# Questions tagged [co.combinatorics]

Questions related to combinatorics and discrete mathematical structures

155 questions
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### Regularity Lemma for Sparse Graphs

Szemeredi's Regularity Lemma says that every dense graph can be approximated as a union of $O(1)$ many bipartite expander graphs. More accurately, there's a partition of most vertices into $O(1)$ sets ...
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### How good is greedy in average?

Given a family ${\cal F}\subset 2^E$ of (feasible solutions), the maximization problem on ${\cal F}$ is, for every weighting $x:E\to \{0,1,\ldots\}$ of ground elements, to compute the maximum weight ...
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### 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 ...
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### Bipartite vertex separator

Are there any common approaches for finding a vertex separator in a bipartite graph $G = (V_1, V_2, E)$ where the selected vertices are constrained to come from one partition of the graph? I have a ...
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### k-uniform k-partite hypergraph matching in polynomial time

I have what seems like an elementary question, but google didn't throw up any answers for it. I would appreciate any pointers users here may provide. Please note that I have also asked this question ...
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### Fair and “robust” fallback permutations

The following is a fun problem we stumbled into today. We have work we wish to distribute on machines $1..n$. Each piece of work is given a list of machines to try, in order. If any machine fails, ...
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### More elementary proof of coloring theorem for d x d^2 rectangles

The following is known: For all $c$, for all $c$-colorings of $N\times N$ there exists a $d \times d^2$ rectangle ($d \ge 2$) such that all four corners are the same color. The proof uses the Poly-...
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### Decomposition based on approximate separators in graph

Suppose I want to find vertex subset $S$ of graph $G=(V,E)$ such that any simple closed walk that visits vertices both in $S$ and in $V\backslash S$ has length $\ge g$ The idea is to relax ...
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### Constructing a large bit-vector set with the following property

I would like to construct a set $S\subseteq\{0,1\}^{2n}$ that satisfy the property: $$\forall x\neq y\in S\ \ \exists k\in [n]:\forall i,j\in[n], \sum_{t=k+i}^{2n}x_t\neq\sum_{t=k+j}^{2n}y_t$$ In ...
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### 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 $q(q+1)/2 + (q-1)/2$ ...
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### 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. M^+=\max\limits_{(U_1,U_2)}\left\{\sum_{e=\{u,v\}\...
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### 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 $S$. ...
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### 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 ...
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### Detailed Materials on Sunflowers

Jukna's book Extremal Combinatorics explains basic results about sunflowers. Is there a more thorough survey of various sunflower results?
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### Sampling small separators

Is there a tractable way to define an approximate distribution over small vertex separators of the graph? I'm looking for something along the lines of [Bayati,2008], a way to turn a single "this is a ...
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### 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-...
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 ...
### How to construct an $(n,m,k)$ separating set''?
This problem is probably known under some other name, if anyone has seen it before, a reference will be great. Given $n,m,k$ (for $m,k\ll n$), a $(n,m,k)$ separating set is a set of $n$-sized binary ...