In the mathematical field of graph theory, Hall-type theorems for hypergraphs are several generalizations of Hall's marriage theorem from graphs to hypergraphs. Such theorems were proved by Ofra Kessler, Ron Aharoni, Penny Haxell, Roy Meshulam, and others.
Preliminaries Hall's marriage theorem provides a condition guaranteeing that a bipartite graph admits a perfect matching, or - more generally - a matching that saturates all vertices of . The condition involves the number of neighbors of subsets of . Generalizing Hall's theorem to hypergraphs requires a generalization of the concepts of bipartiteness, perfect matching, and neighbors.
1. Bipartiteness: The notion of a bipartiteness can be extended to hypergraphs in many ways (see bipartite hypergraph). Here we define a hypergraph as bipartite if it is exactly 2-colorable, i.e., its vertices can be 2-colored such that each hyperedge contains exactly one yellow vertex. In other words, can be partitioned into two sets and , such that each hyperedge contains exactly one vertex of . A bipartite graph is a special case in which each edge contains exactly one vertex of and also exactly one vertex of ; in a bipartite hypergraph, each hyperedge contains exactly one vertex of but may contain zero or more vertices of . For example, the hypergraph with and is bipartite with and
2. Perfect matching: A matching in a hypergraph is a subset of , such that every two hyperedges of are disjoint. If is bipartite with parts and , then the size of each matching is obviously at most . A matching is called -perfect (or -saturating) if its size is exactly . In other words: every vertex of appears in exactly one hyperedge of . This definition reduces to the standard definition of a -perfect matching in a bipartite graph.
3. Neighbors: Given a bipartite hypergraph and a subset of , the neighbors of are the subsets of that share hyperedges with vertices of . Formally:
For example, in the hypergraph from point 1, we have: and and Note that, in a bipartite graph, each neighbor is a singleton - the neighbors are just the vertices of that are adjacent to one or more vertices of . In a bipartite hypergraph, each neighbor is a set - the neighbors are the subsets of that are "adjacent" to one or more vertices of .
Since contains only subsets of , one can define a hypergraph in which the vertex set is and the edge set is . We call it the neighborhood-hypergraph of and denote it:
Note that, if is a simple bipartite graph, the neighborhood-hypergraph of every contains just the neighbors of in , each of which with a self-loop.
Insufficiency of Hall's condition Hall's condition requires that, for each subset of , the set of neighbors of is sufficiently large. With hypergraphs this condition is insufficient. For example, consider the tripartite hypergraph with edges:{ {1, a, A}, {2, a, B} }Let Every vertex in has a neighbor, and itself has two neighbors: But there is no -perfect matching since both edges overlap.
One could try to fix it by requiring that contain at least disjoint edges, rather than just edges. In other words: should contain a matching of size at least . The largest size of a matching in a hypergraph is called its matching number and denoted by (thus admits a -perfect matching if and only if ). However, this fix is insufficient, as shown by the following tripartite hypergraph:{ {1, a, A}, {1, b, B}, {2, a, B}, {2, b, A} }Let Again every vertex in has a neighbor, and itself has four neighbors: Moreover, since admits a matching of size 2, e.g. However, H does not admit a -perfect matching, since every hyperedge that contains 1 overlaps every hyperedge that contains 2.
Thus, to guarantee a perfect matching, a stronger condition is needed. Various such conditions have been suggested.
Aharoni's conditions: largest matching
Let be a bipartite hypergraph (as defined in 1. above), in which the size of every hyperedge is exactly , for some integer . Suppose that, for every subset of , the following inequality holds:
In words: the neighborhood-hypergraph of admits a matching larger than . Then admits a -perfect matching (as defined in 2. above).
This was first conjectured by Aharoni. It was proved with Ofra Kessler for bipartite hypergraphs in which and for . It was later proved for all -uniform hypergraphs.
In simple graphs For a bipartite simple graph , and Aharoni's condition becomes:
Moreover, the neighborhood-hypergraph (as defined in 3. above) contains just singletons - a singleton for every neighbor of . Since singletons do not intersect, the entire set of singletons is a matching. Hence, the number of neighbors of . Thus, Aharoni's condition becomes, for every subset of :
This is exactly Hall's marriage condition.
Tightness The following example shows that the factor cannot be improved. Choose some integer . Let be the following -uniform bipartite hypergraph:
is the union of (where is the set of hyperedges containing vertex ), and: For each in contains disjoint hyperedges of size : contains hyperedges of size :
Note that edge in meets all edges in .
This does not admit a -perfect matching, since every hyperedge that contains intersects all hyperedges in for some .
However, every subset of satisfies the following inequality
since contains at least hyperedges, and they are all disjoint.
Fractional matchings The largest size of a fractional matching in is denoted by . Clearly . Suppose that, for every subset of , the following weaker inequality holds:
It was conjectured that in this case, too, admits a -perfect matching. This stronger conjecture was proved for bipartite hypergraphs in which .
Later it was proved that, if the above condition holds, then admits a -perfect fractional matching, i.e., . This is weaker than having a -perfect matching, which is equivalent to .
Haxell's condition: smallest transversal
A transversal (also called vertex-cover or hitting-set) in a hypergraph is a subset of such that every hyperedge in contains at least one vertex of . The smallest size of a transversal in is denoted by .
Let be a bipartite hypergraph in which the size of every hyperedge is at most , for some integer . Suppose that, for every subset of , the following inequality holds:
In words: the neighborhood-hypergraph of has no transversal of size or less.
Then, admits a -perfect matching (as defined in 2. above).
In simple graphs For a bipartite simple graph so , and Haxell's condition becomes:
Moreover, the neighborhood-hypergraph (as defined in 3. above) contains just singletons - a singleton for every neighbor of . In a hypergraph of singletons, a transversal must contain all vertices. Hence, the number of neighbors of . Thus, Haxell's condition becomes, for every subset of :
This is exactly Hall's marriage condition. Thus, Haxell's theorem implies Hall's marriage theorem for bipartite simple graphs.
Tightness The following example shows that the factor cannot be improved. Let be an -uniform bipartite hypergraph with:
[so ]. where: [so contains hyperedges]. for [so contains hyperedges].
This does not admit a -perfect matching, since every hyperedge that contains 0 intersects every hyperedge that contains 1.
However, every subset of satisfies the following inequality:
It is only slightly weaker (by 1) than required by Haxell's theorem. To verify this, it is sufficient to check the subset , since it is the only subset for which the right-hand side is larger than 0. The neighborhood-hypergraph of is where:
for
One can visualize the vertices of as arranged on an grid. The hyperedges of are the rows. The hyperedges of are the selections of a single element in each row and each column. To cover the hyperedges of we need vertices - one vertex in each row. Since all columns are symmetric in the construction, we can assume that we take all the vertices in column 1 (i.e., for each in . Now, since contains all columns, we need at least additional vertices - one vertex for each column All in all, each transversal requires at least vertices.
Algorithms Haxell's proof is not constructive. However, Chidambaram Annamalai proved that a perfect matching can be found efficiently under a slightly stronger condition.
For every fixed choice of and , there exists an algorithm that finds a -perfect matching in every -uniform bipartite hypergraph satisfying, for every subset of :
In fact, in any -uniform hypergraph, the algorithm finds either a -perfect matching, or a subset violating the above inequality.
The algorithm runs in time polynomial in the size of , but exponential in and .
It is an open question whether there exists an algorithm with run-time polynomial in either or (or both).
Similar algorithms have been applied for solving problems of fair item allocation, in particular the santa-claus problem.
Aharoni–Haxell conditions: smallest pinning sets We say that a set of edges pins another set of edges if every edge in intersects some edge in . The width of a hypergraph , denoted , is the smallest size of a subset of that pins . The matching width of a hypergraph , denoted , is the maximum, over all matchings in , of the minimum size of a subset of that pins . Since contains all matchings in , the width of H is obviously at least as large as the matching-width of .
Aharoni and Haxell proved the following condition:Let be a bipartite hypergraph. Suppose that, for every subset of , the following inequality holds:[in other words: contains a matching such that at least disjoint edges from are required for pinning ]. Then, admits a -perfect matching.They later extended this condition in several ways, which were later extended by Meshulam as follows:Let be a bipartite hypergraph. Suppose that, for every subset of , at least one of the following conditions hold: or
Then, admits a -perfect matching.
In simple graphs In a bipartite simple graph, the neighborhood-hypergraph contains just singletons - a singleton for every neighbor of . Since singletons do not intersect, the entire set of neighbors is a matching, and its only pinning-set is the set itself, i.e., the matching-width of is , and its width is the same:
Thus, both the above conditions are equivalent to Hall's marriage condition.
Examples We consider several bipartite graphs with and The Aharoni–Haxell condition trivially holds for the empty set. It holds for subsets of size 1 if and only if each vertex in is contained in at least one edge, which is easy to check. It remains to check the subset itself.
Here Its matching-width is at least 2, since it contains a matching of size 2, e.g. which cannot be pinned by any single edge from . Indeed, H admits a -perfect matching, e.g. Here Its matching-width is 1: it contains a matching of size 2, e.g. but this matching can be pinned by a single edge, e.g. The other matching of size 2 is but it too can be pinned by the single edge While is larger than in example 1, its matching-width is smaller - in particular, it is less than . Hence, the Aharoni–Haxell sufficient condition is not satisfied. Indeed, does not admit a -perfect matching. Here, as in the previous example, so the Aharoni–Haxell sufficient condition is violated. The width of is 2, since it is pinned e.g. by the set so Meshulam's weaker condition is violated too. However, this does admit a -perfect matching, e.g. which shows that these conditions are not necessary.
Set-family formulation Consider a bipartite hypergraph where The Hall-type theorems do not care about the set itself - they only care about the neighbors of elements of . Therefore can be represented as a collection of families of sets where for each in , the set-family of neighbors of . For every subset of , the set-family is the union of the set-families for in . A perfect matching in is a set-family of size , where for each in , the set-family is represented by a set in , and the representative sets are pairwise-disjoint.