# The Outer Limits of Contention Resolution on Matroids and Connections to the Secretary Problem

Contention resolution schemes have proven to be a useful and unifying abstraction for a variety of constrained optimization problems, in both offline and online arrival models. Much of prior work restricts attention to product distributions for the input set of elements, and studies contention resolution for increasingly general packing constraints, both offline and online. In this paper, we instead focus on generalizing the input distribution, restricting attention to matroid constraints in both the offline and online random arrival models. In particular, we study contention resolution when the input set is arbitrarily distributed, and may exhibit positive and/or negative correlations between elements. We characterize the distributions for which offline contention resolution is possible, and establish some of their basic closure properties. Our characterization can be interpreted as a distributional generalization of the matroid covering theorem. For the online random arrival model, we show that contention resolution is intimately tied to the secretary problem via two results. First, we show that a competitive algorithm for the matroid secretary problem implies that online contention resolution is essentially as powerful as offline contention resolution for matroids, so long as the algorithm is given the input distribution. Second, we reduce the matroid secretary problem to the design of an online contention resolution scheme of a particular form.

## Authors

• 8 publications
• ### Matroid Secretary is Equivalent to Contention Resolution

We show that the matroid secretary problem is equivalent to correlated c...
03/06/2021 ∙ by Shaddin Dughmi, et al. ∙ 0

• ### Online Stochastic Matching, Poisson Arrivals, and the Natural Linear Program

We study the online stochastic matching problem. Consider a bipartite gr...
03/24/2021 ∙ by Zhiyi Huang, et al. ∙ 0

• ### Random Order Contention Resolution Schemes

Contention resolution schemes have proven to be an incredibly powerful c...
04/07/2018 ∙ by Marek Adamczyk, et al. ∙ 0

• ### Greedy Bipartite Matching in Random Type Poisson Arrival Model

We introduce a new random input model for bipartite matching which we ca...
05/02/2018 ∙ by Allan Borodin, et al. ∙ 0

• ### Optimal Online Contention Resolution Schemes via Ex-Ante Prophet Inequalities

Online contention resolution schemes (OCRSs) were proposed by Feldman, S...
06/25/2018 ∙ by Euiwoong Lee, et al. ∙ 0

• ### An Efficient Online-Offline Method for Elliptic Homogenization Problems

We present a new numerical method for solving the elliptic homogenizatio...
07/19/2020 ∙ by Yufang Huang, et al. ∙ 0

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## 1 Introduction

The notion of a contention resolution scheme (CRS) abstracts a familiar task in constrained optimization: converting a (random) set-valued solution which is ex-ante (i.e., on average) feasible for a packing problem to one which is ex-post (i.e., always) feasible. Unlike randomized rounding algorithms more broadly, which in general may be catered to both the constraint and objective function at hand, a contention resolution scheme is specific only to the constraints of the problem, and preserves solution quality in a manner which is largely agnostic to the objective function111In its most general form, a CRS approximately preserves all linear objective functions simultaneously, whereas a monotone CRS approximately preserves all submodular objectives [10]. — element by element. Since they were formalized by Chekuri et al. [10], CRSs have been connected to a variety of online and offline computational tasks, including rounding the solutions of mathematical programs [10], online mechanism design and stochastic probing [17, 1], and prophet inequalities [17, 25].

Starting with [10], prior work defines an (offline) contention resolution scheme for a set system — where is a ground set of elements and is a downwards-closed family of feasible sets

— as an algorithm which takes as input the marginal probabilities

of a product distribution supported on as well as a random set of active elements, and must output a feasible subset of . The contention resolution scheme is -competitive if holds for all product distributions of interest — typically those with marginals

in the convex hull of indicator vectors of

(ex-ante feasibility). In online contention resolution schemes, first explored by Feldman et al. [17] and subsequently by Adamczyk and Włodarczyk [1] and Lee and Singla [25], the active elements arrive sequentially and the decision to include an element in must irrevocably be made online.

The existing literature on (offline and online) contention resolution has restricted attention to ex-ante-feasible and given product distributions, and varied the set system (e.g. matroids, knapsacks, and their intersections), all the while drawing connections to applications such as stochastic online problems, approximation algorithms, mechanism design, and prophet inequalities. In this paper, we restrict our attention to matroid constraints,222Though some of our results hold beyond matroids; we discuss those in the conclusion section. and instead focus on generalizing the class of input distributions. Our main goal is to understand the power and limitations of contention resolution, offline and online, in the presence of correlations in the input distribution and without regard to ex-ante feasibility. A secondary goal is to understand how knowledge of the distribution influences contention resolution. In pursuit of both goals, we draw connections between contention resolution and the secretary problem on matroids, shedding light on challenges posed by the matroid secretary conjecture in the process.

### Results

Our first set of results develops an understanding of offline contention resolution on matroids. We begin with a characterization of the class of -uncontentious distributions: those distributions permitting -competitive offline contention resolution for a given matroid. Most notably, we show that a distribution is -uncontentious if and only if it satisfies a family of inequalities, one for each subset of the ground set. Moreover, we observe that our inequality characterization is the natural generalization of the matroid base covering theorem (see e.g. [31]) from covering a set of elements to covering a distribution over sets of elements. In other words, we show that contention resolution is the natural distributional generalization of base covering. Leveraging our characterization, we establish some basic closure properties of the class of uncontentious distributions, and present some examples of uncontentious distributions exhibiting negative and positive correlation between elements. Finally, we examine whether knowledge of the distribution is essential to contention resolution, and exhibit an impossibility result: any contention resolution scheme which has nontrivial guarantees for all -uncontentious distributions cannot be prior-independent, in that it cannot make do with a finite number of samples from the distribution, even for very simple matroids.

Our second set of results concerns online contention resolution on matroids in the random arrival model, and in particular its connection to the matroid secretary problem. First, we show that a competitive secretary algorithm for a matroid implies that online contention resolution is essentially as powerful as offline contention resolution for that same matroid: a -competitive secretary algorithm implies that any -uncontentious distribution permits -competitive online contention resolution. Second, we provide evidence that contention resolution might hold the key to resolving the matroid secretary conjecture. As our most technically-involved result, we show that the random set of improving elements in a weighted matroid — as originally defined by Karger [20] — is O(1)-uncontentious. Since the improving elements can be recognized online, and moreover hold a constant fraction of the weighted rank of the matroid in expectation, our result can be loosely interpreted as a reduction from the matroid secretary problem to online contention resolution. There is one major caveat to this interpretation of our result, however: not only does the set of active (improving) elements arrive online, but so does the description of the uncontentious distribution from which that set is drawn.

### Additional Discussion of Related Work

#### Contention Resolution Schemes

Contention resolution schemes were introduced by Chekuri et al. [10], motivated by the problem of maximizing a submodular function subject to packing constraints. In particular, offline CRS were used to transform a randomized rounding algorithm which respects the packing constraints ex-ante to one which respects them ex-post, at the cost of the competitive ratio of the CRS. Their focus — like that of all related prior to ours — was on product input distributions, in which case the optimal competitive ratio of an offline CRS was shown to equal the worst-case correlation gap (first studied by [2, 7]) of the weighted rank function associated with the packing constraint. The characterization result of [10] result forms the basis for ours.

Online contention resolution was first studied by Feldman et al. [17], and applied to a number of online selection problems. They show that simple packing constraints — such as matroids, knapsacks, and matchings — permit constant competitive online contention resolution schemes even when elements arrive in an unknown and adversarial order. Moreover, they show how to combine competitive online schemes for different constraints in order to yield competitive online schemes for their intersection. Lee and Singla [25] obtain optimal online CRS in both the known adversarial-order model as well as the random-arrival model. Adamczyk and Włodarczyk [1] restrict attention to the random-arrival model, and obtain a particularly elegant algorithm and argument based on martingales, as well as improved competitive ratios for intersections of matroids and knapsacks.

#### Prophet Inequalities

Contention resolution is intimately tied to prophet inequality problems, also known as Bayesian online selection

problems. In the traditional model for these problems, independent real-valued random variables with known distributions arrive online in a known but adversarial order, and the goal is to select a subset of the variables with maximum sum, subject to a packing constraint. An

-competitive algorithm for a Bayesian online selection problem is also referred to as a prophet inequality with ratio , for historical reasons. Krengel, Sucheston, and Garling [22, 23] proved the first (classical) single-choice prophet inequality with ratio for selecting a single variable (i.e., a 1-uniform matroid packing constraint). Motivated by applications in algorithmic mechanism design, more recent work (e.g. [18, 3, 8, 32]) pursued prophet inequalities for more general packing constraints. Of particular note is the work of Kleinberg and Weinberg [21], who proved an optimal prophet inequality with ratio for matroids. Also notable is polylogarithmic prophet inequality for general packing constraints due to Rubinstein [28]. The (easier) variant of Bayesian online selection problems in which the variables arrive in a uniformly random order has also received recent interest, resulting in improved prophet inequalities for various packing constraints [15, 14, 5].

It was shown by Feldman et al. [17] that an online CRS yields a prophet inequality with the same competitive ratio, and in the same arrival model. A weak converse is also true, as shown by Lee and Singla [25]: a stronger form of prophet inequality — in particular one which competes against the ex-ante relaxation of the Bayesian online selection problem — yields an online CRS with the same competitive ratio and in the same arrival model.

#### Beyond Known Product Distributions

The vast majority of work on contention resolution or prophet inequalities, and all such work discussed thus far, restricts attention to known product distributions, and crucially exploits the product structure and knowledge of the distribution. We note the few exceptions next.

Rinott et al. [27] and Samuel-Cahn [29] show that the single-choice prophet inequality, and some slight generalizations, continue to hold for negatively dependent random variables. It is known [19] that there is no single-choice prophet inequality with ratio better than the number of variables in the presence of arbitrary positive correlation. Moreover, we are unaware of any nontrivial positive results for a class of distributions exhibiting positive correlation, in either prophet inequality or contention resolution models. We note that whereas [25] and [1] use specially-crafted correlated distributions as benchmarks, their results and techniques do not appear to shed light on contention resolution or prophet inequalities in the presence of correlation more generally.

Some work has relaxed the requirement that the distributions be known in prophet inequality problems. Azar et al. [4] study prophet inequality problems when only a single sample is given from each distribution, and obtain constant competitive ratios for a variety of constraints. Wang [30] obtains an optimal algorithm for the single-choice prophet inequality, with ratio , in the same single-sample model. Correa et al. [11] study the single-choice prophet inequality with i.i.d. variables drawn from an unknown distribution, and characterize the relationship between the competitive ratio and the number of samples available from the distribution.

#### Secretary Problems

In a generalized secretary problem, a set of adversarially chosen variables arrive online in a random order, and the goal is to select a subset of the variables with maximum sum subject to a packing constraint. The (classical) single-choice secretary problem, corresponding to a 1-uniform matroid constraint, was introduced and solved by Dynkin [13]. The matroid secretary problem was introduced by Babaioff et al. [6], and has since spawned a long line of work. Constant-competitive algorithms have been discovered for most natural matroids and for some alternative models – see Dinitz [12] for a semi-recent survey — though the general conjecture remains open. The state of the art for the general matroid secretary problem is a -competitive algorithm due to Lachish [24], which was henceforth simplified by Feldman et al. [16].

## 2 Preliminaries

### 2.1 Matroid Theory Basics

We use standard definitions from matroid theory; for details see [26, 31]. A matroid consists of a ground set of elements, and a family of independent sets, satisfying the three matroid axioms. A weighted matroid consists of a matroid together with weights on the elements. We use the standard notions of a dependent set, circuit, flat, and minor in a matroid. We denote the rank of a matroid as , and the rank of a set of elements in as , or when is clear from context. Overloading notation, we use to denote the weighted rank of a set — the maximum weight of an independent subset of — in the weighted matroid , though we omit the superscript when the matroid is clear from context. We note that both rank and weighted rank are submodular set functions on the ground set of the matroid. For and , we denote the restriction of to as , deletion of as , and contraction by as .

When is clear from context, and , we use to denote the vector indicating membership in . We often reference the matroid polytope of a matroid , defined as the convex hull of , or equivalently as the family of satisfying for all .

Throughout this paper we assume that any weighted matroid has distinct weights. This assumption is made merely to simplify some of our proofs, and — using standard tie-breaking arguments — can be shown to be without loss of generality in as much as our results are concerned. Under this assumption, we define as the (unique) maximum-weight independent subset of of minimum cardinality (excluding zero-weight elements), and we omit the superscript when the matroid is clear from context.

### 2.2 The Matroid Secretary Problem

In the matroid secretary problem, originally defined by [6] there is matroid with nonnegative weights on the elements. Elements arrive online in a uniformly random order , and an online algorithm must irrevocable accept or reject an element when it arrives, subject to accepting an independent set of . The algorithm is given at the outset (as an independence oracle), but the weights are chosen adversarially before the order is drawn and then are revealed online. The goal of the online algorithm is to maximize the expected weight of the accepted set of elements. We say that an algorithm is -competitive for a class of matroids if for every matroid in that class and every adversarial choice of , the expected weight of the accepted set (over the random choice of and any internal randomness of the algorithm) is at least an fraction of the maximum weight of an independent set of .

The matroid secretary conjecture, posed by [6], postulates that the matroid secretary problem admits an (online) algorithm which is constant-competitive for all matroids.

### 2.3 Miscellaneous Notation and Terminology

We denote the natural numbers by , and the nonnegative real numbers by . Given a set with weights , and a subset , we use the shorthand . We use as shorthand for the set . For a set , we use to denote the family of distributions over , and to denote the family of subsets of .

###### Definition 2.1.

Given a finite set and a distribution supported on , we define the vector of marginals of by . We refer to as the marginal probability of in .

###### Definition 2.2.

Given a finite set and , let be the distribution of the random set which includes each element of independently with probability . Equivalently, is the product distribution over subsets of with all marginal probabilities equal to .

###### Definition 2.3.

Given a finite set and a vector , let be the distribution of the random set which includes each element independently with probability . Equivalently, is the product distribution over subsets of with marginals .

## 3 Understanding Contention Resolution

### 3.1 The Basics of Contention Resolution

The definitions below are parametrized by a given matroid .

###### Definition 3.1.

A contention resolution map (CRM) is a randomized function from to with the property that for all . Such a map is -competitive for a distribution if, for , we have for all .

The following is known from Chekuri et al. [10].

###### Theorem 3.2 ([10]).

Every product distribution with marginals in admits an -competitive CRM.

###### Definition 3.3.

An online random-order contention resolution map (henceforth online CRM for short) is a contention resolution map which can be implemented as an algorithm in the online random-arrival model. In the online random-arrival model, is presented to the algorithm in a uniformly random order , and at the th step the algorithm learns whether is active — i.e., — and if so must make an irrevocable decision on whether to include in .

The following is known from Lee and Singla [25].

###### Theorem 3.4 ([25]).

Every product distribution with marginals in admits a -competitive online CRM.

### 3.2 Uncontentious Distributions and their Characterization

As shorthand, we refer to distributions which permit competitive (offline) CRMs as uncontentious.

###### Definition 3.5.

Fix a matroid . For , we say that a distribution is -uncontentious if it admits an -competitive contention resolution map.

For convenience, we also refer to a random set as -uncontentious if its distribution is -uncontentious. We prove the following characterization of uncontentious distributions.

###### Theorem 3.6.

Fix a matroid , and let be a distribution supported on . The following are equivalent for every .

1. [label=()]

2. is -uncontentious (i.e., admits an -competitive contention resolution map).

3. For every weight vector , the following holds for :

 E[rankw(R)]≥1αE[w(R)]
4. For every , the following holds for :

 E[|R∩F|]≤αE[rank(R∩F)]
###### Proof.

Property (a) implies property (c) by applying an -CRM to , noting that is necessarily an independent subset of .

 E[rank(R∩F)] ≥E[|ϕ(R)∩F|] =∑i∈FPr[i∈ϕ(R)] ≥1α∑i∈FPr[i∈R] =1αE[|R∩F|].

Property (c) implies property (b) by a summation argument. Sort and number the elements in decreasing order of weights , where denotes the weight of . Denote , and let , and . Recalling that the greedy algorithm computes the maximum weight independent subset of a matroid, we get:

 E[rankw(R)] =E[n∑i=1wi(rank(R∩Ei)−rank(R∩Ei−1))] Invoking the greedy algorithm on M|R =E[n∑i=1(wi−wi+1)rank(R∩Ei)] Reversing order of integration ≥1αE[n∑i=1(wi−wi+1)|R∩Ei|] Invoking (c) and linearity of expectations =1αE[n∑i=1wi(|R∩Ei|−|R∩Ei−1|)] Reversing order of integration. =1αE[w(R)].

Property (b) implies property (a) by a duality argument identical to that presented in [10]. We present a self-contained proof here. Let denote the marginals of . The distribution is -uncontentious if the optimal value of the following LP, with variables and for each deterministic CRM , is at least

The dual of the preceding LP is the following

 minimizeμsubject to∑i∈EwiPrR∼D[i∈ϕ(R)]≤μ,for all CRM ϕ.∑i∈Ewixi=1w⪰0

It is not hard to see that, at optimality, the binding constraint on corresponds to the CRM which maps each set to its maximum weight independent subset according to weights . It follows that the optimal value of the dual, and hence the primal, equals the minimum over all weight vectors of the ratio . (b) implies that this quantity is at least , as needed.

We note that the equivalence between (a) and (b) is essentially implicit in the arguments of [10]. Condition (c) is the most notable part of Theorem 3.6, in no small part because it is reminiscent of the matroid base covering theorem (see e.g., [31]). This theorem can equivalently be stated as follows: a (deterministic) set in a matroid can be covered by (i.e., expressed as a union of) independent sets if and only if for all . In light of part (c) of Theorem 3.6, a set of elements can be covered by independent sets if and only if the point distribution on is -uncontentious. Therefore, we can interpret contention resolution as a distributional generalization of base covering.

### 3.3 Elementary Properties of Uncontentious Distributions

###### Proposition 3.7.

Fix a matroid . Every -uncontentious distribution for has marginals .

###### Proof.

Let and . From Theorem 3.6 (c), for every set of ground set elements we have

 ∑i∈Fxi=E[|R∩F|]≤αE[rankM(R∩F)]≤α rankM(F).

These are the inequalities describing . ∎

###### Proposition 3.8.

Fix a matroid. A mixture of -uncontentious distributions is -uncontentious.

###### Proof.

Follows directly from Theorem 3.6 (b) and linearity of expectations. ∎

###### Proposition 3.9.

Fix a matroid , and let be a minor of , with . If a random set is -uncontentious in , then is also -uncontentious in .

###### Proof.

An independent set of is also independent in . Therefore, the proposition follows by simply applying the same CRM in the context of the larger matroid . ∎

###### Proposition 3.10.

Fix a matroid. Let be an -uncontentious random set, and let for some . The random set is -uncontentious as well.

###### Proof.

We use Theorem 3.6 (b). For any weight vector , submodularity of the weighted rank function implies that . It follows that . ∎

We note that Proposition  3.10 is tight for constant and (where by constant, we mean independent of the size of the matroid). In particular, the random set cannot be guaranteed to be -uncontentious for a constant . To see this, consider the a -uniform matroid with elements , and the following -uncontentious random set : For every singleton we have , and .

### 3.4 Examples of Uncontentious Distributions

We now present some examples of uncontentious distributions in order to develop a feel for them. As mentioned previously, and shown in [10], every product distribution with marginals in the matroid polytope is -uncontentious. More generally, if a distribution satisfies a certain strong notion of negative correlation, defined in [9], then it also is -uncontentious.

###### Proposition 3.11.

Fix a matroid , and let be a distribution with marginals . Assume that satisfies the property of increasing submodular expectations: for every submodular function we have .333In fact, it suffices for to satisfy the (weaker) property of increasing expectations for matroid rank functions (or, equivalently, their weighted sums). It follows that is -uncontentious.

###### Proof.

This is immediate by combining Theorem 3.6 (b) with the property of increasing submodular expectations and the fact that is -uncontentious. ∎

As shown in [9], the property of increasing submodular expectations is stronger than the following standard notion of negative correlation for : For all sets , and .444A natural question is whether negative correlation suffices for the distribution to be -uncontentious. This is open as far as we know. However, we can show that there are distributions exhibiting positive correlation which are also uncontentious for specific matroids. We now list some examples of uncontentious distributions exhibiting positive correlation.

###### Example 3.12.

Let be a -uniform matroid with elements where . Let the random set be empty with probability , and a uniformly random base of otherwise.

It is clear that is -uncontentious, since it is supported on the family of independent sets. However, for each distinct pair of elements and , we have , yet .

###### Example 3.13.

Let be a matroid with pairwise-disjoint bases . For each , let with probability , and let with the remaining probability . The set is -uncontentious, as evidenced by the CRM and a simple calculation. However, for and with , we have and yet .

### 3.5 Contention Resolution Schemes, Universality, and Prior Dependence

A contention resolution scheme (CRS) for a matroid and class of distributions is an algorithm which takes as input a (possibly partial) description of a distribution and a sample , and outputs satisfying . In effect, is a collection of contention resolution maps , one for each . In much of the prior work on contention resolution, was taken to be the class of product distributions with marginals in , and each is described completely via its marginals . In such a setting, the notion of a CRS offers little beyond the notion of a CRM, as each distribution gets its own dedicated CRM. More generally, however, we allow to be an arbitrary class of distributions, and we allow the description to be partial and/or random; for example, may be described by independent samples from .

Next, we set the stage by defining some desirable contention resolution schemes, and establish some limitations on their existence.

###### Definition 3.14.

Fix a matroid. For , an -Universal -competitive CRS is a CRS which is -competitive for the class of -uncontentious distributions.

By definition, there exists an (offline) -universal -competitive CRS for every and every matroid. The notion of a universal scheme becomes more interesting when we restrict dependence on the prior, as per the following definitions.

###### Definition 3.15.

Fix a matroid. A contention resolution scheme is said to be prior-independent if it is not given a complete description of as input, but rather is given a set of independent samples from . When the number of samples is , we say is a prior-independent -sample scheme. The number of samples may be function of the size of the matroid. If , we say the scheme is oblivious: the scheme consists of a single contention resolution map.

We now show that, if a scheme is universal, it cannot be prior-independent with any finite number of samples, even for very simple matroids.

###### Theorem 3.16.

Let be the 1-uniform matroid on elements. For every finite , and every , there does not exist a -competitive -universal CRS for which is prior independent with samples.

To prove Theorem 3.16, we first show that a prior-independent universal scheme implies the existence of an oblivious universal scheme; then we show that an oblivious universal scheme does not exist for the uniform matroid. This is captured in the two following lemmas.

###### Lemma 3.17.

Fix a matroid . If there exists a -competitive -universal CRS which is prior-independent with samples, then there exists an oblivious -competitive -universal scheme .

###### Proof.

Let be any -uncontentious distribution. Let , and let be the mixture of with the point distribution on the empty set with proportions and respectively. By Proposition 3.8 and the fact that the point distribution on the empty set is -uncontentious, it follows that is -uncontentious.

The CRS induces a CRM on the distribution , and by assumption is -competitive for . Since is prior-independent with samples, its induced CRM is a mixture over CRMs , where is a random vector of samples from . With probability at least , we have . For to be -competitive, in particular when with probability it is queried with a draw , a simple calculation shows that must be -competitive for for . As tends to , tends to , and a basic analytic argument implies that is -competitive for . Since was chosen arbitrarily among -uncontentious distributions, and does not depend on , it follows that the oblivious scheme with for every is -competitive and -universal.

###### Lemma 3.18.

The 1-uniform matroid with elements does not admit an oblivious -competitive -universal CRS for any .

###### Proof.

Let be the ground set of the matroid, and fix such that . An oblivious CRS consists of a single CRM . There exists at least one element such that . Let , and consider the following random set : For each we have with probability , and with the remaining probability . The random set is -uncontentious: consider the CRM with for , and . However, our original CRM is no better than -competitive for , since its probability of selecting is no more than . ∎

## 4 A (prior-dependent) Online Universal CRS from a Secretary Algorithm

We show that competitive matroid secretary algorithms imply that every contention resolution scheme can be made online without much loss in the competitive ratio. We prove the following.

###### Theorem 4.1.

Suppose that there is a -competitive online algorithm for the secretary problem on matroid . It follows that every -uncontentious distribution admits an online -competitive contention resolution map. In other words, for every there exists an online -competitive -universal contention resolution scheme for .

Let , and let . Recall that an online CRM operates in the following model: a set of active elements and a random permutation are (independently) sampled by nature, then arrive online in order . When arrives, it is revealed whether , and if so the online CRS must determine whether to select . The online CRM must only select an independent subset of .

Suppose we are given a secretary algorithm for with competitive ratio . Without loss of generality, we assume that selects only non-zero weight elements. Consider the following online CRM for , parametrized by a weight vector . When element arrives, if then it is presented to with weight , and if then it is presented to with weight . selects precisely the elements selected by .

###### Lemma 4.2.

For every distribution , we have

 ER∼D[w(ϕw(R))]≥1γER∼D[rankw(R)]
###### Proof.

Condition on the choice of , and let if and otherwise. are presented to in a uniformly random order, with weights , and is the set of elements selected by . Since is -competitive, it follows that . Since and , we are done. ∎

###### Lemma 4.3.

If is -uncontentious, then

 ER∼D[w(ϕw(R))]≥1γαER∼D[w(R)]
###### Proof.

Combining the previous lemma with Theorem 3.6 (b). ∎

Recall that we are assuming for now that we know the -uncontentious distribution , and we can design an online CRM accordingly. will be a random mixture of the maps described above; in particular, we will show that there exists a distribution over weight vectors such that the (randomized) online CRM which samples upfront then invokes is an online -CRM for .

For each element , let . For each weight vector and , let . For each distribution over weight vectors and element , let . Let be the family of all inclusion probabilities achievable by some online CRM of the form . It is immediate that , and hence is a convex subset of .

An online -CRM for of the form exists if and only if intersects with the upwards closed convex set

. Suppose for a contradiction that this intersection is empty; by the separating hyperplane theorem, this implies that there exists

such that for all . In particular, . Since and , we get a contradiction with Lemma 4.3. This concludes the proof of the theorem.

## 5 From Contention Resolution to a Secretary Algorithm?

One might hope that online contention resolution is equivalent to the secretary problem on matroids. In particular, does a competitive universal online CRS imply a competitive secretary algorithm? We make partial progress towards this question. In particular, we reduce the secretary problem to online contention resolution on a particular uncontentious distribution derived from the matroid and sample of its elements: the distribution of “improving elements”, as originally defined by Karger [20] for purposes different from ours.

###### Definition 5.1.

Fix a matroid with weights , and let . The random set of improving elements with parameter is sampled as follows: Let , and let . Equivalently, is the set of elements in which are not spanned by higher weight elements in . Another equivalent definition is .

The maximum-weight independent subset of the improving elements is -approximately optimal in expectation:

###### Fact 5.2.

Fix a weighted matroid , and let be the random set of improving elements with parameter . Each element of is in with probability . It follows that .

Note that the random set of improving elements does not follow a product distribution. In fact, elements are (weakly) positively correlated in general, as illustrated by the special case of the -uniform matroid on with weights , and : we get with probability for , and with probability . As our main result in this section, we nevertheless show that the random set of improving elements is uncontentious.

###### Theorem 5.3.

Let be a matroid with weights , and let . The random set of improving elements with parameter is -uncontentious.

Theorem 5.3 and Fact 5.2, taken together, essentially reduce the matroid secretary problem to online contention resolution for the distribution of the random set improving elements, with one caveat we will discuss shortly. In particular, consider the following blueprint for a secretary algorithm:

1. Let be the first elements arriving online.

2. Let be a sample of the set of improving elements with parameter .

3. After observing , the elements of arrive online in random order and are presented as such to an online contention resolution algorithm, along with their membership status in . Note that membership in can be determined “on the spot” as required for online contention resolution.555Technically, a CRM requires that elements of — rather than merely — be presented in uniform random order along with their membership status in . This is easily accomplished by appropriately interleaving the elements of — none of which are in — among the elements of .

Now given a -competitive -universal online CRS, we set and obtain a -competitive secretary algorithm. However, the following caveat prevents us from proving a formal theorem of this form: we cannot provide the online CRS with a complete description of the prior distribution. In particular, the distribution of improving elements — while fully described by the weighted matroid and the parameter — can not be fully described to the contention resolution algorithm prior to its invocation, since entries of are revealed online. As such, we learn both the sample and the distribution gradually as elements arrive. An oblivious universal online CRS would resolve this difficulty, but unfortunately we proved in Theorem 3.16 that such a CRS can not exist even for simple matroids and even offline. A reduction from the matroid secretary problem to contention resolution must therefore require the CRS to be partially prior independent, or make do with a non-universal CRS which focuses on the class of improving element distributions, or both. We leave exploration of these possibilities for future work, and discuss them further in the Conclusion section.

### 5.1 Proof of Theorem 5.3

Let , , and be as in Definition 5.1. We prove that is uncontentious by leveraging (c) from Theorem 3.6. In particular we will show that, for arbitrary .

 E[rank(R∩F)]≥pE[|R∩F|]

We break this up into three lemmas, outlined and proved below.

###### Lemma 5.4.
 E[rank(R∩F)]≥(1−p)E[|F∩OPTw(S∪F)|]
###### Proof.

Let , and note that . We condition on the random variable and show that the following holds conditionally

 E[rank(R∩F)]≥(1−p)|F∩OPTw(T⊎F)|]. (1)

Take . We will show that is in , and hence is in , with probability . Since and , it follows from the matroid axioms that . With probability we also have , in which case by definition.

Since is an independent set, (1) follows. ∎

###### Lemma 5.5.
 |F∩OPTw(S∪F)|≥|F∩OPTw(S)|
###### Proof.

We prove this by induction on a set with , initialized to at the base case. Consider how the value of changes as we add elements of to one by one. When adding an element to , there are three cases:

• : In this case, and .

• is not spanned by , and : In this case, , and therefore .

• is spanned by , and : In this case, elementary application of the matroid axioms implies that for some . Since , it follows that is either equal to or exceeds it by , depending on whether .

###### Lemma 5.6.
 E[|OPTw(S)∩F|]≥p1−pE[|R∩F|]
###### Proof.

For each , we will show that , which suffices.

Take , and let . Conditioning on , there are two cases:

• : It follows that and , with certainty.

• : With probability we have and therefore and . With the remaining probability we have and therefore and .

In both cases, the conditional probability that is at least times the conditional probability that . The lemma follows. ∎

## 6 Conclusions and Open Problems

In this paper, we begin an exploration of the power and limitations of contention resolution beyond known product distributions, as well as its connections to secretary problems. We hope that our results will enable future applications of contention resolution, and highlight a new possible approach to resolving the matroid secretary conjecture. We identify several open questions towards this agenda.

• Can the result of Theorem 4.1 be shown unconditionally; i.e., can we show a competitive universal online CRS for matroids without assuming the matroid secretary conjecture? Prior work which designs online CRSs for product distributions seems to crucially exploit the product structure of the distribution, so new ideas appear to be required.

• Recalling the caveat to our results from Section 5, can a tighter connection be made between the secretary problem and contention resolution? Is there a natural model of contention resolution on matroids which permits a reduction both from and to the matroid secretary problem?

• Our impossibility result of Theorem 3.16, paired with the caveat to our results from Section 5, suggest the following question: Is there a competitive prior-independent (or even oblivious) CRS for the class of improving element distributions? A positive answer would resolve the matroid secretary conjecture. A possibly even easier question is whether there is a competitive prior-independent (or even oblivious) CRS for ex-ante-feasible product distributions.

• Do more general set systems permit a characterization with a finite set of inequalities, a-la Theorem 3.6?

We restricted our attention to matroids in the paper, though some notes are in order on extensions of our results to more general constraints. In the characterization of Theorem 3.6, the equivalence of (a) and (b) holds for a general downwards-closed set systems, and is implicit in the arguments of [10]. The equivalence with (c) exploits the matroid structure, however. Theorem 4.1 also holds for general downwards-closed set systems, and our proof does not invoke the matroid assumption. The results and arguments of Section 5, in particular Theorem 5.3, heavily rely on the matroid structure and do not appear to be easily extensible beyond matroids. We leave further extensions of our results beyond matroids for future work.

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