Questions related to combinatorics and discrete mathematical structures

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5
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42 views

Upper bound on Euler characteristic of downward closed family

(Definition: $\mathcal{F}$ is called downward closed if for any $A \in \mathcal{F}$ and $B \subseteq A$ it holds that $B \in \mathcal{F}.$) Let $\mathcal{F}$ be a downward closed family of subsets of ...
0
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0answers
48 views

Notion of 'order type' for non-well ordered sets?

Given a well-ordered poset $P = (S,\leq)$, it is possible to define its order type as the supremum of the order types of its linear extensions; this is well-defined as shown in 'Well-partial orderings ...
2
votes
1answer
77 views

Faithful functors vs forgetful functors: exact category-theoretic defs?

In category theory, a functor between two categories $C,D$ is a map $F$ that assigns to each object (resp. morphism) $x$ of $C$ a corresponding object (resp. morphism) $F(x)$ of $D$ by respecting the ...
10
votes
0answers
89 views

Sylver Coinage Game

A game in which the players alternately name positive integers that are not sums of previously named integers (with repetitions being allowed). The person who names 1 (so ending the game) is the ...
1
vote
0answers
66 views

Restoring symmetry in certain combinatorial bijections?

I'm interested in two 'natural bijections' that involve labeled forests and Young tableaux. Let me give the definition for labeled forests. By this, we mean a pair $\cal{F} = (F,f)$ where $F$ is an ...
2
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0answers
54 views

Existence of bijective proofs involving Yamanouchi words?

The following notion comes from algebraic combinatorics and might have some connections to language/permutation theory. Fix an integer $k$. Let us call a $k$-Yamanouchi word a sequence $u$ of integers ...
2
votes
1answer
78 views

Upperbound the order of P3-free partition of P4-free graphs

A graph $G$ is $P_k$-free if and only if $G$ does not have an induced subgraph isomorphic to a path of $k$ vertices. Thus, $P_2$-free graphs are exactly independent sets (or stable sets). $P_4$-free ...
7
votes
3answers
160 views

On the size of P4-transversals of graphs

A subset $T$ of vertices of a graph $G$ is called a $P_4$-transversal if $T$ intersects every $P_4$ of $G$. In the context of this question, we consider $P_4$ as an induced path on 4 vertices. ...
0
votes
1answer
109 views

Various Parameters on Hyper graph

Numerous parameters are defined and considered in the graph theory. I am interested in analogy of these parameters in theory of $hyper$ $graphs$. Is there some survey or book or lecture notes about ...
20
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2answers
1k views

Representing OR with polynomials

I know that trivially the OR function on $n$ variables $x_1,\ldots, x_n$ can be represented exactly by the polynomial $p(x_1,\ldots,x_n)$ as such: $p(x_1,\ldots,x_n) = 1-\prod_{i = ...
7
votes
1answer
406 views

Connections between the Erdos Discrepancy Problem and (Theoretical) CS?

Recently there have been some new results on computer-based experimental study of the Erdos Discrepancy Problem (EDP) (via SAT solvers, cited below). This problem has been cited and studied by several ...
13
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0answers
284 views

$NP$-completeness of recognizing the difference of two permutations

Shor stated, in his comment to anonymous moose's answer to this question Can you identify the sum of two permutations in polynomial time?, that it is $NP$-complete to identify the difference of two ...
5
votes
1answer
81 views

Worst case ratio between minimum clique cover and maximum independent set

The maximum independent set problem gives a lower bound for the minimum clique cover problem. This is easy to see because given any clique cover together with an independent set, any two vertices ...
3
votes
0answers
101 views

Building a “balanced” universal set

A set of length $n$ binary vectors $\mathcal{U}=\{u_1,..,u_r\}$ is called $(n,k)$-universal if for all $S\subset [n], |S|=k$, $|\mathcal{U}_{|S}|=2^k$, i.e. for every subset of indices of size $k$, ...
4
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0answers
72 views

Constructing a small (n,k)-Covering Matrices family

Let ${\cal A}$ be a set of $k\times n $ matrices over ${\mathbb F}_2$. We call ${\cal A}$ a (n,k)-covering, if for every subset of columns $I=(i_1,\ldots,i_k)\subseteq [n]$, there is a matrix $A \in ...
6
votes
1answer
216 views

Constructing a k-perfect permutations family

I'm looking for a set of permutations over $n$ elements $\mathcal{P}=\{P_1,P_2,...,P_r\}$ of minimal size such that for every ordered subset of size $k$, $S=<x_1,x_2,...,x_k>, (x_i \in [n])$, ...
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0answers
52 views

Upper bound on the number of vertex-transitive graphs

Is there a known upper bound on the number of vertex transitive graphs on $n$ vertices?
13
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0answers
137 views

Is there a regular tree language in which the average height of a tree of size $n$ is neither $\Theta(n)$ nor $\Theta(\sqrt{n})$?

We define a regular tree language as in the book TATA: It is the set of trees accepted by a non-deterministic finite tree automaton (Chapter 1) or, equivalently, the set of trees generated by a ...
3
votes
0answers
90 views

Concentration of Stationary Distribution on Random Directed Graphs

We consider a random directed graph with fixed out-degree $d$. Each vertex chooses $d$ neighbors with replacement, uniformly and independently. Self-loops and multiple arcs are allowed in this model. ...
9
votes
0answers
97 views

Computing weighted sums of binomial coefficients

This question is a reformulation of Complexity for computing weighted number of paths on integer lattice Is there any way to compute in $o(n^2)$ all $n$ sums $\sum_{0\leq i \leq j} a_i\binom{j}{i}$ ...
3
votes
0answers
102 views

An identity about the Majority function?

Let $f:\{-1,1\}^n \rightarrow \{-1,1\}$ be any boolean function. Let $Maj_n$ represent the majority function. Let $\langle f,g \rangle = E[f(x)g(x)]$ and $\mathcal{I}(f) = E_x[\# i, s.t. f(x)\neq f(x ...
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0answers
59 views

Nearly-uniformally sampling lattice points from a basis that lie on the interior of a polytope

Definitions: Consider a polytope $P \subset \mathbb{R}^n$ with a nonempty interior to be $P : \{x \in \mathbb{R}^n | Ax \le B\}$ for appropriate real $n \times m$ matrix $A$ and $m \times 1$ vector ...
0
votes
1answer
112 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). ...
7
votes
3answers
478 views

Counting the Number of Simple Paths in Undirected Graph

How can I go about determining the number of unique simple paths within an undirected graph? Either for a certain length, or a range of acceptable lengths. Recall that a simple path is a path with no ...
3
votes
0answers
113 views

Subset of a bipartite graph with maximal number of minimal unmatched vertices

Given a bipartite graph $G = (U \sqcup V, E)$ with sets of vertices $U$ and $V$ and edge set $E \subseteq U \times V$, a matching $M$ is a subset of $E$ whose edges have no common vertices: for all ...
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vote
0answers
78 views

Complexity for computing weighted number of paths on integer lattice

Our input is a $(n+1)\times (n+1)$ table filled with some value (integer) for each leftmost and bottom cell $l_i,b_i$ as in the figure. We wish to compute the value of all upper and rightmost cells ...
2
votes
2answers
115 views

Polynomial time construction of families of pairwise nonhomomorphic graphs

Is it true that for all $n$ there are $n$ pairwise nonhomomorphic graphs with $poly(n)$ vertices? Is there a polynomial time algorithm for constructing such families of graphs?
2
votes
1answer
155 views

Dividing users with certain files into 2 equal groups

I am framing a particular combinatorial question using users and files for better understanding. Let there be a universe of files $F$ = $\{f_1, f_2,\ldots,f_n\}$ and $2k$ users $\{u_1, u_2,\ldots, ...
5
votes
1answer
143 views

Kruskal-Katona Theorem with Majority?

I am interested in the following problem which seems like an extension of the Kruskal-Katona Theorem. Let $A_k \subseteq \{0,1\}^n$ be a subset of the hypercube such that every element in $A$ has ...
0
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0answers
39 views

Different estimators for uniform convergence of means/averages to expectations

In uniform converge results of means or averages to their expectations (think of the typical results involving VC-dimension, covering numbers, Pseudo-dimension, fat shattering dimension, ...) , the ...
2
votes
0answers
90 views

Geometry on a space of polynomial functions

I am considering some geometric concepts in a space of functions.But I am not sure the concept I consider is already defined in some references. Let $P_{1},...,P_{N}:\mathbb{F}_{2}^{n}\rightarrow ...
5
votes
0answers
151 views

Lemma about intersecting sets

I use the following Lemma and I wonder whether it is known in literature. If you look at the proof it feels like it should be known from combinatorics or extremal graph theory. Lemma. Let $A$ be an ...
5
votes
0answers
102 views

Maintain mex with efficient union

Do you know of any data structure $S[A]$, that maintains a (finite) set $A \subset \mathbb{Z}_{\geq0}$ of non-negative integers, subject to the following operations: Given $S[A],$ calculate minimal ...
5
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0answers
146 views

Local Hamiltonian and combinatorial search problems

I was going through the PhD thesis of Daniel Nagaj. At the beginning of chapter two he indicated a relation between the local Hamiltonian perspective of adiabatic quantum computation and combination ...
4
votes
1answer
170 views

Colouring achieving simple discrepancy bound?

Given a hypergraph $H$ with $n$ vertices and $m$ edges, one of the simplest inequalities on the discrepancy of $H$ is $\text{disc}(H) \le \sqrt{2n \ln (2m)}$. This is usually proved by mixing ...
2
votes
0answers
68 views

Bits required to encode difference between number of subgraphs with odd number of edges and number of subgraphs with even number of edges

This question was originally posted on mathoverflow. Having obtained no answers after one month, I've decided to cross-post it here. Let $H = ( V, E )$ be a $k$-uniform connected hypergraph, with ...
7
votes
1answer
229 views

Erdos conjecture, number of cliques, Turan`s graph

Erdos&Stone conjectured in 1946-08 that there are at least $ck-1$ (k+1)-cliques in $G=(V,E)$ whenever $|V|=ck$, $|E|-1$ is the edge-count in Turan's $T(|V|,k)$, i.e., $|E|= 1+\lfloor ...
-1
votes
1answer
107 views

Representation suitable for reconstruction of a tree with bounded degree

I am dealing with reconstruction of molecular graphs for which unlabelled rooted trees with maximum degree 4 are fair approximations. In particular, I would like to encode a small tree (assume number ...
4
votes
0answers
61 views

Residual for transitive hull

I work in the algebra $R$ of reflexive, transitive relations over some set $S$, ordered by subset inclusion. This is a complete lattice, with intersection as g.l.b. and transitive hull as l.u.b., i.e. ...
12
votes
1answer
242 views

Beigel-Tarui transformation of ACC cricuits

I am reading the appendix about ACC lower bounds for NEXP in Arora and Barak's Computational Complexity book. http://www.cs.princeton.edu/theory/uploads/Compbook/accnexp.pdf One of the key lemmas is a ...
16
votes
0answers
372 views

Tiling a rectangle with the fewest squares

Consider this problem: Find a tiling of an $m \times n$ rectangle by minimum number of integer-sided squares. Is there any polynomial time (in $m$ and $n$) algorithm to do this? What is the best ...
7
votes
2answers
491 views

Find all pairs of values that are close under Hamming distance

I have a few million 32-bit values. For each value, I want to find all other values within a hamming distance of 5. In the naive approach, this requires $O(N^2)$ comparisons, which I want to avoid. ...
6
votes
1answer
155 views

correlation in an almost independent set of random variables

Suppose I have a set of $n$ binary random variables $X_1, \ldots, X_n$ that sit on a line, and assume that $\Pr(X_i=0)=\delta$ for all $i$. In addition, assume that any two subsets of variables that ...
-1
votes
2answers
301 views

Intersection between sets

Assume that we have $p$ sets, with given sizes: $m_1,m_2,...,m_p$. The (distinct) elements in each set are taken from $N$ elements (where $m_1,m_2,...,m_p \le N$). A combination is defined as an ...
6
votes
2answers
146 views

Given a set of distances (no info regarding what points the distances correspond to) from a complete graph, is the realization of the graph unique?

There are $n$ points in $R^2$ (i.e. the 2D real space). We can think of them as a complete graph where edge weights correspond to the distance between points. Let $D$ be the distance matrix between ...
0
votes
1answer
105 views

Finding Tours through Near-Hamiltonian Paths?

Say I have a connected graph. I want to find a tour that visits each vertex at least once. It's not always possible, though, for there to be a solution if there is a bridge in the graph. Is there a ...
0
votes
0answers
52 views

On introductive material of set systems

I am reading Jukna's textbook "Extremal Combinatorics", and I want to some introductive material various theory of set systems including optimizational area like matroid theory. Are there a lecture ...
1
vote
2answers
235 views

What's the relation between the dominating set and vertex cover?

I wonder if the minimal dominating set is always a subset of the minimal vertex cover in any graph. If so, what's the proof?
1
vote
2answers
79 views

Monomial decomposition as product of smaller monomials

Is there an algorithm to get all possible decompositions of a monomial as the product of smaller monomials ? For example $x^2y^3$ is $x∗xy^3$, $x^2∗y^3$, $xy∗xy^2$, and so on. For a monomial in one ...
10
votes
1answer
938 views

Efficient algorithm for existence of permutation with differences sequence?

This question is motivated by this post, Can you identify the sum of two permutations in polynomial time? , and my interest in computational properties of permutations. A differences sequence $a_1, ...