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★ Matching in hypergraphs - 1710s in new york ..



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★ Matching in hypergraphs

In graph theory, a hypergraph matching is a set of hypergraphs in which every two hypergraphs are disjoint. This is an extension of the concept of matching in the graph.

                                     

1. Definition. (Определение)

Recall that a hypergraph H is a pair of V, E, where V is the set of vertices, and E is the set of subsets of V, called hyperages. Each hyperedge can contain one or more vertexes.

A match in H is a subset M of E such that every two hypercurses e 1 and e 2 in M have an empty intersection and no common vertex.

The matching number of the hypergraph H is the largest matching size in H. it is often denoted V H {\displaystyle \nu H}.

As an example, let V be the set {1.2.3.4.5.6.7}. consider a 3-uniform hypergraph on a V hypergraph in which each hyperedge contains exactly 3 vertices. Let H-3 be a uniform hypergraph with 4 hyperreferences:

{ {1.2.3}, {1.4.5}, {4.5.6}, {2.3.6} }

Then H admits several correspondences of size 2, for example:

{ {1.2.3}, {4.5.6} } { {1.4.5}, {2.3.6} }

However, in any subset of 3 hyperfields, at least two of them intersect, so there is no correspondence of size 3. Hence, the corresponding number H is 2.

                                     

2. Correspondence in the graph as a special case. (Соответствие в графе как частный случай)

A graph without self-loops is just a 2-uniform hypergraph: each edge can be considered as a set of two vertices that it connects. For example, this 2-uniform hypergraph is a graph with 4 vertices {1.2.3.4} and 3 edges:

{ {1.3}, {1.4}, {2.4} }

According to the above definition, a correspondence in a graph is a set of M edges, such that every two edges in M have an empty intersection. This is equivalent to saying that no two edges in M are adjacent to the same vertex, this is exactly the definition of matching in the graph.

                                     

3. Partial approval. (Частичное одобрение)

Fractional matching in a hypergraph is a function that assigns a fraction in each hyperedge, so that for each vertex v in V, the sum of the fractions of hyperedges containing v is no more than 1. a Match is a special case of fractional matching in which all fractions are equal to either 0 or 1. The size of a fractional match is the sum of fractions of all hyperedges.

The fractional correspondence number of a hypergraph H is the largest size of the fractional correspondence in H. it is often denoted by V * H {\displaystyle \nu ^{*}H}.

Since the mapping is a special case of fractional mapping, for each hypergraph H:

compliance is the number of H ≤ fractional matching-a number of H,

In characters:

ν H ≤ ν ∗ H. {\displaystyle \nu H\leq \nu ^{*}H.}

In General, a fractional matching number can be greater than the matching number. The Zoltan Furedi theorem provides upper bounds for the fractional match-numberH / match-numberH ratio:

  • If each hyperedge in H contains at most r vertices, then V * H / V h ≤ r− 1 / r {\displaystyle \nu ^{*}H / \nu H\leq r-1 / r}. In particular, in a simple graph V * H / V H ≤ 3 / 2 {\displaystyle \nu ^{*}H / \nu H\leq 3 / 2}.
  • Inequality is sharp: let H r-R be a uniform finite projective plane. Then V h r = 1 {\displaystyle \nu H_{r}=1}, since every two hyperreferences intersect, and V *H r = r− 1 / r {\displaystyle \nu ^{ * } H_{r}=r-1 / R} by a fractional correspondence that assigns a weight of 1 / r to each hyperemesis, this is a correspondence, since each vertex is contained in R hyperemeses, and its size is r -1 / r, since R 2 - r exist 1 hyperemia. Consequently, the ratio is exactly equal to r -1 / r.
  • If H-R-partite vertices are divided into r parts and each hyperedge contains a vertex from each part, then V * H / V H ≤ r − 1. {\displaystyle \nu ^{*}H / \nu H\leq r-1.} In particular, in the bipartite graph V * H = V H {\displaystyle \nu ^{*}H=\nu H}.
  • The inequality is sharp: let H r be a truncated projective plane of order r -1. Then V H r - = 1 {\displaystyle \nu H_{r -} =1} since every two hyperreferences intersect, and V * H r- = r-1 {\displaystyle \nu ^{*}H_{r -}=r-1} by a fractional correspondence that assigns a weight of 1 / r to each hyperreference, there are r 2 - r hyperreferences.
  • If r is such that r is a uniform finite projective plane that does not exist, for example, r =7, then a stronger inequality holds: ν ∗ H / ν h ≤ r − 1 {\displaystyle \nu ^{*}H / \nu H\leq r-1}.


                                     

4. Perfect match. (Идеальный матч)

A match M is called perfect if each vertex v in V is contained in exactly one hyperedge M. This is a natural extension of the concept of perfect correspondence in the graph.

A fractional correspondence of M is called perfect if for each vertex v in V, the sum of the fractions of hyperedges in M containing v is exactly 1.

Consider a hypergraph H in which each hyperedge contains at most n vertices. If H admits a perfect fractional correspondence, then its fractional correspondence number is at least |V| / n. if each hyperedge in H contains exactly n vertices, then its fractional correspondence number is exactly |V|/ n.this is a generalization of the fact that in a graph the size of the ideal correspondence is |V|/2.

Given a set V of vertices, a set e of subsets V is called balanced if the hypergraph V, E admits a perfect fractional correspondence.

For example, if V = {1.2.3, a, b, c} and E = { {1, a}, {2, a}, {1, b}, {2, b}, {3, c}}, then E is balanced with a perfect fractional correspondence { 1 / 2, 1 / 2, 1 / 2, 1 / 2, 1 }.

There are various sufficient conditions for the existence of an ideal correspondence in a hypergraph:

  • The ideal correspondence in hypergraphs of high degree is sufficient conditions analogous to Diracs theorem on Hamiltonian cycles based on the degree of vertices.
  • Halls theorems for hypergraphs are sufficient conditions analogous to Halls marriage theorem based on sets of neighbors.
                                     

5. Intersecting hypergraph. (Пересекающийся гиперграф)

A hypergraph H = V, E is called intersecting if every two hyperrels in E have a common vertex. In the intersecting graph, there is no match with two or more hyperedges, so V H = 1 {\displaystyle \nu H=1}.

                                     

6. Balanced set-family. (Сбалансированный набор-семья)

A family of sets E over a basic set V is called balanced with respect to V If the hypergraph H = V, E admits a perfect fractional correspondence.

For example, consider the set of vertices V = {1.2.3, a, b, c} and the set of edges E = {1-a, 2-a, 1-b, 2-b, 3-c}. E is balanced because there is a perfect fractional correspondence with weights {1 / 2, 1 / 2, 1 / 2, 1 / 2, 1}.

                                     

7. The calculation for the maximum matching. (Расчет для максимального соответствия)

The problem of finding the correspondence of the maximum cardinality in a hypergraph, thus calculating V H {\displaystyle \nu H}, is NP-complete even for 3-homogeneous hypergraphs see 3-dimensional matting. This is in contrast to the case of simple 2-homogeneous graphs, in which the maximum power correspondence can be calculated in polynomial time.

                                     

8. Compliance and coverage. (Соответствие требованиям и охват)

A vertex cover in a hypergraph H = V, E is a subset T of V such that every hyperedge in E contains at least one vertex it is also called a transverse or impact set and is equivalent to a cover set. This is a generalization of the concept of vertex cover in a graph.

The vertex cover number of a hypergraph H is the smallest vertex cover size in H. it is often denoted τ H {\displaystyle \tau H}, for transverse.

A fractional vertex cover is a function that assigns a weight to each vertex in V such that for each hyperedge e in E, the sum of the vertex shares in e is at least 1. a Vertex cover is a special case of a fractional vertex cover in which all weights are either 0 or 1. The size of a fractional vertex cover is equal to the sum of fractions of all vertexes.

The fractional number of the vertex cover of a hypergraph H is the smallest size of the fractional vertex cover in H. it is often denoted τ ∗ H {\displaystyle \tau ^{*}H}.

Since vertex cover is a special case of fractional vertex cover, for every hypergraph H:

fractional vertex-cover number H ≤ vertex-cover number H.

The duality of linear programming implies that for every hypergraph H:

fractional matching number H = fractional vertex covering number H.

Hence, for every hypergraph H:

ν H ≤ ν ∗ H = τ ∗ H ≤ τ H {\displaystyle \nu H\leq \nu ^{*}H=\tau ^{*}H\leq \tau H}

If the size of each hyperge in H is at most r, then the Union of all hypergeases in maximum matching is a vertex cover. if there were an uncovered hyperge, we could add it to the match. Therefore:

τ H ≤ r ν H {\displaystyle \tau H\leq r\cdot \nu H}.

This inequality is a hard one: equality holds, for example, when V contains vertices r V H r-1 {\displaystyle r \ cdot \nu H r-1}, And E contains all subsets of vertices r.

However, in General case τ ∗ H <, r V H {\displaystyle \tau ^{*}H

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Pino - logical board game which is based on tactics and strategy. In general this is a remix of chess, checkers and corners. The game develops imagination, concentration, teaches how to solve tasks, plan their own actions and of course to think logically. It does not matter how much pieces you have, the main thing is how they are placement!

online intellectual game →