Matching in hypergraphs
In graph theory, a matching in a hypergraph is a set of hyperedges, in which every two hyperedges are disjoint. It is an extension of the notion of matching in a graph.
Definition
Recall that a hypergraph H is a pair, where V is a set of vertices and E is a set of subsets of V called hyperedges. Each hyperedge may contain one or more vertices.A matching in H is a subset M of E, such that every two hyperedges e1 and e2 in M have an empty intersection.
The matching number of a hypergraph H is the largest size of a matching in H. It is often denoted by.
As an example, let V be the set. consider a 3-uniform hypergraph on V. Let H be a 3-uniform hypergraph with 4 hyperedges:
Then H admits several matchings of size 2, for example:
However, in any subset of 3 hyperedges, at least two of them intersect, so there is no matching of size 3. Hence, the matching number of H is 2.
Matching in a graph as a special case
A graph without self-loops is just a 2-uniform hypergraph: each edge can be considered as a set of the two vertices that it connects. For example, this 2-uniform hypergraph represents a graph with 4 vertices and 3 edges:By the above definition, a matching in a graph is a set M of edges, such that each 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 a matching in a graph.
Fractional matching
A fractional matching in a hypergraph is a function that assigns a fraction in to each hyperedge, such that for every vertex v in V, the sum of fractions of hyperedges containing v is at most 1. A matching is a special case of a fractional matching in which all fractions are either 0 or 1. The size of a fractional matching is the sum of fractions of all hyperedges.The fractional matching number of a hypergraph H is the largest size of a fractional matching in H. It is often denoted by.
Since a matching is a special case of a fractional matching, for every hypergraph H:
matching-number ≤ fractional-matching-number ;In symbols:
In general, the fractional matching number may be larger than the matching number. A theorem of Zoltán Füredi provides upper bounds on the ratio fractional-matching-number / matching-number:
- If each hyperedge in H contains at most r vertices, then. In particular, in a simple graph,.
- * The inequality is sharp: Let Hr be the r-uniform finite projective plane. Then since every two hyperedges intersect, and by the fractional matching that assigns a weight of 1/r to each hyperedge. Therefore the ratio is exactly r-1+1/r.
- If r is such that the r-uniform finite projective plane does not exist, then a stronger inequality holds:.
- If H is r-partite, then In particular, in a bipartite graph,.
- * The inequality is sharp: Let Hr- be the truncated projective plane of order r-1. Then since every two hyperedges intersect, and by the fractional matching that assigns a weight of 1/r to each hyperedge.
Perfect matching
A fractional matching M is called perfect if for every vertex v in V, the sum of 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 matching, then its fractional matching number is at least |V|/n. If each hyperedge in H contains exactly n vertices, then its fractional matching number is at exactly |V|/n. This is a generalization of the fact that, in a graph, the size of a perfect matching is |V|/2.
Given a set V of vertices, a collection E of subsets of V is called balanced if the hypergraph admits a perfect fractional matching.
For example, if V = and E =, then E is balanced, with the perfect fractional matching.
There are various sufficient conditions for the existence of a perfect matching in a hypergraph:
- Hall-type theorems for hypergraphs - presents sufficient conditions analogous to Hall's marriage theorem, based on sets of neighbors.
- Perfect matching in high-degree hypergraphs - presents sufficient conditions analogous to Dirac's theorem on Hamiltonian cycles, based on degree of vertices.
- Keevash and Mycroft developed a geometric theory for hypergraph matching.
Intersecting hypergraph
Balanced set-family
A set-family E over a ground set V is called balanced if the hypergraph H = admits a perfect fractional matching.For example, consider the vertex set V = and the edge set E =. E is balanced, since there is a perfect fractional matching with weights.
Computing a maximum matching
The problem of finding a maximum-cardinality matching in a hypergraph, thus calculating, is NP-complete even for 3-uniform hypergraphs. This is in contrast to the case of simple graphs in which computing a maximum-cardinality matching can be done in polynomial time.Matching and covering
A vertex-cover in a hypergraph H = is a subset T of V, such that every hyperedge in E contains at least one vertex of T. It is a generalization of the notion of a vertex cover in a graph.The vertex-cover number of a hypergraph H is the smallest size of a vertex cover in H. It is often denoted by, for transversal.
A fractional vertex-cover is a function assigning a weight to each vertex in V, such that for every hyperedge e in E, the sum of fractions of vertices 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 the sum of fractions of all vertices.
The fractional vertex-cover number of a hypergraph H is the smallest size of a fractional vertex-cover in H. It is often denoted by.
Since a vertex-cover is a special case of a fractional vertex-cover, for every hypergraph H:
fractional-vertex-cover-number ≤ vertex-cover-number.Linear programming duality implies that, for every hypergraph H:
fractional-matching-number = fractional-vertex-cover-number.Hence, for every hypergraph H:If the size of each hyperedge in H is at most r then the union of all hyperedges in a maximum matching is a vertex-cover. Therefore:
.This inequality is tight: equality holds, for example, when V contains vertices and E contains all subsets of r vertices.
However, in general, since ; see Fractional matching above.
Ryser's conjecture says that, in every r-partite r-uniform hypergraph:
.Some special cases of the conjecture have been proved; see Ryser's conjecture.
Kőnig's property
A hypergraph has the Kőnig property if its maximum matching number equals its minimum vertex-cover number, namely if. The Kőnig-Egerváry theorem shows that every bipartite graph has the Kőnig property. To extend this theorem to hypergraphs, we need to extend the notion of bipartiteness to hypergraphs.A natural generalization is as follows. A hypergraph is called 2-colorable if its vertices can be 2-colored so that every hyperedge contains at least one vertex of each color. An alternative term is Property B. A simple graph is bipartite iff it is 2-colorable. However, there are 2-colorable hypergraphs without Kőnig's property. For example, consider the hypergraph with V = with E = all triplets =. It is 2-colorable, for example, we can color blue and white. However, its matching number is 1 and its vertex-cover number is 2.
A stronger generalization is as follows. Given a hypergraph H = and a subset V of V, the restriction of H to V' is the hypergraph whose vertices are V, and for every hyperedge in e in E that intersects V', it has a hyperedge e
A simple graph is bipartite iff it has no odd-length cycles. Similarly, a hypergraph is balanced iff it has no odd-length circuits. A circuit of length k in a hypergraph is an alternating sequence, where the vi are distinct vertices and the ei are distinct hyperedges, and each hyperedge contains the vertex to its left and the vertex to its right. The circuit is called unbalanced if each hyperedge contains no other vertices in the circuit. Claude Berge proved that a hypergraph is balanced iff it does not contain an unbalanced odd-length circuit. Every balanced hypergraph has Kőnig's property.
The following are equivalent:
- Every partial hypergraph of H has the Kőnig property.
- Every partial hypergraph of H has the property that its maximum degree equals its minimum edge coloring number.
- H has the Helly property, and the intersection graph of H is a perfect graph.
Matching and packing
The problem of finding a maximum vertex-packing in a graph is equivalent to the problem of finding a maximum matching in a hypergraph:
- Given a hypergraph H =, define its intersection graph Int as the simple graph whose vertices are E and whose edges are pairs such that e1,e2 have a vertex in common. Then every matching in H is a vertex-packing in Int and vice-versa.
- Given a graph G =, define its star hypergraph St as the hypergraph whose vertices are E
' and whose hyperedges are the stars of the vertices of G. Then every vertex-packing in G is a matching in St and vice-versa. - Alternatively, given
a g raph G =, define its clique hypergraph Cl as the hypergraph who se vertices are the cliques of G, and for each vertex v ' in V ', there is a hypere dge in Cl containing all cliques in G that contain v '. Then aga in, every vertex-packing in G is a matching in Cl and vice-versa. Note that Cl cannot be constructed from G in polynomial time, so it cannot be used as a reduction for proving NP-hardness. But it has some theoretical uses.