A hypergraph H is a pair of sets (V, E), where V is the set...

A hypergraph $H$ is a pair of sets $(V, E)$, where $V$ is the set of vertices and $E$ is the set of hyperedges. Every hyperedge in $E$ is a subset of $V$. In particular, an $r$-uniform hypergraph is one where the size of each edge is $r$. For example, a 2 -uniform hypergraph is just a standard graph. A dominating set in a hypergraph $H$ is a set of vertices $S \subset V$ such that $e \cap S \neq \emptyset$ for every edge $e \in E$. That is, $S$ hits every edge of the hypergraph. Let $H=(V, E)$ be an $r$-uniform hypergraph with $n$ vertices and $m$ edges. Show that there is a dominating set of size at most $n p+(1-p)^{r} m$ for every real number $0 \leq$ $p \leq 1$. Also, show that there is a dominating set of size at most $(m+n \ln r) / r$.

A hypergraph $H$ is a pair of sets $(V, E)$, where $V$ is the set of vertices and $E$ is the set of hyperedges. Every hyperedge in $E$ is a subset of $V$. In particular, an $r$-uniform hypergraph is one where the size of each edge is $r$. For example, a 2 -uniform hypergraph is just a standard graph. A dominating set in a hypergraph $H$ is a set of vertices $S \subset V$ such that $e \cap S \neq \emptyset$ for every edge $e \in E$. That is, $S$ hits every edge of the hypergraph.

Let $H=(V, E)$ be an $r$-uniform hypergraph with $n$ vertices and $m$ edges. Show that there is a dominating set of size at most $n p+(1-p)^{r} m$ for every real number $0 \leq$ $p \leq 1$. Also, show that there is a dominating set of size at most $(m+n \ln r) / r$.

Key Concepts

Hypergraph

A hypergraph is a generalization of a graph where each edge, called a hyperedge, can connect any number of vertices rather than just two. This concept broadens the scope of graph theory and allows the modeling of more complex relationships between entities.

Uniform Hypergraph

An r-uniform hypergraph is a special type of hypergraph where every hyperedge has exactly r vertices. This property simplifies many combinatorial arguments by ensuring that every hyperedge has a fixed size, which can be particularly useful in averaging and counting techniques.

Dominating Set

In the context of hypergraphs, a dominating set is a subset of vertices such that every hyperedge contains at least one vertex from this set. This concept is important because it represents a way of 'covering' or 'hitting' all hyperedges, and finding a small dominating set is a common problem in combinatorial optimization and theoretical computer science.

Probabilistic Method

The probabilistic method is a powerful technique in combinatorics and discrete mathematics used to prove the existence of a structure by showing that, under a random construction, the probability of the structure having a desired property is positive. This method often involves random selections and expectations to bound the size or number of certain configurations, such as dominating sets.

Expectation and Parameter Optimization

In analyses using the probabilistic method, expected values are computed to derive average-case bounds. By selecting vertices independently with a certain probability and then optimizing over this parameter, one can obtain tight bounds for the size of the dominating set. This involves balancing the probability of covering hyperedges and the cost in terms of the number of vertices added.

Randomized Construction

The randomized construction involves creating a set by independently choosing each vertex with a fixed probability. Afterwards, any hyperedge that remains uncovered is 'fixed' by adding an arbitrary vertex from the hyperedge to the dominating set. This two-step approach allows obtaining an upper bound on the size of the dominating set through expected values, showcasing the interplay between randomness and combinatorial structure.

Transcript

00:04 We were asked to prove a theorem of oistine or, i think that's how he pronounced his name.

00:16 So we're given that g, which is the pair of ve, is a bipartite graph, and bipartition is v1, v2, and that a is going to be a subset of set v1.

00:50 We're asked to show that the maximum number of vertices of v1, that are the endpoints of a matching, equals the order of v1 minus the maximum for all subsets a of v of v of d of a, where defecency of a is the order of a minus the number of neighbors of the subset a.

01:32 So to prove this, let b be the maximum for all subsets a of v, this should in particular be, say, v1, of the deficiency of a, and let capital b be the subset of v1, such that b is equal to the deficiency of a, and let capital b be the subset of v1 such that b is equal to the deficiency of capital b.

02:57 And so it follows that b is equal to, by definition of deficiency, the magnitude of b minus the number of neighbors of the subset b.

03:19 And since nb is the number of neighbors of b, we have that a matching has at most nb edges with endpoints in b.

04:06 Because if we had more than nb edges with end points in b, well, that would be impossible, because then we would have more edges than we'd have neighbors.

04:32 And we have that any matching can have at most v1 minus b edges, with their end points in in the set v1 minus b, since v1 is the number of elements in v1 and b is the number of elements in b.

05:39 And so the total number of edges, this is going to be the number of neighbors of b plus and then v1 minus b.

06:16 And we have that doing some rearranging.

06:22 This is equal to v1 plus little b.

06:29 The deficiency of matrix b should be v1 minus b, my mistake.

06:49 And so it follows that a matching in g has at most v1.

06:54 Minus b edges.

07:17 And so the matching, the edges of a matching can touch at most v1 minus b vertices v1...

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