Perfect fractional matchings in k-out hypergraphs

Extending the notion of (random) $k$-out graphs, we consider when the $k$-out hypergraph is likely to have a perfect fractional matching. In particular, we show that for each $r$ there is a $k=k(r)$ such that the $k$-out $r$-uniform hypergraph on $n$ vertices has a perfect fractional matching with high probability (i.e., with probability tending to $1$ as $n\to \infty$) and prove an analogous result for $r$-uniform $r$-partite hypergraphs. This is based on a new notion of hypergraph expansion and the observation that sufficiently expansive hypergraphs admit perfect fractional matchings. As a further application, we give a short proof of a stopping-time result originally due to Krivelevich.


Introduction
Hypergraphs constitute a far-reaching generalization of graphs and a basic combinatorial construct but are notoriously difficult to work with. A hypergraph is a collection H of subsets ("edges") of a set V of "vertices." Such an H is r-uniform (or an r-graph) if each edge has cardinality r (so 2-graphs are graphs). A perfect matching in a hypergraph is a collection of edges partitioning the vertex set. For any r > 2, deciding whether an r-graph has a perfect matching is an NP-complete problem [17]; so instances of the problem tend to be both interesting and difficult. Of particular interest here has been trying to understand conditions under which a random hypergraph is likely to have a perfect matching. The most natural model of a random r-graph is the "Erdős-Rényi" model, in which each r-set is included in H with probability p, independent of other choices. One is then interested in the "threshold," roughly, the order of magnitude of p = p r (n) required to make a perfect matching likely. Here the graph case was settled by Erdős and Rényi [7,8], but for r > 2 the problem-which became known as Shamir's Problem following [6]-remained open until [16]. In each case, the obvious obstruction to containing a perfect matching is existence of an isolated vertex (that is, a vertex contained in no edges), and a natural guess is that this is the main obstruction. A literal form of this assertion-the stopping time version-says that if we choose random edges sequentially, each uniform from those as yet unchosen, then we w.h.p. 1 have a perfect matching as soon as all vertices are covered. This nice behavior does hold for graphs [3], but for hypergraphs remains conjectural (though at least the value it suggests for the threshold is correct).
An interesting point here is that taking p large enough to avoid isolated vertices produces many more edges than other considerations-e.g., wanting a large expected number of perfect matchings-suggest. This has been one motivation for the substantial body of work on models of random graphs in which isolated vertices are automatically avoided, notably random regular graphs (e.g., [22]) and the k-out model. The generalization of the latter to hypergraphs, which we now introduce, will be our main focus here.
The k-out model. For a ("host") hypergraph H on V , H(k-out) is the random subhypergraph ∪ v∈V E v , where E v is chosen uniformly from the k-subsets of H v := {A ∈ H : v ∈ A} (or-but we won't see this- The k-out model for H = K n,n (the complete bipartite graph) was introduced by Walkup [21], who showed that w.h.p. K n,n (2-out) is Hamiltonian, so in particular contains a perfect matching, and Frieze [12] proved the nonbipartite counterpart of the matching result, showing that K 2n (2-out) has a perfect matching w.h.p. (Hamiltonicity in the latter case turned out to be more challenging; it was studied in [9,13,4] and finally resolved by Bohman and Frieze [2], who proved K n (3-out) is Hamiltonian w.h.p.). The idea of a general host G was introduced by Frieze and T. Johansson [11]; see also e.g., Ferber et al. [10] for (inter alia) a nice connection with G n,p .
For hypergraphs the k-out model seems not to have been studied previously (random regular hypergraphs have been considered, e.g., in [5]). Here the two most important examples would seem to be H = K (r) n (the complete r-graph on n vertices) and H = K [n] r (the complete r-partite r-graph with n vertices in each part). It is natural to expect that for each of these there is some k = k(r) for which H(k-out) has a perfect matching w.h.p.. Note that, while almost certainly correct, these are likely to be difficult, as either would imply the aforementioned resolution of Shamir's Problem; still, we would like to regard the following linear relaxations as a small step in this direction. (Relevant definitions are recalled in Section 2.) Theorem 1. For each r, there is a k such that w.h.p. K (r) n (k-out) admits a perfect fractional matching and w ≡ 1/r is the only fractional cover of weight n/r. Theorem 2. For each r, there is a k such that w.h.p. H = K [n] r (k-out) admits a perfect fractional matching and each minimum weight fractional cover of H is constant on each block of the r-partition.
Our upper bounds on the k's are quite large (roughly r r ), but in fact we don't even know that they must be larger than 2 (though this sounds optimistic), and we make no attempt to optimize. In the more interesting case of (ordinary) perfect matchings, consideration of the expected number of perfect matchings shows that k does need to be be at least exponential in r.
We will make substantial use of the next observation (or, in the r-partite case, of the analogous Proposition 6, whose statement we postpone), in which the notion of expansion may be of some interest. Recall that an independent set in a hypergraph is a set of vertices containing no edges.
there is some edge meeting X but not Y . Then H has a perfect fractional matching. If, moreover we replace "<" by "≤" in (1), then w ≡ 1/r is the only fractional cover of weight n/r.
It's not hard to see that for r > 2 the proof of this can be tweaked to give the stronger conclusion even under the weaker hypothesis. (For r = 2 this is clearly false, e.g., if G is a matching.) Related notions of expansion (respectively stronger than and incomparable to ours) appear in [18] and [14]. An additional application of Proposition 3, given in Section 4, is a short alternate proof of the following result of Krivelevich [18].
Theorem 4. Let {H t } t≥0 denote the random r-graph process on V in which each step adds an edge chosen uniformly from the current non-edges, let T denote the first t for which H t has no isolated vertices. Then H T has a perfect fractional matching w.h.p..
Outline. Section 2 includes definitions and brief linear programming background. Section 3 treats K

Preliminaries
Except where otherwise specified, H is an r-graph on V = [n]. As usual, we use [t] for {1, 2, . . . , t} and X t for the collection of t-element subsets of X. Throughout we use log for ln and take asymptotics as n → ∞ (with other parameters fixed), pretending (following a common abuse) that all large numbers are integers and assuming n is large enough to support our arguments.
We need to recall a minimal amount of linear programming background (see e.g., [20] for a more serious discussion). For a hypergraph H, a fractional (vertex) cover is a map w : V → [0, 1] such that v∈e w(v) ≥ 1 for all e ∈ H; the weight of a cover w is |w| = v w(v); and the fractional cover number, τ * (H), is the largest such weight. Similarly a fractional matching of H is a ϕ : H → [0, 1] such that e∋v ϕ(e) ≤ 1 for all v ∈ V ; the weight of such a ϕ is defined as for fractional covers; and the fractional matching number, ν * (H), is the maximum weight of a fractional matching.
In this context, LP-duality says that ν * (H) = τ * (H) for any hypergraph. For r-graphs the common value is trivially at most n/r (e.g., since w ≡ 1/r is a fractional cover). A fractional matching in an r-graph is perfect if it achieves this bound; that is, if ϕ e = n/r (equivalently e∋v ϕ e = 1 ∀v, which would be the definition of perfection in a nonuniform H).
Finally, given H we say a nonempty X ⊆ V is λ-expansive if for all Y ⊆ V \ X of size at most λ|X|, there is some edge meeting X but not Y .

Proofs of Proposition of and Theorem 1
Proof of Proposition 3. It is enough to show that if w is a fractional cover with t 0 := 1/r − min v w(v) > 0, then |w| ≥ n/r, with the inequality strict if we assume the stronger version of (1). We give the argument under this stronger assumption; for the weaker, just replace the few strict inequalities below by nonstrict ones. Given w as above, set, for each t > 0, Since w is a fractional cover, each edge meeting W t must also meet W t/(r−1) (or the weight on the edge would be less than 1); so, since W t is independent, the hypothesis of with the inequality strict if t ∈ (0, t 0 ]. Thus, We should perhaps note that the converse of Proposition 3 is not true in general (failing, e.g., if r > 2 and H is itself a perfect matching). But in the graphic case (r = 2) the converse is true (and trivial), and the proposition provides an alternate proof of the following characterization, which is [19,Thm. 2.2.4] (and is also contained in [1, Thm. 2.1], e.g.).
Corollary 5. A graph has a perfect fractional matching iff |N (I)| ≥ |I| for all independent I.
(where N (I) is the set of vertices with at least one neighbor in I).

Proof of Theorem 2
As in the proof of Theorem 1 we first show that the conclusions of Theorem 2 are implied (deterministically) by sufficiently good expansion and then show that K [n] r (k-out) w.h.p. expands as desired. We take V = V 1 ∪ · · · ∪ V r to be our r-partition (so |V i | = n ∀i) and below always assume H ⊆ K [n] r . Proposition 6. Suppose ε ∈ (0, 1/2) and λ > 2r 2 are fixed and H satisfies: for any i ∈ [r], T ⊆ V i , U j ⊆ V j for j = i and U = ∪ j =i U j , there is an edge meeting T but not U provided either (i) |T | ≤ εn and |U j | ≤ λ|T | ∀j = i, or (ii) |T | ≥ εn and |U j | ≤ (1 − ε)n ∀j = i.
Then H admits a perfect fractional matching, and every minimum weight fractional cover of H is constant on each V i .
Proof. Define a balanced assignment to be a w : V → R with v∈V i w(v) = 0 and w(e) ≥ 0 for all e ∈ H. We claim that (under our hypotheses) the only balanced assignment is the trivial w ≡ 0. To get Proposition 6 from this, let f be a minimum weight fractional cover, and let w f (v) = f (v) − u∈V i f (u)/n, for each i and v ∈ V i . Then w f is a balanced assignment: v∈V i w f (v) = 0 is obvious and nonnegativity holds since f (e) ≥ 1 and, by minimality, v∈V f (v) ≤ n. Thus w f ≡ 0, implying f is as promised.
Proof of Theorem 2. Set λ = 4r 3 , ε = (2rλ) −1 and k = 2rε −r (so k is a little more than r 4r ). We show that w.h.p. H = K [n] r (k-out) is as in Proposition 6. As earlier, let B(X, Y ) be the event that every edge meeting X meets Y .
Suppose first that T and U are fixed with |U i | = λ|T | ≤ λεn. Then Summing over choices of T and U bounds the probability that H violates the assumptions of the proposition for some T and U as in (i) by

Proof of Theorem 4
We now turn to our proof of Theorem 4, for which we work with the following standard device for handling the process {H t }.
Provided the ξ S 's are distinct, this defines the discrete process {H t } in the natural way, namely by adding edges S in the order in which their associated ξ S 's appear in [0, 1]. We will work with the following quantities, where γ = ε log n for some small fixed (positive) ε and g is a suitably slow ω(1).
• Λ = min{λ : G(λ) has no isolated vertices}; Preview. With the above framework, our assignment is to show that G(Λ) has a perfect matching w.h.p.. Perhaps the nicest part of this-and the point of coupling the different G(λ)'s-is that, so long as Λ ∈ [σ, β], which we will show holds w.h.p., the desired assertion on G(Λ) follows deterministically from a few properties ((b)-(d)) of Lemma 5.1) involving G(σ), G(β) or both; so by showing that the latter properties hold w.h.p. we avoid the need for a union bound to cover possibilities for Λ. Production of the fractional matching is then similar to (though somewhat simpler than) what happens in [18]: the relatively few vertices of W Λ (and some others) are covered by an (ordinary) matching, and the hypergraph induced by what's left has the expansion needed for Proposition 3. (b) α(G(σ)) < Z; (c) no β-edge meets W σ more than once and no u ∈ W σ lies in more than one β-edge meeting N \ {u}; (d) each X ⊆ V \ W σ of size at most Z is r-expansive in G(σ).
The proofs of (c) and (d) are similarly routine but take a little longer. Aiming for (c), set p = P(ζ ≤ γ), where ζ is binomial with parameters n−2 r−1 and σ. Since µ := Eζ ∼ log n, a standard large deviation estimate (e.g., [15, Thm. 2.1]) gives where ϕ(x) = (x + 1) log(x + 1) − x for x ≥ −1 and δ ≈ ε log(1/ε). Failure of the first assertion in (c) implies existence of S ∈ K (r) n and (distinct) u, v ∈ S with S ∈ G(β) and u, v ∈ W σ . The probability that this occurs for a given S, u, v is less than βp 2 (the p 2 bounding the probability that each of u, v lies in at most γ edges not containing the other), so the probability that the assertion fails is less than n r r 2 βp 2 ∼ nr(log n)p 2 = o(1).
For (d) it is enough to bound (by o(1)) the probability that for some (nonempty) X ⊆ V of size x ≤ Z and Y ⊆ V \ X of size rx, there are at least γx/r σ-edges meeting both X and Y .
For given X, Y the expected number of such edges is less than x · rx n−2 r−2 σ < xr 2 Z log n n−1 =: bx.
(The first inequality is a significant giveaway for small x, but we have lots of room.) So, again using [15, Thm. 2.1], we find that the probability of (3) is Proof of Theorem 4. By Lemma 5.1 it is enough to show that if (a)-(d) of the lemma hold then G(Λ) has a perfect fractional matching; so we assume we have these conditions and proceed (working in G(Λ)). According to (c) (and the definition of Λ), G(Λ) admits a matching, M , covering W σ (each edge of which contains exactly one vertex of W σ ). Let W be the set of vertices covered by M (so W consists of W σ plus some subset of N \ W σ ), and H = G(Λ) − W (as usual meaning that the edges of H are the edges of G(Λ) that miss W ). It is enough to show that H has a perfect fractional matching, which will follow from Proposition 3 if we show each independent set X of H is (r − 1)-expansive. (4) Proof. Since such an X is also independent in G(σ), (b) gives |X| ≤ Z, and (d) then says X is r-expansive in G(σ), a fortiori in G(Λ). On the other hand, since X ∩ W σ = ∅, (c) guarantees that the β-edges (so also the Λ-edges) meeting X and not contained in V (H) can be covered by some U ⊆ W of size at most |X| (namely, (c) says each x ∈ X lies in at most one such edge). It follows that the Λ-edges meeting X that do belong to H cannot be covered by (r − 1)|X| vertices of V (H) \ X.