of two) in a way that, by the end of it, everyone would have played against the rest of ...
4
votes
3answers
98 views
Sum of unique elements in all sub-arrays of an array
Given an array $A$, sum the number of unique elements for each sub-array of $A$. If $A = \{1, 2, 1, 3\}$ the desired sum is $18$.
Subarrays:
...
0
votes
1answer
43 views
Different iterations of regular expressions
A four-part question dealing with formal languages and regular expressions:
How many basic regular expressions (using only the rules 0/ϵ, 1/∅, *, +, and •) are there to match a given string?
How ...
1
vote
1answer
40 views
No. of subsets whose element multiply to give a square number
I have been given an array whose elements lie between [1,70] and the size of array [1,10^5].
I have to find the total number of subsets whose all elements multiply to give a perfect square number.
...
2
votes
0answers
9 views
Maximal number of rounds we can do distributing 64 diners on 8 groups in different ways if they can't meet each other more than once?
N=64 hungry diners come to a buffet.
We sit them at 8 different (s=8 people at each table) tables so that they get to know each other while they eat.
After a while we distribute them over the tables ...
2
votes
1answer
30 views
Count number of the ways to fill a N-lengthed binary string
From the problem, count the number of ways to fill a binary string of length $N$ with at least one $1$'s consecutive sequence of length $K$ and other $1$'s consecutive sequences have length no more ...
2
votes
0answers
61 views
Maximum number of non-overlapping rectangles where each contains a minimum number of points
Given n points and 0 < p < n, find the maximum number k of rectangles such that each rectangle contains at least p points and no two rectangles overlap. Each point is distinct from every other ...
15
votes
8answers
2k views
Cardinality of the set of algorithms
Someone in a discussion brought up that (he reckons) there can be at least continuum number of strategies to approach a specific problem. The specific problem was trading strategies (not algorithms ...
3
votes
1answer
48 views
How to generate a binary string so to minimize weighted bitwise distance?
I've got an array of weights, for example
[1, 0, 3, 5]
The distance between two strings is defined as a sum of weights for different bits, like this:
...
3
votes
1answer
50 views
Primary/Secondary On Call Rotations
I'm trying to setup a primary and secondary on call schedule with 6 people.
I'm trying to schedule people to be on call, so that they will be paired with everyone and they have the longest "break" ...
2
votes
0answers
34 views
Inexact cover, or cover with gaps
Dancing Links: wikipedia article, research paper is an implementation of algorithm X for exact cover problem.
In the Knuth's research papaer, linked above it is shown how Polymino problem (that is ...
4
votes
1answer
71 views
Integer Problem Solving with two boolean selection variables
I am trying to solve a two dimensional combinatorial problem.
Hereis my input space {{RA1,RA2},{RB1,RB2},{RC1,RC2}} and i have to choose two out of three elements{A,B,C} and one out of two possible ...
4
votes
1answer
620 views
Making 5-tuples out of their respective 4-tuples
Let's take integer numbers from [1..36]. We can generate 376992 different (order is not important) five-number-combinations like (1,3,5,7,12), etc. Such five-number-combinations always have five ...
0
votes
2answers
613 views
Given n numbers How to find out a set of numbers whose sum equal to a certain given number
I am given an list of numbers and A number-s. I need to find out the collection(s) of numbers from the list of numbers whose sum corresponds to the given number s.
...
1
vote
1answer
18 views
Finding the maximum possible size of S, where S is a set of pairwise-disjoint subsets of the list, and every element of S sums to k
Say I had a list of numbers in the range of 1-20 for example: [5, 16, 17, 3, 2, 14, 4, 9, 11, 19], and an integer k (let's say k = 40)
How would I find the maximum possible size of S, where S is a ...
2
votes
1answer
97 views
Which algorithm can calculate an optimal allocation of students to projects?
I am trying to find an algorithm which calculates an optimal and stable allocation of $n$ students to $m$ projects, where each student strictly ranks all projects by preference. The available projects ...
2
votes
1answer
22 views
How many different strongly connected graphs can be created given n nodes?
Given some fixed number of nodes $n$, which we will number 1 to $n$ in order to tell them apart, how many different strongly connected graphs can be created? Multiple edges with the same starting and ...
4
votes
2answers
91 views
How to model and solve this problem?
I have a matrix $P \in M_n(\mathbb N)$, where
$$ P =
\begin{bmatrix}
0 & P_{12} & \ldots & P_{1n}\\
P_{21} & 0 & \ldots & P_{2n}\\
\vdots & \vdots & \ddots &...
2
votes
1answer
25 views
How to find splitting point in [0,1] to maximize sum of sign function?
Given two $n$-number arrays $a_1, a_2, \ldots,a_n \in [0,1] $ and $b_1, b_2, \ldots,b_n \in [0,1] $. We would like to find the real number $x^{*} \in (0,1)$ s.t:
$x^{*} =\arg\max_{x} \sum_{i=1}^n \...
