Questions tagged [co.combinatorics]
Questions related to combinatorics and discrete mathematical structures
591
questions
3
votes
1answer
168 views
Level $k$ bounds in Analysis of Boolean functions
In Ryan O'Donnell's book Analysis of Boolean functions, following Corollary 9.25 the following appears:
If $f\colon \{-1,1\}^n \to \{0,1\}$, and we have $\mathbb{E}[f] = \alpha$, then for any integer ...
3
votes
1answer
139 views
Dual to hypercontractive inequality
Recall the hypercontractive inequality:
Let $\rho = \sqrt{\frac{p-1}{q-1}}$, then $||T_\rho(f)||_q \leq ||f||_p$
In https://www.cs.cmu.edu/~odonnell/papers/analysis-survey.pdf it is stated that the ...
4
votes
1answer
215 views
How tight is the XOR lemma?
The XOR lemma states that if you have a distribution $D$ on $\{0,1\}^n$, and all the Fourier coefficients of $2^n D$ are small, then it is close in $L_1$ to the uniform distribution. Specifically, ...
5
votes
1answer
256 views
Re-packing of integers into fixed-size bins
We are given a constant $B \geq 3$, a set of positive integers $I$ s.t. $\forall i \in I, i \leq B$ and $\sum_{i \in I} = N$,
and 2) a packing of these numbers into $N/B > 1$ bins of size $B$.
An ...
-1
votes
1answer
58 views
Chomsky-Schutzenberg Hierarchies explained for physicist (general) [closed]
I am classically trained in physics, however I have been interested in the use of information theory in studying some classical systems.
As someone who is somewhat unfamiliar with the language of ...
0
votes
0answers
46 views
choosing the best subset where the metric is based on pairwise relationship
I want to model the competition between agents.
Say there is a set of agents, and at each time step each agent $i \in \{0,1,\cdots, n-1\}$ can select a nonnegative number $x_i$ from a given range $[0,...
1
vote
0answers
79 views
Shortest s-t Path with a covering constraint
Instance: an undirected graph $G=(V,E)$ with edge-weights $w:E\to{\mathbb{R}}$;
a source $s\in V$ and a sink $t\in V$;
a ground set $X=\{x_1, ..., x_k\}$, and for every $v\in V$ a corresponding ...
5
votes
1answer
225 views
The asymptotic behavior of a recurrence related to stable matchings
I would like to provide asymptotic estimates for a sequence defined (for n a power of 2) as follows:
$$a_1 = 1, a_2 = 2$$
$$a_n = 3a_{n/2}^2 - 2a_{n/4}^4$$
Apparently, Knuth was able to prove that ...
2
votes
0answers
116 views
Sensitivity and Low-Degree Approximation under Non-Uniform Distribution
I am searching for generalizations of analysis of Boolean functions when the input strings are distributed according to a general non-uniform distribution, possibly with arbitrary dependencies between ...
2
votes
1answer
85 views
Placing color boxes on a colored image such that color consistency is maximized
I have encountered the following challenging problem that I think to be a non straightforward generalization of the Knapsack problem.
Given an image with black background that contains blobs whose ...
0
votes
0answers
63 views
Lower bound for permutation generator
I'm interested in a problem akin to combinatorial circuits, but in terms of complexity. Apologies for missing the correct terminology, I'll appreciate any corrections.
Given $n$ inputs numbered $1 ......
1
vote
1answer
68 views
How to play the following game? (placing balls into bins)
Let $n,\ell\in\mathbb N$ for some $n\gg \ell\gg 1$.
The goal is to pick two sequences of numbers, $x_1,\ldots,x_\ell$ and $y_1,\ldots,y_\ell$ such that $$\Sigma_{i=1}^\ell x_i = n\quad{}\mbox{and}\...
4
votes
1answer
176 views
Which (almost) balanced Boolean function has smallest “total” influence
The well known Kahn–Kalai–Linial (KKL) Theorem says that for any Boolean function $f\colon \{-1,1\}^n \xrightarrow{} \{-1,1\}$
$$
\max_{i \in [n]} \{\mathbf{Inf}_i[f] \} \geq \mathop{\bf Var}[f] \cdot ...
3
votes
2answers
156 views
Compactness of domino tilings
I've read in Lemma 2 of the paper 1 that if every square region of the plane admits a tiling, then the whole plain admits a tiling, but the proof is omitted. This sounds like a compactness property, ...
2
votes
0answers
38 views
Time complexity of finding a point of infinite order on a rank 1 elliptic curve over Q
As an outsider, it sounds like a lot of progress has been made on understanding rank 1 elliptic curves over Q. Much of the BSD conjecture is known for rank 1, and Heegner points provide a way in ...
4
votes
0answers
69 views
Optimal scheduling with delay constraints
Suppose you have $K$ servers numbered $\{1,2,...,K\}$. Playing server $i$ provides a value of $v_i > 0$. However, once you play server $i$, you are not allowed to play it for the next $n_i$ time-...
3
votes
0answers
69 views
Complexity of counting Wang tiles
Consider the question of counting Wang tilings on a torus. The decision version of this problem is known to be NP-complete. Is the counting version #P-complete?
1
vote
0answers
124 views
Star seperators to explain computational complexity of algorithms on a class of graphs?
A lot of NP-hard optimization problems on graphs which are perfect become solvable in polynomial time. Unfortunately, the class of graphs that arise in my problem are not perfect. The graphs can be ...
4
votes
1answer
92 views
Stable order on binary strings
I need some order on binary strings such that if I have a small (but superlinear in their length) number of sufficiently different strings, the order will stay the same if I change a few bits in the ...
0
votes
1answer
143 views
Subset Sum Problem and hard looking instances that are not really hard
I have been working in a subset sum solver (some new approach) and while working on the time complexity analysis I found what I describe below. Maybe this could explain why some "hard looking" ...
4
votes
1answer
114 views
Applications of Christol theorem
I'm looking forward to know about applications of Christol theorem mentioned in Jefrrey Shallit's Number theory and formal languages. One of them is purely algebraic: if $f, g \in \mathbb{F}_q[[z]]$ ...
1
vote
1answer
82 views
Pulling a graph across a partition
I am looking for the name for a particular graph property, if it has been studied, and efficient algorithms for computing it, if they exist. I realise that this may be a well known property that I am ...
9
votes
0answers
147 views
Random unbalanced bipartite graphs are good small set expanders
My question is about small set expansion properties of random unbalanced bipartite graphs.
Fix a positive $\delta<1/2$, and a positive integers $n,m,d$. Let us call a bipartite graph $\mathcal{G}$...
2
votes
0answers
134 views
Smallest disjoint union chain containing a sequence of sets
Let $\mathcal{A}=\{A_1,\ldots,A_n\}$ be a family of sets, we have the property that $A_1=\emptyset$, and one can obtain $A_i$ from $A_{i-1}$ by adding or deleting a single element.
A family $\...
3
votes
0answers
73 views
Linear optimization over intersection of totally unimodular matrices
I am currently dealing with a problem of the following form
\begin{alignat}{2}
&\underset{x, y \in \mathbb{R}^n}{{\text{min}}} && e^T x \nonumber\\
&\text{sub to} \hspace{0.05in}&&...
1
vote
0answers
103 views
Growth of random square lattice trees
Consider the problem of growing a random tree on a $L\times L$ square lattice of initially disconnected vertices, starting from an isolated vertex on one of the corners of the lattice and proceeding ...
6
votes
1answer
121 views
Average-case analogue of Small-bias Spaces
Recall that an $\epsilon$-biased space is a set $S \subset \{0,1\}^n$ such that for every non-zero linear test $\alpha \in \{0,1\}^n \setminus \{0\}^n$, the expected bias
$$| \mathbb{E}_{x \in S} [ (-...
3
votes
1answer
121 views
Mutually exclusive replacement paths, an existing problem?
A replacement path is a simple path allocated to an edge $e \in G$
that connects the endpoints of $e$ in $G \setminus \{e\}$.
The problem
An undirected graph $G$ is given and the task is to allocate ...
4
votes
1answer
106 views
Probability of a $k$-path in a random graph
Assume that $G\in G(n,p)$; if $p=\frac{\ln n +\ln \ln n +c(n)}{n}$,
the following fact is well known:
\begin{eqnarray}
Pr [G\mbox{ has a Hamiltonian cycle}]=
\begin{cases}
1 & (c(n)\...
1
vote
1answer
107 views
Total flow using minimum number of edges on a bipartite network
If I have a set of sources $S$ with total capacity $C$ and a set of sinks $T$ with the same capacity $C$, but not necessarily the same cardinality, is there an efficient way to find the minimum number ...
1
vote
0answers
59 views
Counting vertex covers on a chain of k nodes that do not contain a sub-chain of length >=3
By a "chain of k nodes", I mean k nodes lined up like a linked-list: o-o-o.....-o . By "do not contain a sub-chain of length >=3", I mean that no cover should contain two edges that shares a node. ...
7
votes
1answer
262 views
Distinguishability a set of permutations
Given integers $d<n$, find the largest $k$ such that there exists a set of $k$ permutations $\sigma_1,\cdots,\sigma_k$ on $[n]$, such that any size-$d$ subset $T\subseteq [n]$ is ``distinguishable''...
1
vote
0answers
31 views
Inductive definition of language operators like the set of all permutations of a word came from the shuffle operator
Let $X$ be a finite alphabet. Given two words $u, v \in X^{\ast}$ the shuffle operator is defined to be
$$
u || v := \{ u_1 v_1 u_2 v_2 \ldots u_n v_n : 1 \le i \le n, u_i, v_i \in X^{\ast}, u = u_1 \...
1
vote
2answers
264 views
Is perfect matching for bipartite graph with no cycles unique?
Given a balanced bipartite graph that satisfies Hall's theorem (is non singular) then it shown that it has at least one perfect matching.
My question is if the balanced bipartite graph is also ...
1
vote
0answers
152 views
How to represent boolean tree + algorithm as a mathematical formula
In programming say you have a boolean tree like this:
...
-1
votes
1answer
149 views
Application of the inequality with expectations
Let $\Vert\cdot\Vert$ is a norm in $R^n$. Let $x_1,\dots,x_N$ non-independent Rademacher random variables random variables (variables which are uniform on $\{-1, 1\}$). . By $E$ we denote an ...
1
vote
0answers
75 views
Nim-game variant - weird ending condition
the game is structured like this:
two players
the players alternate moves
4 heaps $h_1,h_2,h_3,h_4$ with sizes $n_1,n_2,n_3,n_4$
at each move, the player can either remove one or two elements from ...
4
votes
2answers
301 views
NP-hardness of finding 0-1 vector to maximize rows of {-1, +1} matrix
Consider the following discrete optimization problem: given a collection of $m$-dimensional vectors $\{ v_1, \dots, v_n \}$ with entries in $\{-1, +1\}$, find an $m$-dimensional vector $x$ with ...
3
votes
2answers
3k views
How to solve the Shortest Hamiltonian Path problem on Sparse Graphs?
Problem: Given a positive-weighted undirected graph, find the shortest path (in terms of total sum of edges) that visits each node exactly once.
For a subset $S$ of nodes and a node $i\in S$, let $D[...
-3
votes
1answer
146 views
Continous work distribution algorithm with failover
Imagine there's a system where there's N workers and M units of work, for example, N ≤ 64, M = 256.
Is there an algorithm that ...
-3
votes
1answer
124 views
Proving that a random permutation generator is not fair [closed]
If I'm generating random permutations using the following algorithm:
...
3
votes
0answers
77 views
Finding the longest sub-permutation with bounded inversion number
Given a permutation $\sigma\in S_n$ and a positive integer $k$, find the largest integer $0\le m\le n$ such that there exist a subset $I\subseteq [n], |I|=m$, satisfying that the restriction of $\...
4
votes
1answer
1k views
Is Hartmanis-Stearns conjecture settled by this article?
The paper
"On the computational complexity of algebraic numbers: the Hartmanis--Stearns problem revisited"
by Boris Adamczewski, Julien Cassaigne, Marion Le Gonidec
https://arxiv.org/abs/1601....
9
votes
1answer
203 views
Finding subgraphs with high treewidth and constant degree
I am given a graph $G$ with treewidth $k$ and arbitrary degree, and I would like to find a subgraph $H$ of $G$ (not necessarily an induced subgraph) such that $H$ has constant degree and its treewidth ...
11
votes
2answers
149 views
Does the first order theory of a finite structure have bounded quantifier rank?
Let $\mathfrak{A} $ be any finite structure.
Does its first order theory $ \mathfrak{T} := \mathfrak{TH}(\mathfrak{A}) $ have bounded quantifier rank, in the sense that there is a $ q\in\mathbb{N} $ ...
3
votes
1answer
67 views
Reference: Cancellability of the Dyck congruence
I consider the Dyck congruence $\equiv$ on a parenthesis alphabet $\Sigma = \{a, \bar a, b, \bar b\}$, i.e. the least congruence on $\Sigma^*$ such that $a \bar a \equiv \varepsilon $ and $b \bar b \...
3
votes
1answer
322 views
Relation between variance and mutual information
Given two discrete random variables $X,Y$ such that $X,Y \in \mathbb{R}$ and $0 \leq X,Y \leq 1$, is it true that $$|\text{Cov}[X,Y] \leq \sqrt{\frac{1}{2} \text{I}[X,Y]}|. $$
This bound may be ...
6
votes
1answer
124 views
Directed graph with bounded in-deg can be partitioned in a balanced way
I want to prove that for all $n$, there exists a constant $c(n)$ such that if $G=(V,E)$ is a directed graph with in-degree bounded by $n$, it is possible to partition the set of vertices $V$ into two ...
4
votes
0answers
60 views
Short supersequence for many permutations
A shortest supersequence $c_n$ of all permutations on $[n]$ has length $\Theta(n^2)$: see this question on Mathoverflow. What if we force $c_n$ to be short? How many permutations can it cover?
Let's ...
2
votes
1answer
71 views
Almost regular subhypergraph of hypergraph with large minimal degree
I am interested in knowing whether the following conjecture is true or not:
For every $d \geq 1$ there exist constants $\delta,M_0 > 0$ such that the following holds for all $M \geq M_0$.
...
