Questions related to combinatorics and discrete mathematical structures

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PROOF OF GOLDBACH'S CONJECTURE [closed]

Every even integer > 2 is the sum of two prime numbers & equivalent Each odd integer > 5 is the sum of three prime numbers USING THE SIEVE OF ERATOSTHENES ...
-3
votes
0answers
22 views

Cheapest route in graph under time constraints

Here is my question: I have to find the cheapest path on a graph containing vertices as cities and edges as flights. Each flight has associated with it: departure time,arrival time,city of departure ...
8
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0answers
230 views

Is the complexity of this covering problem known?

Let $G=(V,E)$ be a graph. A vertex set $X\subseteq V$ is called critical if $X\neq\emptyset$ and no vertex in $V\setminus X$ is adjacent to exactly one vertex in $X$. The problem is to find a vertex ...
4
votes
1answer
113 views

Special properties of bipartite expanders

It is well known that expanders, and often the special case of bipartite expanders, have found many uses in derandomization, coding, etc. However, I am curious if there are any special properties of ...
24
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5answers
3k views

Is it a rule that discrete problems are NP-hard and continuous problems are not?

In my computer science education, I increasingly notice that most discrete problems are NP-complete (at least), whereas optimizing continuous problems is almost always easily achievable, usually ...
3
votes
1answer
111 views

Hereditary Discrepancy

It is known that it is NP-hard to distinguish whether a set system has discrepancy $0$ or $O(\sqrt{n})$, given that the set system has $n$ elements and $m=O(n)$ sets. In general if it is so hopeless ...
5
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0answers
64 views

Applications of small Kakeya sets over finite fields

It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$. For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ ...
4
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0answers
82 views

Thresholds for overlapping sets

A common scenario is that one has a well-ordered universe, and one wishes to answer queries of the form "how many elements are at most x?". If one has d elements, then one can pick logd thresholds in ...
11
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4answers
421 views

Additive combinatorics applications in algorithm design

I'm reading surveys by Trevisan and Lovett on applications of additive combinatoric in TCS. The majority of these applications fall under computational complexity, e.g., lower bounds. I wonder if ...
3
votes
1answer
47 views

Emptiness of complement of subspace arrangement

Given $k$ affine subspaces in $\{0,1\}^n$, consider the problem of testing whether their union covers all of $\{0,1\}^n$. What's the complexity of this problem? P.S.: It seems that this can be ...
6
votes
1answer
165 views

On the notion of positive rank

The positive rank of a square matrix is defined in Theorem $3$ of "Expressing Combinatorial Optimization Problems by Linear Programs" by Mihalis Yannakakis as follows: given a $n\times n$ matrix $A$, ...
4
votes
2answers
333 views

Weird claim of graphclasses about complexity of domination

EDIT this got 'fixed' on graphclasses, as per answers/comments, so you might not reproduce it, unless you have their earlier database, which is publicly available via sage - http://sagemath.org. ...
2
votes
0answers
105 views

What is the most “unbalanced” vector between two given vectors of numbers?

Let $\mathbb{R}_+$ be the set of non-negative real numbers. Let $m$ be a positive integer and $\leq_m$ the product ordering on $\mathbb{R}_+^m$. That is, $\leq_m$ is the partial ordering on ...
4
votes
1answer
115 views

Addding edges to spanning tree without destroying planarity

Given a graph $G=(V,E)$ with n vertices, m edges, and the maximum degree $\Delta$. Let $T$ be a spanning tree of $G$. Let $E_c \subseteq E - E(T)$ be the maximum number of edges that we can add to $T$ ...
3
votes
2answers
235 views

computing maximal bit density over a FSM

let $L$ be a regular language defined by a FSM over binary symbols $\{0,1\}$. consider a function $f(x)$ on words/ strings that computes "bit density", defined as the number of $1$'s in a word ...
1
vote
0answers
110 views

Does this problem have a name? Finding a subset intersecting all subsets of a family

I am pretty sure this problem has a name, or it can be reworded so it does. We are given a set $X$ and a family $\mathcal F$ of subsets of $X$, the problem is to find a subset $B$ such that $B\cap ...
1
vote
0answers
110 views

Scheduling to maximize idle time

In the context of scheduling maintenance jobs on arcs of a flow network I came across the problem to schedule jobs, indexed by $j$, and given by triples $(r_j,d_j,p_j)$ of (integer) release time, due ...
3
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1answer
99 views

Generalization of Beck's theorem

Beck's theorem is a classical result in discrete geometry which describes about the geometry of points in the plane. The result states that a finite collections of points in the plane fall into one of ...
2
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1answer
117 views

What is the standard name for the function which inflates a string by duplicating each of its characters?

Given a string $s$ over some alphabet, I'd like to use the proper nomenclature/notation for the operation/function $f$ which inflates $s$ by independently duplicating each of its characters. For ...
4
votes
2answers
216 views

Is there notation for converting a multi-set to a set?

Suppose we have a multi-set $S$. For example, $S = \{ 1,2,2,3 \}$. Suppose we also have a set $T$, e.g., $T=\{1,2,3\}$. I would like to say, compactly, that $S$, when its duplicates are removed, is ...
1
vote
0answers
59 views

Discrepancy of Hadamard type matrix

Let $H$ be $\{-1,+1\}$ Hadamard matrix of size $2$ and $J$ be the same size all $1$ matrix. Let $W$ be $\frac{H+J}{2}$. Is the discrepancy of $W^{\otimes k}$ atmost $\sqrt{k^{-1}}$?
2
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0answers
79 views

Recent insights on algorithms for 1D bin packing

This is just a general question on recent algorithms for the 1D bin packing problem. I just want to collect some information on this issue, so I’m grateful for any information. Especially heuristics ...
3
votes
1answer
98 views

Number of different longest common substrings

Given an alphabet $\Sigma$ of size $k$ and two strings $w_1,w_2\in \Sigma^n$ of length $n$. The longest common substring problem asks for a longest string in the set $A(w_1,w_2)$ of all common ...
7
votes
3answers
187 views

How many sets of vectors can be represented as the solutions of a Horn-SAT instance?

Let the solution space of a SAT instance be the set of Boolean vectors of satisfying assignments of $\{0,1\}$ to the variables (that result in the formula evaluating to TRUE). In other words, a ...
6
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0answers
68 views

Approximating a max-cut's intersection with other cuts

For the purposes of this question, a cut in a graph $G$ is the edge-set $\delta (S)\subseteq E(G)$ between some vertex-set $S$ and its complement. A max cut is one with at least as many edges as any ...
8
votes
1answer
183 views

A curious asymmetry in good characterizations

There are quite a few theorems, mostly in graph theory and combinatorial optimization, that are often referred to as good characterizations. They typically put a property in $NP\cap co-NP$, by showing ...
2
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0answers
42 views

Minimizing a general submodular pseudo boolean function

Are there algorithms that minimize a general submodular pseudo boolean function (PBF) without first transforming it to a quadratic pseudo boolean function (QPBF)?
2
votes
1answer
75 views

Weighted furthest point voronoi diagrams

I found that Weighted nearest neighbor voronoi diagrams are widely studied and there are optimal algorithms for that. But I could not find anything on Weighted furthest point voronoi diagrams !! But ...
5
votes
0answers
132 views

lower bound for difference between max cut and min cut

Let $G=(V,E)$ be a graph on $n$ vertices with edge weights $w=(w_e)_{e\in E}$. Let $M^+$ and $M^-$ be the maximum and the minimum cut values, i.e. ...
4
votes
0answers
161 views

Is this graph polynomial known? Can it be efficiently computed?

Consider a connected simple graph $G$ with $n$ vertices and $m$ edges. View each edge $\ell$ as a transposition $t_{\ell}$ acting on the set of vertices. [To be more explicit, given an edge $\ell$ ...
6
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0answers
94 views

General Results for Complicated Constraint Satisfaction Problem

Consider the following problem: on a finite two-dimensional grid (say the grid points are vertices of a graph), I need to color the vertices in such a way to satisfy the existence and nonexistence of ...
11
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0answers
192 views

Sign patterns for Fourier coefficients of Boolean functions

Given a sequence of real numbers $(a_i)$, the sign-pattern sequence $(s_i)$ is defined by $s_i = +$ if $a_i \geq 0$ and $s_i = -$ otherwise. For a boolean function $f: \{0,1\}^n \to \{0,1\}$, ...
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votes
2answers
79 views

Arrangements of Objects

Suppose there are $n$ bins each having $k$ objects. Assume that capacity of each bin is also $k$. Now we want to rearrange the objects such that each bin contains $k$ objects but this time if $x,y$ ...
2
votes
0answers
60 views

Name for the “stronger submodularity” property in cut function

Let $f:2^V \rightarrow \mathbb{R}$ be a set function over $V$ that satisfies the following: $f(A \cap B) + f(A \cup B) \le f(A) + f(B)$ $f(A \backslash B) + f(B \backslash A) \le f(A) + f(B)$. ...
10
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0answers
132 views

Is RAMSEY COLORING in $NC$?

Say that a function $f$ is a RAMSEY COLORING if on unary input $n$, it returns a complete graph on $n$ vertices with its edges colored red and blue without a monochromatic clique of size $10\log n$. ...
5
votes
0answers
70 views

Perfect hashing family variation - injectivity on $r$ disjoint sets

We denote by $[t]$ the set $\{1,2,\ldots,t\}$. A $(n,k)$-perfect hashing family is a set of functions $H=\{h_i:[n]\to[k]\}$ such that for every set $S\subset [n], |S|\leq k$, there exists some $h_S ...
0
votes
0answers
80 views

Regularlity lemma and exceptional set

Regularity lemma is one of the most basic tools in algorithmic thoery and pure math theory like additive number theory and combinatorics. This lemma is stated two ways: the existence of a partion ...
28
votes
4answers
872 views

What is simplest polynomial algorithm for PLANARITY?

There are several algorithms that decide in polynomial time whether a graph can be drawn in the plane or not, even many with a linear running time. However, I could not find a very simple algorithm ...
4
votes
2answers
651 views

Sum of products of all combinations?

We are given a list $S$ containing $n$ numbers $S=(s_1,\ldots, s_n)$. Let $S \choose k$ be the set of all possible $k$-combinations from $S$ (i.e. size $k$ subsets of $S$). We want to compute the ...
8
votes
4answers
505 views

Polynomial problems in graph classes defined by forbidden induced cyclic subgraphs

Crossposted from MO. Let $C$ be a graph class defined by a finite number of forbidden induced subgraphs, all of which are cyclic (contain at least one cycle). Are there NP-hard graph problems ...
0
votes
2answers
129 views

Determining the distribution of results of a simple algorithm

Setting Consider repeating the following process on the numbers $N=\{1, 2, 3, \ldots, n\}$: Pick an integer $k \in N$, uniformly at random. Pick a subset of $k$ elements from $N$, uniformly at ...
4
votes
0answers
68 views

What is the smallest deterministic construction of an ordered perfect hashing family?

A $(n,k)$-perfect hashing family is a family of functions $H=\{h_i:[n]\to[k]\}$ such that for every set $S\subset [n], |S|\leq k$, there exists some $h_S \in H$ such that $H_S$ is injective on $S$. ...
9
votes
1answer
178 views

Random restrictions and the connection to total influence of Boolean functions

Say we have a Boolean function $f:\{-1,1\}^n\rightarrow \{-1,1\}$ and we apply $\delta$-random restriction on $f$. In addition, say that the decision tree $T$ that computes $f$ shrinks to size $O(1)$ ...
1
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0answers
143 views

Time-inhomogeneous Markov Chains

I'm trying to find out what is known about time-inhomogeneous ergodic Markov Chains where the transition matrix can vary over time. All textbooks and lecture notes I could find initially introduce ...
2
votes
1answer
173 views

Literature for Generalized Load Balancing

i am looking for literature on this kind of problem. $$ \begin{align} \min_x \max_k &\quad \sum_{i,j} x_{ij}c_{ijk}\\ \text{subject to}&\\ &\sum_j x_{ij}=1,&& \forall i\in\mathcal ...
10
votes
0answers
232 views

A bijection between ordered lambda terms and rooted planar maps?

Consider the following recurrence in two parameters $n$ and $k$: \begin{aligned} NF(0,k) &= 0 \\ NF(n,k) &= Neu(n,k) + NF(n-1,k+1) \\ Neu(n,k) &= [n=1 \wedge k=1] + ...
2
votes
1answer
75 views

Upper bound to number of closed itemsets

Given a set $I$ of $n$ items, and a collection $D$ of $m<2^n$ subsets of $I$, a closed itemset is a subset $A$ of $I$ that is contained in strictly more elements of $D$ than any of its proper ...
8
votes
1answer
173 views

Multidimensional arithmetic progression variant

For $\vec{d} \in \mathbb{N}^n$, let $Q(\vec{d}) \subset \mathbb{N}^n$ be the set of vertices of the $n$-dimensional cube scaled in the direction of the $i$-th coordinate by $d_i$, i.e. $Q(\vec{d} = ...
6
votes
3answers
186 views

Partition of $\{(i, j)\ |\ i \neq j \leq n\}$

Consider the set of $n^2 - n$ pairs $\{(i, j)\ |\ i \neq j \leq n\}$. Call a partition $(P_1,\ldots,P_r)$ of this set valid if for each $P_t$, the sets $\{i\ |\ \exists k: (i,k) \in P_t\}$ and ...
0
votes
0answers
105 views

Edges in a graph with girth greater than 4

According to the following paper by Füredi, the maximum number of edges for a graph with $n$ vertices is upper bounded by $O(n^{\frac{3}{2}})$, where the leading co-efficient of the term ...