Packing in a hypergraph explained
In mathematics, a packing in a hypergraph is a partition of the set of the hypergraph's edges into a number of disjoint subsets such that no pair of edges in each subset share any vertex. There are two famous algorithms to achieve asymptotically optimal packing in k-uniform hypergraphs. One of them is a random greedy algorithm which was proposed by Joel Spencer. He used a branching process to formally prove the optimal achievable bound under some side conditions. The other algorithm is called the Rödl nibble and was proposed by Vojtěch Rödl et al. They showed that the achievable packing by the Rödl nibble is in some sense close to that of the random greedy algorithm.
History
The problem of finding the number of such subsets in a k-uniform hypergraph was originally motivated through a conjecture by Paul Erdős and Haim Hanani in 1963. Vojtěch Rödl proved their conjecture asymptotically under certain conditions in 1985. Pippenger and Joel Spencer generalized Rödl's results using a random greedy algorithm in 1989.
Definition and terminology
In the following definitions, the hypergraph is denoted by H=(V,E). H is called a k-uniform hypergraph if every edge in E consists of exactly k vertices.
is a
hypergraph packing if it is a subset of edges in H such that there is no pair of distinct edges with a common vertex.
is a
(
,
)-good hypergraph if there exists a
such that for all
and
and both of the following conditions hold.
D(1-\epsilon)\leqdeg(x)\leqD(1+\epsilon)
where the degree
of a vertex
is the number of edges that contain
and the
codegree
of two distinct vertices
and
is the number of edges that contain both vertices.
Theorem
There exists an asymptotic packing P of size at least
for a
-uniform hypergraph under the following two conditions,
- All vertices have the degree of
in which
tends to infinity.
- For every pair of vertices shares only
common edges.
where
is the total number of vertices. This result was shown by Pippenger and was later proved by Joel Spencer. To address the asymptotic hypergraph packing problem, Joel Spencer proposed a random greedy algorithm. In this algorithm, a branching process is used as the basis and it was shown that it almost always achieves an asymptotically optimal packing under the above side conditions.
Asymptotic packing algorithms
There are two famous algorithms for asymptotic packing of k-uniform hypergraphs: the random greedy algorithm via branching process, and the Rödl nibble.
Random greedy algorithm via branching process
Every edge
is independently and uniformly assigned a distinct real "birthtime"
. The edges are taken one by one in the order of their birthtimes. The edge
is accepted and included in
if it does not overlap any previously accepted edges. Obviously, the subset
is a packing and it can be shown that its size is
almost surely. To show that, let stop the process of adding new edges at time
. For an arbitrary
, pick
such that for any
-good hypergraph
where
denotes the probability of vertex
survival (a vertex survives if it is not in any edges in
) until time
. Obviously, in such a situation the expected number of
surviving at time
is less than
. As a result, the probability of
surviving being less than
is higher than
. In other words,
must include at least
vertices which means that
.
To complete the proof, it must be shown that
. For that, the asymptotic behavior of
surviving is modeled by a continuous branching process. Fix
and begin with Eve with the birthdate of
. Assume time goes backward so Eve gives birth in the interval of
with a unit density
Poisson distribution. The probability of Eve having
birth is
. By conditioning on
the birthtimes
are independently and uniformly distributed on
. Every birth given by Eve consists of
offspring all with the same birth time say
. The process is iterated for each offspring. It can be shown that for all
there exists a
so that with a probability higher than
, Eve has at most
descendants.
A rooted tree with the notions of parent, child, root, birthorder and wombmate shall be called a broodtree. Given a finite broodtree
we say for each vertex that it survives or dies. A childless vertex survives. A vertex dies if and only if it has at least one brood all of whom survive. Let
denote the probability that Eve survives in the broodtree
given by the above process. The objective is to show
and then for any fixed
, it can be shown that
. These two relations complete our argument.
To show
, let
. For
small,
f(c+\Deltac)-f(c) ≈ -(\Deltac)f(c)Q+1
as, roughly, an Eve starting at time
might have a birth in time interval
all of whose children survive while Eve has no births in
all of whose children survive. Letting
yields the differential equation
. The initial value
gives a unique solution
. Note that indeed
.
To prove
, consider a procedure we call History which either aborts or produces a broodtree. History contains a set
of vertices, initially
.
will have a broodtree structure with
the root. The
are either processed or unprocessed,
is initially unprocessed. To each
is assigned a birthtime
, we initialize
. History is to take an unprocessed
and process it as follows. For the value of all
with
but with no
that has already been processed, if either some
has
and
with
or some
have
with
and
, then History is aborted. Otherwise for each
with
add all
to
as wombmates with parent
and common birthdate
. Now
is considered processed. History halts, if not aborted, when all
are processed. If History does not abort then root
survives broodtree
if and only if
survives at time
. For a fixed broodtree, let
denote the probability that the branching process yields broodtree
. Then the probability that History does not abort is
. By the finiteness of the branching process,
, the summation over all broodtrees
and History does not abort. The
distribution of its broodtree approaches the branching process distribution. Thus
.
The Rödl nibble
In 1985, Rödl proved Paul Erdős’s conjecture by a method called the Rödl nibble. Rödl's result can be formulated in form of either packing or covering problem. For
the covering number denoted by
shows the minimal size of a family
of
-element subsets of
which have the property that every
-element set is contained in at least one
.
Paul Erdős et al. conjecture was
.
where
. This conjecture roughly means that a tactical configuration is asymptotically achievable. One may similarly define the packing number
as the maximal size of a family
of
-element subsets of
having the property that every
-element set is contained in at most one
.
Packing under the stronger condition
In 1997, Noga Alon, Jeong Han Kim, and Joel Spencer also supply a good bound for
under the stronger codegree condition that every distinct pair
has at most one edge in common.
For a k-uniform, D-regular hypergraph on n vertices, if k > 3, there exists a packing P covering all vertices but at most
. If
k = 3 there exists a packing
P covering all vertices but at most
.
This bound is desirable in various applications, such as Steiner triple system.A Steiner Triple System is a 3-uniform, simple hypergraph in which every pair of vertices is contained in precisely one edge. Since a Steiner Triple System is clearly d=(n-1)/2-regular, the above bound supplies the following asymptotic improvement.
Any Steiner Triple System on n vertices contains a packing covering all vertices but at most
.
This has subsequently improved to
[1] and
[2] See also
References
Notes and References
- Keevash . Peter . Peter Keevash . Pokrovskiy . Alexey . Sudakov . Benny . Benny Sudakov . Yepremyan . Liana . 2022-04-15 . New bounds for Ryser's conjecture and related problems . Transactions of the American Mathematical Society, Series B . en . 9 . 8 . 288–321 . 10.1090/btran/92 . free . 2330-0000. 20.500.11850/592212 . free .
- Montgomery . Richard . 2023 . A proof of the Ryser-Brualdi-Stein conjecture for large even n . math.CO . 2310.19779.