Choice functions in the intersection of matroids

We prove a common generalization of two results, one on rainbow fractional matching and one on rainbow sets in the intersection of two matroids: Given $d = r \lceil k \rceil - r + 1$ functions of size $k$ that are all independent in each of $r$ given matroids, there exists a rainbow set of $supp(f_i)$, $i \leq d$, supporting a function with the same properties.


Introduction
Let F = (F 1 , . . . , F m ) be a family (namely, a multiset) of sets. A (partial) rainbow set for F is the image of a partial choice function. Namely, it is a set of the form R = {x i 1 , x i 2 , . . . , x i k }, where 1 ≤ i 1 < i 2 < . . . < i k ≤ k, and x i j ∈ F i j (j ≤ k). Here it is assumed that R is a set, namely that the elements x i j are distinct. There are many theorems of the form "under some conditions there exists a rainbow set satisfying a prescribed condition". For example, the case where the condition is being full (representing all F ′ i s) is the subject of Hall's marriage theorem. The following theorem of the first author and Berger [1], which generalizes a result of Drisko [6] belongs to this family, and is a forefather of the results in the present paper: Theorem 1.1. Any family of 2k − 1 matchings of size k in a bipartite graph G have a rainbow matching of size k.
(Drisko's slightly narrower result was formulated in the language of Latin rectangles.) In [2] it was conjectured that almost the same is true in general graphs, namely that in any graph 2k matchings of size k have a rainbow matching of size k, and that for odd k the Drisko bound suffices -2k − 1 matchings of size k have a rainbow matching of size k. This is far from being solved (in [2] the bound 3k − 2 was proved), but in [3] a fractional version of the conjecture was proved, in a more general setting. Recall that ν * (F ) denotes the largest total weight of a fractional matching in a hypergraph H. Theorem 1.2 (Aharoni, Holzman and Jiang [3]). Let m be a real number, let H be an r-uniform hypergraph and let q ≥ ⌈rk⌉ be an integer. Then any family E 1 , ..., E q of sets of edges in H satisfying ν * (E j ) ≥ k for all j ≤ q has a rainbow set F of edges with ν * (F ) ≥ k. If H is r-partite then it suffices to assume that q ≥ r⌈k⌉ − r + 1 to obtain the same conclusion.
Drisko's theorem is a special case, since in bipartite graphs ν * = ν. The integral version of the theorem is false for r > 2. For example, the four matchings of size 2 in the complete 2 × 2 × 2 3-partite hypergraph do not have a rainbow matching of size 2, which shows that 3k − 2 matchings of size k do not suffice. In [4,13] bounds are studied in the integral case.
Kotlar and Ziv proved a matroidal generalization of Theorem 1.1: Theorem 1.3 (Kotlar and Ziv [9]). Let M 1 , M 2 be two matroids on the same vertex set V . Then any 2k − 1 sets E 1 , E 2 , . . . , E 2k−1 of size k in M 1 ∩ M 2 have a rainbow set of size k belonging to M 1 ∩ M 2 .
Theorem 1.1 is obtained by taking M 1 and M 2 to be the two partition matroids whose parts are (respectively) the stars in the two sides of the bipartite graph.
The aim of this paper is to prove a matroidal generalization of the r-partite case of Theorem 1.2, along the lines of Theorem 1.3. For this purpose we need a matroidal generalization of the notion of fractional matchings. This involves the familiar notion of matroid polytopes.
Edmonds [7] proved that all vertices of P (M) are integral, and that this is true also for the intersection of two matroids. This is a corollary of another theorem of Edmonds, the two matroids intersection theorem [7].
Our main result is: . . , M r be matroids on the same ground set V , and let k be a real number. Let d = r⌈k⌉ − r + 1. Let f 1 , . . . , f d be non-negative real valued functions belonging to i≤r P (M i ), satisfying Then there exists a function f ∈ i≤r P (M i ) such that supp(f ) is a rainbow set of (F 1 , . . . , F d ), and |f | ≥ k. Theorem 1.3 follows. Let E i , i ≤ 2k − 1 be sets as in that theorem. Applying Theorem 1.6 to the functions χ E i , i ≤ 2k − 1 (here χ S is the characteristic function of the set S), yields a function f ∈ P (M 1 ) ∩ P (M 2 ) with |f | ≥ k whose support is a rainbow set for the E i 's. The function f is a convex combination of vertices of P (M 1 ) ∩ P (M 2 ), and since in this combination all coefficients are positive, the supports of these vertices are contained in supp(f ). Among these there is at least one vertex g with |g| ≥ |f |. By Theorem 1.5 g is integral, namely a 0, 1 function, meaning that it is a characteristic function of a set as in the conclusion of Theorem 1.3.
To obtain the r-partite case of Theorem 1.2 from Theorem 1.6, choose the matroids M i , i ≤ r to be the partition matroids on i≤d E i defined by the stars in the i-th side V i of the hypergraph. Namely, a set is independent in M i if it does not contain two edges meeting in V i . Then a function belongs to i P (M i ) if and only if it is a fractional matching. The condition ν * (E j ) ≥ k means that there exists a fractional matching Applying Theorem 1.6 then yields a fractional matching f whose support is rainbow with respect to the sets E j .

A Topological Tool
The proof of Theorem 1.6 closely follows the footsteps of the proof in [3] of Theorem 1.2, but some further devices are needed.
A complex is a downward-closed collection of sets, called faces. Let C be a complex on a vertex set V . A face σ of C is called a collapsor if it is contained in a unique maximal face. The operation of removing from C all faces containing a collapsor σ is then called a collapse, and if |σ| ≤ d then the operation is called a d-collapse. We say that C is d-collapsible if it can be reduced to ∅ by a sequence of d-collapses. Wegner [14] observed that a d-collapsible complex is d-Leray, meaning that the homology groups of all induced complexes vanish in dimensions d and higher.
Our main tool will be a theorem of Kalai and Meshulam [8]. For a complex C let C c be the collection of all non-C-faces (namely, C c := 2 V \ C).
In fact, this is a special case of the main theorem in [8]. The way to derive it from the original theorem can be found in [3].
We will use Theorem 2.1 to reduce Theorem 1.6 to a topological statement. To state this, we first extend the definition of the fractional matching number ν * to our matroidal setting. For each W ⊆ V , let For a positive real k let X k be the simplicial complex of all sets W ⊆ V with ν * (W ) < k.
Theorem 1.6 follows from Theorem 2.2. Indeed, as X k is (r ⌈k⌉ − r)collapsible, by Theorem 2.1 any r ⌈k⌉ − r + 1 sets not in X k contain a rainbow set not in X k . Since F ∈ X k means that some f ∈ i≤r P (M i ) supported on F satisfies |f | ≥ k, Theorem 1.6 follows.

Proof of Theorem 2.2
A non-negative function c : Note that the rank function rk M of a matroid M is submodular [15].
Note that excluding the A = ∅ inequality does not change the polytope.
We shall use the acronym PDS for "positive, decreasing and submodular". As in [3], we shall consider perturbations of X k . For this purpose, we shall need the following: By linear programming duality, ν * a,b (W ) is equal to Given a positive real number k, let X a,b,k be the simplicial complex consisting of all sets W ⊆ V for which ν * a,b (W ) < k.
Theorem 2.2 is the special case of Theorem 3.3 obtained by fixing every b i (A) = 1 and a = 1. Theorem 3.3 applies since the constant-1 function is PDS. Here, X k = X a,b,k , a = b = 1, and k ≤ ⌈k⌉ − 1, yielding that X k is (r ⌈k⌉ − r)-collapsible.
We prove Theorem 3.3 by induction on |X a,b,k |. Note that |X a,b,k | > 1 , since X a,b,k contains at least one nonempty set.
Following a crucial idea from [3], we may assume that generically, for every W ⊆ V there is a unique function h on [r] × 2 V attaining the minimum in the program defining τ * a,b (W ). For, the set of all b = (b 1 , . . . , b r ) for which the optimum is not uniquely attained is the union of finitely many hyperplanes. By Lemma 3.2, it is possible to perturb the b i 's so as to avoid these hyperplanes, in a fashion sustaining the value of b. If the perturbation is sufficiently small, X a,b,k stays unaffected. Now, we choose any W ∈ X a,b,k such that: ( †) ν * a,b (W ) = k, and W is inclusion-minimal among all such sets.
We prove that removing all supersets of W is an elementary r k abcollapse in X a,b,k . This requires the three claims (♦), (♣), and (♠) as follows, which together will constitute the remainder of the proof of Theorem 3.3.

(♦)
W is contained in a unique facet.
To prove (♦), we follow [3], but reproduce the argument for completeness. Let W + := {v ∈ V : W ∪ {v} ∈ X a,b,k }. Let v ∈ W + be arbitrary. By maximality of k, we know ν * a,b (W ∪ {v}) = ν * a,b (W ) = k, and hence τ * a,b (W ∪ {v}) = τ * a,b (W ) = k. By our assumed perturbations, there exists a unique function h on [r]×2 V attaining the minimum defining τ * a,b (W ). Since the function h ′ witnessing τ * a,b (W ∪ {v}) = k is also feasible for τ * a,b (W ), it follows that h ′ = h, so h must satisfy the additional constraint i∈[r],A∋v h(i, A) ≥ a v for v. Since this is true for every v ∈ W + , the function h satisfies the constraints for all vertices in W ∪ W + , witnessing τ * a,b (W ∪ W + ) = k. Thus W ∪ W + ∈ X a,b,k is the unique facet containing W , giving (♦).
The proof of (♣) is the main place where new arguments are needed, beyond those appearing in [3]. These appear in Lemma 3.4, Theorem 3.5 and Lemma 3.
so the set {χ A | A ∈ F f i } consists of w i linearly independent vectors. We can take advantage of these w i sets as follows. Recall that χ S denotes the indicator vector of S. We use the term "chain of length r of sets" for a collection of r distinct non-empty sets, totally ordered by inclusion. Proof. We proceed by induction on t. It is obvious when t = 1. For t ≥ 2, we may assume that there exists a chain ∅ = A 1 · · · A t−1 of length t − 1. Since {χ S : S ∈ F } spans a t-dimensional space, there exists a non-empty set A ∈ F such that χ A ∈ U := span({χ A i : i < t}). If A ⊆ A t−1 , then letting A t = A t−1 ∪ A yields the desired chain of length t. Thus we may assume A ⊆ A t−1 . For strictly between A i−1 and A i , so its addition forms the desired chain. We may thus assume that there is no such B i .
Let S = {i ≤ t : B i ⊆ A}. By the above assumption A = i∈S B i . Hence We wish to show that each F f i satisfies the condition of Lemma 3.4, namely it is closed under intersections and unions. Indeed, for the usual matroid polytopes, it is a well-known fact (see Lemma 3.6 below). Extending this to skew matroids first requires the following result.
Theorem 3.5. If c, r are nonnegative submodular functions on a lattice of sets, c is decreasing and r is increasing, then c · r is submodular.
This may be folklore, and it can be derived from a similar fact on the product of convex functions (see e.g. 3.32 of [5], ascribing the result to an observation of Lovász [10]). The only explicit reference we could find is in a question answered in [12]. For completeness we provide a proof here.
Proof. We wish to show that, for any A, B ⊆ V , We shall show that D S D R (cr) is non-positive for any sets S, R. To see this, write: gives us the product rule D R (cr)(S) = c(S∪R)D R r(S)+(D R c(S))(r(S)). Letting T R h(X) denote h(X ∪ R) for any h, this says Applying this twice gives All four products above are non-positive, as can be seen from the following: • c, r ≥ 0 by nonnegativity, • D R c, D T c ≤ 0 as c decreasing, • D T r, D R r ≥ 0 as r increasing. Then F is closed under intersections and unions.
The second inequality is the submodularity of c · rk. The first and last inequalities follow from the fact that f ∈ P c (M). Since equality should hold throughout, it follows that A ∪ B, A ∩ B ∈ F . Lemma 3.6 enables application of Lemma 3.4 to F := F f i . We obtain as f (v) > 0 for each v ∈ W . We may rewrite this as Since ranks are integers, it follows that rk M i (A w i ) ≥ w i . Thus in fact, for each i ∈ [r]: and by integrality w i ≤ k ab . So we conclude which proves (♣).
(♠) Suppose W satisfies ( †) and let X ′ be the complex obtained by removing from X a,b,k all faces containing W . Then there exists a ′ ∈ R V + , satisfying r k a ′ b ≤ r k ab , for which The proof of (♠) follows a parallel argument in [3]. We claim that there is some ǫ > 0 for which X ′ = X a ′ ,b,k is satisfied by the objective coefficients a ′ defined coordinate-wise by: First consider any W ′ ⊆ V that wasn't even in X a,b,k to begin with, so that ν * a,b (W ′ ) ≥ k. The feasibility regions for ν * a,b (W ′ ) and ν * a ′ ,b (W ′ ) are the same, so if ǫ is sufficiently small relative to k − k, it follows ν * a ′ ,b (W ′ ) ≥ k, so that W ′ ∈ X a ′ ,b,k either. Next, pick any W ′ ⊆ V previously in X a,b,k , but which contained W so was removed in the collapse. As before, let f be an optimiser for the LP defining ν * a,b (W ), so a · f = k but also supp(f ) = W ⊆ W ′ . This way, f is also feasible for the linear program defining ν * a,b (W ′ ). But whenever a ′ v < a v , e ∈ W and hence f (v) = 0 by minimality of W . Hence ν * a ′ ,b (W ′ ) ≥ a ′ · f = a · f = ν * a,b (W ′ ) = k. Thus W ′ ∈ X a ′ ,b,k . Finally, take some W ′ ⊆ V previously in X a,b,k and not fully containing W . Note W ∩ W ′ W . We wish to show ν * a ′ ,b (W ′ ) < k for deducing W ′ ∈ X a ′ ,b,k , so assume for contradiction ν * a ′ ,b (W ′ ) ≥ k, as witnessed by some g ∈ P b i (M i ), supp(g) ⊆ W ′ with a ′ · g ≥ k. We cannot have supp(g) ⊆ W ∩ W ′ . For otherwise g would also witness ν * a,b (W ∩W ′ ) ≥ a ′ ·g = a·g ≥ k, hence ν * a,b (W ∩W ′ ) = k by maximality of k, and this would contradict inclusion-minimality of W . So there is at least one e 0 ∈ supp(g) \ W . So g(v 0 ) > 0 and a ′ v 0 < a v 0 means v∈W ′ a v g(v) > v∈W ′ a ′ v g(v) ≥ k, still contradicting maximality of k.
So, by inductive hypothesis, X a ′ ,b,k is indeed r k a ′ b -collapsible, and since k < k, we can make ǫ small enough to guarantee r k a ′ b ≤ r k ab .