MATROID SECRETARIES 3 Minors Project’

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1 R (M,w) is at least c OPT(M,w) for all weight functions w : E(M) >0. The following deep conjecture remains open. →

Conjecture 1 (Babaioﬀ, Immorlica, and Kleinberg [2]). For all matroids M, there is a c-competitive matroid secretary algorithm for M, where c is a universal constant independent of M.

The original motivation for considering the matroid secretary problem was mechanism design for online auctions. Here the ‘secretaries’ are agents and their weights correspond to the prices they are willing to pay for auctioned items. It turns out that matroid constraints model many real world situations and they unify many previously considered models. For example, the situation in which an auctioneer wants to sell at most k items corresponds to the case that M is a rank-k uniform matroid. This is known as the multiple-choice secretary problem and was essentially completely solved by Kleinberg [22]. For more on the connection to mechanism design, see [1]. Although Conjecture 1 remains open, steady progress has been made in improving the competitive ratio. Babaioff, Immorlica, and Kleinberg [2] gave an O(log r)-competitive algorithm for all matroids M, where r is the rank of M. This was improved to an O(√log r)-competitive algorithm by Chakraborty and Lachish [3]. The state-of-the-art is an O(log log r)-competitive algorithm, first obtained by Lachish [24]. Using different tools, Feldman, Svensson, and Zenklusen [9] give a simpler O(log log r)-competitive algorithm for all matroids. On the other hand, there exist constant-competitive algorithms for restricted classes of matroids. We now give a partial list of such matroid classes. Graphic matroids have a 2e-competitive algorithm due to Korula and Pál [23]. Cographic matroids have a 3e-competitive algorithm due to Soto [32]. Regular matroids and max-flow min-cut matroids both have 9e-competitive algorithms due to Dinitz and Kortsarz [7]. Our main result is a generalization of all these aforementioned constant-competitive algorithms.

1.1. Our Results. For each prime p, we let Fp denote the field with p elements. A class of matroids is minor-closed if M and N a minor of M implies that N is also in . A properM minor-closed class of∈MF -representable matroids is a minor-closed M p class of Fp-representable matroids that is not equal to the class of all Fp-representable matroids. Our main result is that Conjecture 1 holds for every proper minor-closed class of Fp-representable matroids. Note that graphic matroids, cographic matroids, regular matroids, and max-flow min-cut matroids are all examples of proper minor-closed classes of Fp-representable matroids. Here is our main result. Theorem 2. Suppose that Hypothesis 1 holds and let p be a prime. Then for every proper minor-closed class of Fp-representable matroids, there is a constant c( ) such that every matroid in M has a c( )-competitive matroid secretary algorithm.M M M To be forthright with the reader, we stress that Theorem 2 relies on a structural hypothesis communicated to us by Geelen, Gerards, and Whittle, which has not yet appeared in print. This hypothesis is stated as Hypothesis 1. The proof of Hypothesis 1 will stretch to hundreds of pages, and will be a consequence of their decade-plus ‘matroid MATROID SECRETARIES 3 minors project’. 1 This is a body of work generalising Robertson and Seymour’s graph minors structure theorem [30] to matroids representable over a fixed finite field, leading to a solution of Rota’s Conjecture [14]. See [15] for a discussion of the project. In proving Theorem 2, we develop techniques to handle each of the ingredients that appear in the statement of Hypothesis 1. We believe these techniques may be of independent interest. Also, our proof of Theorem 2 is one of the few instances of using deep results from matroid theory in attacking Conjecture 1. To complement our main result, we also show that Conjecture 1 holds for almost all matroids. For each M, let RB denote the matroid secretary algorithm that chooses a uniformly random basis B of M, and selects precisely the elements of B, ignoring weights. Theorem 3. For asymptotically almost all matroids on n elements, the algorithm RB 3 is (2 + √n )-competitive. The veracity of Theorem 3 is not terribly surprising, although many properties which ‘obviously’ hold for almost all matroids are very difficult to establish. For example, the fact that almost all matroids are k-connected (for each fixed k) was only proved recently [29]. Indeed, the proof of Theorem 3 relies on recent breakthrough results of Pendavingh and van der Pol [28, 29]. One can think of Theorem 2 and Theorem 3 as complementary results. Theorem 3 asserts that the ‘typical’ matroid has a constant-competitive algorithm, so we expect counterexamples to Conjecture 1 to come from structured classes of matroids. On the other hand, Theorem 2 says that restricting to proper-minor closed classes of Fp- representable matroids also cannot produce counterexamples to Conjecture 1. Note that for almost all matroids, the algorithm RB has a better competitive ratio than the best algorithm for the classical secretary problem. Indeed, under a widely believed conjecture in asymptotic matroid theory, we note that there is in fact a (1+ o(1))- competitive algorithm for almost all matroids. Theorem 4. If asymptotically almost all matroids are paving, then thereisa (1+o(1))- competitive matroid secretary algorithm for asymptotically almost all matroids. 1.2. Our Methods. Roughly, Hypothesis 1 asserts that every matroid in a proper minor-closed class of Fp-representable matroids admits a tree-like decomposition into pieces that are ‘close’ to being ‘sparse’ or ‘graph-like’. It is quite interesting that many of the ingredients of Hypothesis 1 have already been considered in the literature. For example, it is already known that ‘sparse’ and ‘graph-like’ matroids (definitions will be given later) have constant-competitive algorithms [32]. Moreover, Dinitz and Kortsarz [7] have a very nice framework for obtaining constant-competitive algorithms for matroids decomposable as 1-, 2-, and 3-sums from matroids for which we already have constant-competitive algorithms. In Theorem 14, we extend this framework to allow tree-decompositions of any fixed ‘thickness’. This is a non-trivial extension, and is necessary since Hypothesis 1 requires ‘thickness’ greater than 3. We also prove that with an appropriate definition of ‘closeness’, if a minor-closed class ′ is close to another minor-closed class for which M M 1We are aware of the problematic family of dyadic matroids recently discovered by Grace and van Zwam [16], but they are not counterexamples to Hypothesis 1. 4 T. HUYNH AND P. NELSON we already have a constant-competitive algorithm, then there is a constant-competitive algorithm for ′ (with a worse competitive ratio). Both of these results are independent of any matroidM structure theory results. For example, plugging in the regular matroid decomposition theorem of Seymour [31] into Theorem 14, we recover the constant- competitive algorithm for regular matroids. Plugging in Hypothesis 1 into Theorem 14 gives Theorem 2.

1.3. Limitations and Related Results. Probably the greatest drawback of the matroid secretary algorithm in Theorem 2 is that it requires knowledge of the entire matroid upfront. This is the same model as originally introduced by Babaioff, Immorlica, and Kleinberg [2], but many of the known matroid secretary algorithms work in the setting where the matroid structure is also revealed online, see [17]. It is a nice open problem whether Theorem 2 holds in this setting. We also note that the existence of constant-competitive algorithms for transversal matroids [5, 21] and laminar matroids [19, 18, 20] is not implied by Theorem 2. Laminar matroids are a minor-closed class, and every laminar matroid is representable over a sufficiently large field (see [11]), but there is no single finite field over which all laminar matroids are representable. The following strengthening of our main result would also imply a constant-competitive algorithm for laminar matroids. Conjecture 5. For every proper minor-closed class of matroids , there exists a constant c( ) such that every matroid in has a c( )-competitiveM matroid secretary algorithm. M M M

This is by no means an exhaustive list of related results. We refer the reader to the survey of Dinitz [6] for more information on the matroid secretary problem.

1.4. Outline of Paper. Section 2 provides a quick review of matroid terminology and notation. In Section 3, we describe the two ‘basic’ classes of matroids that appear in Hypothesis 1. In Section 4 we consider a notion of ‘closeness’ of matroids, and show that constant-competitive algorithms are preserved for matroids that are ‘close’. In Section 5, we introduce our deﬁnition of tree-decompositions for matroids and show that we can obtain constant-competitive algorithms for matroids that have a bounded ‘thickness’ tree-decomposition into pieces for which we already have constant- competitive algorithms. In Section 6, we quickly re-derive the constant-competitive algorithm for regular matroids. We prove our main result (Theorem 2) in Section 7. Finally, in Section 8, we prove our results on random matroids.

2. Matroid Basics For a more thorough introduction to matroid theory we refer the reader to Oxley [27], whose notation and terminology we follow. A matroid is a pair M =(E, ) where E is a ﬁnite ground set and is a collection of subsets of E satisfying I I , •if ∅∈II and J I, then J , and • if I,J∈ I and ⊆I < J , then∈ I there exists x J such that I x . • ∈ I | | | | ∈ ∪{ } ∈ I A set X E is independent if X , and dependent otherwise. The maximal independent⊆ sets are bases, and the minimal∈ I dependent sets are circuits. The rank MATROID SECRETARIES 5

E function of M is the function rM : 2 Z>0 where rM (X) is the size of a largest independent set contained in X. The closure→ of X E, denoted cl (X), is the largest ⊆ M set Y such that rM (Y )= rM (X). The dual of a matroid M, denoted M ∗, is the matroid whose bases are the complements of the bases of M. We now give several examples of matroids. Let G = (V, E) be a graph. The cycle matroid of G, denoted M(G), is the matroid with ground set E and circuits the edge sets of cycles of G. Such matroids are called graphic matroids. Cographic matroids are the duals of graphic matroids. Let F be a field. An F-matrix is a matrix with entries in F. Let A be an F-matrix with its columns labelled by E. We let M[A] be the matroid with ground set E whose independent sets are the subsets of E corresponding to linearly independent columns of A. A matroid M is F-representable if M = M[A] for some F-matrix A. It is binary if it is F2-representable and it is regular if it is F-representable for every field F. A rank-r uniform matroid is a matroid where all the r-subsets are bases. We denote the rank-r uniform with n elements as Ur,n. It is well-known exactly which fields the rank-2 uniform matroids are representable over. We only require the following direction. Lemma 6. U is not F-representable if F 6 k 2. 2,k | | − Proof. Suppose not and let A bea2 k F-matrix which represents U . By performing × 2,k row operations we may assume that A1,1 = A2,2 = 1 and A1,2 = A2,1 = 0. Since every two columns are independent, A1,ℓ = 0 for all ℓ 3,...,k . By column scaling, we may assume that A = 1 for all ℓ 3,...,k6 . Now∈{ note that }A = 0 for all ℓ 3,...,k . 1,ℓ ∈{ } 2,ℓ 6 ∈{ } Since F 0 6 k 3, there exist distinct ℓ,ℓ′ 3,...,k such that A2,ℓ = A2,ℓ′ , which contradicts| \{ that}| every− two columns of A are independent.∈{ }

We now describe a relation on the set of all matroids analogous to the minor relation on graphs. Let M be a matroid with ground set E, and let C and D be disjoint subsets of E. We let M/C D be the matroid with ground set E (C D) and rank \ \ ∪ function rM/C D(X)= rM (X C) rM (C). We say that M/C D is obtained from N by contracting\ C and deleting∪ D.− A matroid N is a minor of\ a matroid M, written N M, if N is isomorphic to M/C D for some choice of C and D. A restriction of M is a minor of M obtained by only deleting\ edges. We write M X to denote M (E X). | \ − The minor relation on matroids generalizes the minor relation on graphs in the following sense. Let G = (V, E) be a graph. It is easy to show (see 3.1.2 and 3.2.1 in [27]) that for all x E, M(G) x = M(G x) and M(G)/x = M(G/x). To give the reader some more∈ intuition, we\ now describe\ what the minor relation means for representable matroids. Let M be a representable matroid and x E(M). It is useful to regard the elements of M as points in a projective space P. Deleting∈ x has an obvious meaning; we simply remove the point x. To contract x, we choose a hyperplane H not containing x, and we project the remaining points of M onto H from x. It is easy to show that the resulting minor is independent of the choice of H. A class of matroids is minor-closed if M and N M implies N . By the previousM two paragraphs, the class of graphic∈ M matroids is minor-closed, and∈ M for every ﬁeld F, the class of F-representable matroids is minor-closed. It follows that the class of regular matroids is also minor-closed. 6 T. HUYNH AND P. NELSON

A proper minor-closed class of F-representable matroids is a minor-closed class of F- representable matroids not equal to the class of all F-representable matroids. In this paper, we are interested in proper minor-closed classes of Fp-representable matroids, where p is a prime. For example, the class of regular matroids is a proper minor-closed class of binary matroids.

3. Sparse and Graph-like Matroids We now introduce the two classes of matroids that appear as ‘pieces’ of the tree- decomposition in Hypothesis 1. Let M be a matroid and N be the matroid obtained from M deleting loops and suppressing parallel elements. Let γ R. We say that M is ∈ γ-sparse if X 6 γrN (X) for all X E(N). To give some intuition to the reader, note that there does| | not exist γ R such⊆ that all graphic matroids are γ-sparse. This is ∈ n because the cycle matroid of the complete graph Kn has 2 elements and rank n 1. On the other hand, all graphic matroids of planar graphs are 3-sparse, since it is well-known− that a planar graph on n vertices contains at most 3n 6 edges. − Soto [32, Theorem 5.2] proved that there is constant-competitive matroid secretary algorithm for γ-sparse matroids. Theorem 7. If M is a γ-sparse matroid, then there is a γe-competitive matroid secretary algorithm for M.

Let k N. A k-column sparse matroid is a matroid isomorphic to M[A], where A has at most∈ k non-zero entries per column. Note that graphic matroids are 2-column sparse. Soto [32, Theorem 5.4] also proved that k-column sparse matroids have constant- competitive matroid secretary algorithms for every ﬁxed constant k. We only need this result for k = 2, and we call 2-column sparse matroids represented frame matroids. We use the current best ratio in [33, Theorem 14]. Theorem 8. If M is a represented frame matroid, then there is a 4-competitive matroid secretary algorithm for M.

4. Lifts and Projections

Let M be a matroid. A loop is an element x such that rM ( x ) = 0, and a free 2 { } element is an element x such that rM ∗ ( x ) =0. Note that if x is a loop, no secretary algorithm can select x. On the other hand,{ } if x is a free element, we may assume every secretary algorithm always selects x.

Let M1 and M2 be matroids on a common ground set E. We say that M1 is a distance- 1 perturbation of M2 if there is a matroid M and a non-loop, non-free element x of M such that M/x, M x = M1, M2 . We say that M/x is a projection of M x and M x is a lift of {M/x (many\ } authors{ call these} elementary projections/lifts). The \perturbation\ distance between two matroids M, M ′ on a common ground set is the minimum t for which there exists a sequence M = M0, M1,...,Mt = M ′ where each Mi is a distance-1 perturbation of Mi 1. We write dist(M, M ′) for this quantity. If M = N, we may take − M = M0 = N, and so dist(M, M) = 0.

2We use free element instead of coloop because it is more common in the computer science literature. MATROID SECRETARIES 7

In this section, we show that the existence of constant-competitive algorithms is robust under a bounded number of lifts/projections. We ﬁrst note that this is true for deleting elements, which can be proved by setting elements to have zero weight appropriately. Lemma 9. If there is a c-competitive matroid secretary algorithm for M, then there is a c-competitive algorithm for every restriction of M. Lemma 10. Let N be a lift of a matroid M. If there is a c-competitive algorithm for M then there is a max(e, 2c)-competitive algorithm for N.

Proof. Let ALGM be a c-competitive algorithm for M. Let L be a matroid and x be a non-loop and non-free of L such that L/x = M and L x = N. Let P be the parallel class of L containing x. Note that each element in P \ x is a loop in M, and hence −{ } will never be selected by ALGM . We specify an algorithm ALGN for N as follows:

as the elements of P x are received, ALGN runs the classical secretary • algorithm to select one.− If {P}= x , then no element is chosen in this way. as the elements of E(M) P are{ } received, ALG passes them to ALG and • − N M selects them as ALGM does. Let I be the set of elements chosen in the ﬁrst way (so I 6 1) and J be the set of elements chosen in the second way. Clearly J is independent| | in N/I and so I J is independent in N. It remains to show that E(w(I J)) > 1 OPT(N,w∪ ). ∪ max(e,2c) Let B be a max-weight basis of (N,w) and let C be the unique circuit of L with x C B x . To analyse ALG we distinguish two cases. { } ⊆ ⊆ ∪{ } N If C > 3, then let C′ = C x and let y be a minimum-weight element | | > 1 − { } of C′. Now w(C′ y ) 2 w(C′) and B y is a basis of M satisfying − { } > − { } 1 > 1 w(B y ) = w(B C′)+ w(C′ y ) w(B C′)+ 2 w(C′) 2 w(B). Therefore −{ } > 1 − −{ } > 1 − > 1 OPT(M,w) 2 OPT(N,w) and so E(w(J)) c OPT(M,w) 2c OPT(N,w). If C < 3, then (since x is a non-loop of L) we have C = x, x′ for some x′ P x . | | { } ∈ −{ } In this case B = x′ J ′ for some independent set J ′ of M. Now I is chosen by an { } ∪ 1 e-competitive secretary algorithm on P x , so E(w(I)) > w(x′). Moreover, since −{ } e B is optimal and x′ B is independent in N for every basis B of M, we have { } ∪ 0 0 w(B x′ ) = OPT(M,w). Therefore −{ } 1 1 E(w(I J)) > w(x′)+ OPT(M,w) ∪ e c 1 1 = w(x′)+ w(B x′ ) e c −{ } > 1 max(e,c) OPT(N,w).

It follows from these two cases that ALGN is max(e, 2c)-competitive for N. Lemma 11. Let N be a matroid obtained from a matroid M by a sequence of t projections. Let L be the set of loops of N. If there is a c-competitive algorithm for M, then there is a c(e + 1)t-competitive algorithm for M L whose output is always independent in N. \ Proof. We proceed by induction on t. The case t = 0 is trivial. Suppose t > 1 and let N ′ be the matroid such that N ′ is obtained from M by t 1 projections and N is − obtained from N ′ by a single projection. Let L′ and L be the set of loops of N ′ and N, t 1 respectively. By induction, there is a c(e +1) − -competitive algorithm for M L′ whose \ 8 T. HUYNH AND P. NELSON

t 1 output is always independent in N ′. Since L′ L, by Lemma 9, there is a c(e + 1) − - ⊆ competitive algorithm ALGM L for M L whose output is always independent in N ′. Let P be a matroid and x be a non-loop\ \ and non-free element of P so that P/x = N and heads e P x = N ′. Let X be the random variable taking the value with probability e+1 \ tails 1 ALG and with probability e+1 . We deﬁne another matroid secretary algorithm N for M L as follows: \ if X = heads, then we run a classical secretary algorithm on N L to select just • one element. \ if X = tails, then we run ALGM L, except we only pretend to hire any element • whose selection would create a\ dependency in N with the elements already chosen.

In both cases, ALGN clearly produces an independent set in N. Fix a weighting w of M L. Let x0 E(M L) have maximum weight, and let I be the set selected by ALGN . If X\ = heads,∈ then E\(w(I) X = heads) > 1 w(x ). Moreover, if J is the set output by | e 0 ALGM L, then J x contains at most one circuit of P . It follows that if X = tails \ ∪{ } then I is obtained from J by removing at most one element, so w(I) > w(J) w(x0). Thus − E(w(I)) = e E(w(I) X = heads)+ 1 E(w(I) X = tails) e+1 | e+1 | > 1 w(x )+ 1 (E(w(J) w(x )) X = tails) e+1 0 e+1 − 0 | = 1 E(w(J) X = tails) e+1 | 1 > t OPT(M L, w). c(e+1) \ It follows that ALG is (e + 1)tc-competitive for M L. N \ In particular, since every independent set in N is an independent set of M L, the \ algorithm ALGN is (e +1)c-competitive for N. Since (e +1)c> max(e, 2c) for c > 1, we can thus combine Lemmas 10 and 11 with an inductive argument to yield the following. Lemma 12. Let t N and let M and N be matroids with dist(M, N) 6 t. If there is a c-competitive algorithm∈ for M, then there is a c(e + 1)t-competitive algorithm for N.

5. Tree-decompositions In this section, we introduce a notion of tree-decompositions of matroids, and give a constant-competitive matroid secretary algorithm for matroids having a ‘bounded- thickness’ tree-decomposition into pieces for which constant-competitive algorithms are known. Note that there already exist various notions of tree-decompositions of matroids, such as branch-decompositions [12]. Therefore, to avoid confusion we use the term thickness for the ‘width’ of our tree-decompositions.

We begin by defining the connectivity function of a matroid. Let M = (E,rM ) be a matroid. For disjoint sets X,Y E, the local connectivity between X and Y is ⊆ M (X,Y ) := rM (X)+ rM (Y ) rM (X Y ). The connectivity function λM of M is the ⊓ E − ∪ function λ : 2 Z> defined by λ (X)= (X, E X). M → 0 M ⊓M − To give the reader some intuition, we now describe what the connectivity function λM means for graphic and representable matroids. Let G = (V, E) be a graph and M = M(G) be the cycle matroid of G. For X E, we let G[X] be the subgraph ⊆ MATROID SECRETARIES 9 of G induced by the edges in X. It is easy to show that if G[X] and G[E X] are − both connected, then λM (X) + 1 is the number of vertices in both G[X] and G[E X] (see [27, Lemma 8.1.7]). Thus, λ (X) is a measure of how ‘connected’ X is to E − X. M − Indeed, it is possible to define graph connectivity using λM (see [27, Section 8.6]). Let M = M[A] for some matrix A with column labels E. For X E we let X be ⊆ h i the subspace generated by the columns corresponding to X. In this case, λM (X) is the dimension of X E X . So again, λM (X) is a measure of how ‘connected’ X is to E X. h i∩h − i − Next, we show that the connectivity function λM is related to the perturbation distance between deletion and contraction minors of M.

Lemma 13. Let M = (E,rM ) be a matroid and let X E. Then M/(E X) is obtained from M X by a sequence of λ (X) projections. ⊆ − | M

Proof. Let I1 X and I2,I3 E X be disjoint independent sets of M so that I1 is a basis for M⊆X, I I is a⊆ basis− for M, and I I is a basis for M X. We have | 1 ∪ 2 2 ∪ 3 \ I3 = rM (E X) (rM (E) rM (X)) = λM (X). Now the matroid N =(M/I2) (X I3) |satisfies| N/I−= M/− (E X−) and N I = M X. The lemma follows. | ∪ 3 − \ 3 | We are now ready to introduce our notion of tree-decompositions. A tree- decomposition of a matroid M is a pair (T, ) where T is a tree and := Xv : v V (T ) is a partition of E(M) indexed byX V (T ). Let e = v v EX(T ) and{ T ∈ } 1 2 ∈ 1 and T2 be the components of T e where vi V (Ti). Let X1 := v V (T1) Xv. We \ ∈ ∈ define the thickness of e, λ(e), to be λM (X1). The thickness of (T, ) is maxe E(T ) λ(e). X S ∈ If, for all e = uv E(T ), we have M (Xu,Xv) = λ(e), then we say (T, ) is a full tree-decomposition∈ of M. For those⊓ familiar with the standard definitionX of tree- decompositions of graphs (see [4, Section 12.3]), fullness is analogous to the fact that the separations displayed by a standard tree-decomposition of a graph are obtained by intersecting adjacent bags of the tree [4, Lemma 12.3.1]. Theorem 14. Let be a class of matroids for which there exists a c-competitive matroid secretary algorithm.M Let k N and let t ( ) be the set of all matroids M with ∈ k M a full tree-decomposition (T, ) of thickness at most k such that M clM (Xv) for each v V (T ). Then there isX an c(e + 1)k-competitive matroid secretary| algorithm∈ M for t ( ).∈ k M Proof. We say that a tree-decomposition (T, ) of a matroid M is an -tree decomposition if M cl (X ) for all v V (TX). For each m > 1, let t ( )M denote | M v ∈M ∈ k,m M the class of matroids in tk( ) having a full -tree-decomposition (T, ) of thickness at most k with V (T ) 6 mM. There is clearlyM a c(e + 1)k-competitive matroidX secretary | | algorithm for every matroid in tk,1( )= . Let m> 1 and suppose inductively that M Mk every matroid in tk,m 1( ) has a c(e + 1) -competitive matroid secretary algorithm. − M Let M tk,m( ), let E = E(M), and let (T, ) be a full -tree-decomposition of M of thickness∈ M at most k with V (T ) 6 m. WeX may assumeM that V (T ) = m, else we are done by the induction hypothesis.| | Let ℓ be a leaf of T and let| e =| ℓu be the edge of T incident with ℓ. Let Xℓ′ and Xu′ be obtained from Xℓ and Xu by moving all elements from X cl (X ) into X . It is easy to check that λ (X′) 6 λ (X ). Since ℓ ∩ M u u M ℓ M ℓ cl (X′ ) = cl (X ), we also have r (E X )= r (E X′). Therefore, M u M u M − ℓ M − ℓ 10 T. HUYNH AND P. NELSON

(X′,X′ )= r (X′)+ r (X′ ) r (X′ X′ ) ⊓M ℓ u M ℓ M u − M ℓ ∪ u = r (X′)+ r (X ) r (X X ) M ℓ M u − M ℓ ∪ u = r (X′) r (X )+ (X ,X ) M ℓ − M ℓ ⊓ ℓ u = r (X′) r (X )+ λ (X ) M ℓ − M ℓ M ℓ = r (X′)+ r (E X ) r (E) M ℓ M − ℓ − M = r (X′)+ r (E X′) r (E) M ℓ M − ℓ − M = λM (Xℓ′).

Again since clM (Xu′ ) = clM (Xu), it follows that M (Xu′ ,Xv) = M (Xu,Xv) for all v / u,ℓ . Therefore, this move preserves the property⊓ of being⊓ a full -tree- decomposition∈ { } of thickness at most k; and so we may assume that X cl (XM)= ∅. ℓ ∩ M u Let M(ℓ) = M X and let M ′(ℓ) = M/(E X ). By Lemma 13, the latter is | ℓ − ℓ obtained from the former by at most λM (Xℓ) 6 k projections. Moreover, since

(X ,X ) = λ (X ), we have λ u (X ) = 0 and so M ′(ℓ)=(M/X ) X ; since ⊓M ℓ u M ℓ M/X ℓ u | ℓ Xℓ clM (Xu)= ∅ it follows that M ′(ℓ) has no loops. ∩ k By Lemmas 9 and 11, there is a c(e +1) -competitive algorithm ALGℓ for M(ℓ) whose output is always independent in M ′(ℓ). Moreover, we see that (T ℓ, X : w V (T ℓ) ) \ { w ∈ \ } is a full -tree-decomposition of M Xℓ with thickness at most k, so M Xℓ tk,m 1( ) M k \ \ ∈ − M and there is thus a c(e+1) -competitive algorithm ALG′ for M X . Deﬁne an algorithm \ ℓ ALG for M by running ALG′ and ALGℓ on the elements of E Xℓ and Xℓ respectively as they are received, choosing all elements chosen by either. −

Let Iℓ and I′ be the sets chosen by ALGℓ and ALG′ respectively. Since I′ is independent in M Xℓ and Iℓ is independent in M ′(ℓ), the set I = Iℓ I′ chosen by ALG is independent in M\. Moreover, for each weighting w of M we have ∪

E(w(I)) = E(w(I′)) + E(w(Iℓ)) 1 > k (OPT(M(ℓ),w X ) + OPT(M X ,w (E X )) c(e+1) | ℓ \ ℓ | − ℓ > 1 c(e+1)k OPT(M,w), since each independent set of M is the union of an independent set of M(ℓ) and one of M X . The theorem follows. \ ℓ 6. Regular Matroids As an easy corollary of Theorem 14, we obtain a short proof that there is an O(1)- competitive algorithm for regular matroids, a result ﬁrst proved by Dinitz and Kortsarz [7]. The constant 9e that they obtain is better than ours by a factor of 1 (e +1)2 4.6. 3 ≈ Corollary 15. There is a 3e(e + 1)2-competitive matroid secretary algorithm for each regular matroid. Proof. By Seymour’s regular matroid decomposition theorem [31], every regular matroid M is obtained from pieces that are either graphic, cographic, or R10 by 1-, 2- or 3-sums. This gives a tree-decomposition (T, ) of thickness at most 2 in M so that each M X X | v MATROID SECRETARIES 11 is either graphic, cographic or R10. Moreover, by performing parallel extensions of the elements to be deleted before each 2-sum and 3-sum, one can construct a matroid M ′ having M as a restriction and a full tree-decomposition (T, ′) of M ′ so that each X M ′ cl ′ (X′ ) is either graphic, cographic or a parallel extension of R . | M v 10 Korula and P´al [23] proved that there is a 2e-competitive matroid secretary algorithm for graphic matroids. Soto [32] proved that there is a 3e-competitive matroid secretary algorithm for cographic matroids. Since R10 is the union of two bases, Theorem 7 implies that each of its parallel extensions has a 2e-competitive algorithm. It follows 2 from Theorem 14 that there is a 3e(e +1) -competitive algorithm for M ′; by Lemma 9 there is also one for M.

7. Proper Minor-Closed Classes In this section we show that, contingent on a certain deep structural hypothesis, there is a constant-competitive algorithm for every proper minor-closed class of matroids representable over a ﬁxed prime ﬁeld. As well as this structural hypothesis, we require one other result, due to Geelen ([13], Theorem 4.3). We denote the matroid with n elements where all k-subsets are independent as Uk,n. Theorem 16. Let q > 2 and n be positive integers. If M is a simple matroid with no 3n U -minor and no M(K )-minor, then M 6 qq r(M). 2,q+2 n | | Since no restriction of such an M has either of the forbidden minors, for such an M q3n we also have X 6 q rM (X) for each X E(M). Combining this theorem with Theorem 7, we| thus| have the following. ⊆ Corollary 17. Let q > 2 and n be positive integers. If M is a matroid with no q3n U2,q+2-minor and no M(Kn)-minor, then there is an eq -competitive matroid secretary algorithm for M.

We now state the structural hypothesis we will use to prove Theorem 2.

Hypothesis 1. Let p be a prime. For every proper minor-closed class of Fp- representable matroids, there exist k, n and t for which every M is aM restriction ∈ M of an F -representable matroid M ′ having a full tree-decomposition (T, ) of thickness p X at most k such that for all v V (T ), if M ′ cl ′ (X ) has an M(K )-minor, then there ∈ | M v n is a represented frame matroid N with dist(M ′ cl ′ (X ), N) 6 t. | M v We can now prove Theorem 2, which we restate here. Theorem 2. Suppose that Hypothesis 1 holds. If p is prime and is a proper minor- M closed subclass of the Fp-representable matroids, then there exists c = c( ) so that every M has a c-competitive matroid secretary algorithm. M ∈M

Proof. Let k, n and t be the integers given for by Hypothesis 1. Let 1 denote the class of matroids having perturbation distanceM at most t from some representedM frame matroid. Let 2 denote the class of matroids with no U2,p+2-minor or M(Kn)- M t minor. By Theorem 8 and Lemma 12, every matroid in 1 has a 4(e +1) -competitive 3nM matroid secretary algorithm. Corollary 17 gives an epp -competitive matroid secretary algorithm for each matroid in . M2 12 T. HUYNH AND P. NELSON

By Lemma 6, U is not F -representable. Thus, by Hypothesis 1, every M 2,p+2 p ∈ is a restriction of a matroid M ′ having a proper tree-decomposition (T, ) of M X thickness at most k so that M cl ′ (X ) for each v V (T ). By | M v ∈ M1 ∪ M2 ∈ Theorem 14, every such M ′ has a c-competitive matroid secretary algorithm, where 3n c =(e+1)k max 4(e + 1)t, epp . By Lemma 9, the same is true for every M . ∈M 8. Asymptotic Results We say asymptotically almost all matroids have a property if the proportion of matroids with ground set 1,...,n having the property tends to 1 as n approaches infinity. We finish our paper{ by showing} that asymptotically almost all matroids have a constant-competitive matroid secretary algorithm. We require two recent results of Pendavingh and van der Pol [28, 29]. Theorem 18 (Theorem 1.3 from [29]). There exists α> 0 so that asymptotically almost 3 all rank-r matroids on n elements have at least 1 α(log n) n bases. − n r 1 Theorem 19 (Theorem 16 from [28]). If β > 2 ln(2), then asymptotically almost all matroids on n elements have rank between n qβ√n and n + β√n. 2 − 2 Recall that RB denotes the algorithm that selects a basis B uniformly at random from the set of all bases of M, and chooses the elements of B as secretaries regardless of the weights. By Theorem 18, nearly every r-set in a typical matroid is a basis, so a uniformly random basis can be sampled in probabilistic polynomial time in almost all matroids by repeatedly choosing a uniformly random r-set until a basis is chosen. We now state and prove a stronger version of Theorem 3. Theorem 3. Let γ > 8 ln(2). For asymptotically almost all matroids M on n elements, the algorithm RB is (2 + γ )-competitive for M. p √n ′ 3 1 γ α(log n) Proof. Let γ′ ( 8 ln(2),γ). Note that for each α R we have > ∈ ∈ 2 − 4√n − n (2+ γ ) 1 for all sufficiently large n. Let n be a positive integer and let M be a matroid √n − p on n elements with ground set E = 1,...,n and rank r. Let E denote the set { } B ⊆ r of bases of M. For each e E let e = B : e B . By Theorems 18 and 19, ∈ B { ∈ B ∈ } 3 R > α(log n) n there exists α so that, asymptotically almost surely, (1 n ) r and ′ ∈ |B| − r > ( 1 γ )n. Thus, almost surely, 2 − 4√n E n > B : e B |Be| |{ ∈ r ∈ }| − r − |B| r n n = + n r − r |B| n r = |B| + 1 . r n n − r ! 3 ′ > 1 α(log n) + 1 γ 1 |B| − n 2 − 4√n − γ 1 > (2 + )− . |B| √n MATROID SECRETARIES 13

Let B0 be a basis chosen uniformly at random from , as per RB. For each e E, we γ 1 B ∈ have P(e B )= / > (2+ )− . Given a weighting w of M, we therefore have ∈ 0 |Be| |B| √n γ 1 γ 1 E(w(B )) = w(e)P(e B ) > (2 + )− w(E) > (2 + )− OPT(M,w). 0 ∈ 0 √n √n e E X∈ RB γ is thus (2 + √n )-competitive.

A rank-r matroid is paving if all its circuits have cardinality at least r. Although few well-known constructions of matroids have this property, it is believed to almost always hold. The following conjecture due to Mayhew, Newman, Welsh, and Whittle [26] is central in asymptotic matroid theory. Conjecture 20. Asymptotically almost all matroids on n elements are paving.

An alternative characterisation is that a matroid M is paving if and only if its truncation to rank r(M) 1 (which we denote as T (M)) is a uniform matroid. The matroid secretary problem− for uniform matroids is known as the multiple-choice secretary problem, and has been essentially completely solved by Kleinberg [22]. In the language of matroids, his result is the following.

5 1 Theorem 21. Let r > 25. There is an algorithm UNI that is (1 )− -competitive for − √r each uniform matroid Ur,n.

We slightly modify UNI to deal with paving matroids. Let PAV be the algorithm that, given a rank-(r> 26) weighted paving matroid (M,w), returns the output of UNI on the rank-(r 1) weighted uniform matroid (T (M),w). Note that, since every independent set in T−(M) is independent in M, this output is always legal.

6 1 Theorem 22. The algorithm PAV is (1 )− -competitive for every paving matroid − √r of rank at least 37.

Proof. Let B be a maximum-weight basis of a rank-r paving matroid M with weights w, and let x B have minimum weight. Now B x is a basis of T (M) and has weight r ∈1 > 1−{ } at least −r w(B), so OPT(T (M),w) (1 r ) OPT(M,w). Let I be the independent set output by UNI on T (M). Now − E > 5 > 5 1 (w(I)) (1 √r 1 ) OPT(T (M),w) (1 √r 1 )(1 r ) OPT(M,w). − − − − − 5 1 > 6 > PAV 6 1 Since (1 √r 1 )(1 r ) 1 √r for all r 37, is thus (1 √r )− -competitive. − − − − − As promised, we ﬁnish with a proof of Theorem 4, which we restate for convenience. Theorem 4. If asymptotically almost all matroids are paving, then thereisa (1+o(1))- competitive matroid secretary algorithm for asymptotically almost all matroids.

n Proof. By Theorem 19, almost all matroids have 37 < r 2 and so, conditioned on Conjecture 20, Theorem 22 implies that PAV is (1+ǫ(n))-competitive≈ for asymptotically almost all matroids on n elements, where ǫ(n) 6√2 . ≈ √n 14 T. HUYNH AND P. NELSON

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