This paper follows in the hallowed TCS tradition of reducing the number of questions without providing any answers. We establish an equivalence between one of the central open problems in online algorithm design, the matroid secretary conjecture, and the increasingly rich and fruitful framework of contention resolution. Specifically, we show that the matroid secretary problem admits a constant-competitive algorithm if and only if matroid contention resolution for general (correlated) distributions is approximately as powerful (up to a constant) in the online random-order model as it is in the offline model. Our result paves the way for application of the many recent advances in contention resolution, and in stochastic decision-making problems more generally, to resolving the conjecture.
The classical (single-choice) secretary problem , and its many subsequent combinatorial generalizations, capture the essence of online decision making when adversarial datapoints arrive in a non-adversarial order. The paradigmatic such generalization is the matroid secretary problem, originally proposed by Babaioff et al. . Here, elements of a known matroid arrive online in a uniformly random order, each equipped with a nonnegative weight chosen at the outset by an adversary. An algorithm for this problem must decide online whether to accept or reject each element, knowing only the weights of the elements which have arrived thus far, subject to accepting an independent set of the matroid. The goal is to maximize the total weight of accepted elements. The matroid secretary conjecture of  postulates the existence of an (online) algorithm for this problem which is constant competitive, as compared to the offline optimal, for all matroids. Though much prior work has designed competitive algorithms for specific classes of matroids, the general conjecture has remained open.
Recent years have seen an explosion of interest in a variety of online decision-making problems of a similar flavor, albeit distinguished from secretary problems in that the uncertainty in the data is stochastic, with known distribution, rather than adversarial. Such models include variants and generalizations of the classical prophet inequality, adaptive stochastic optimization models such as stochastic probing, and what is increasingly emerging as the technical core of such problems: contention resolution. The offline model of contention resolution was introduced by Chekuri et al. , motivated by applications to approximation algorithm design. It has since been extended to various online settings (e.g. [15, 1]), and emerged as the basic technical building block of a number of important results for stochastic decision-making problems (see e.g. [15, 1, 26, 5]).
In contention resolution, elements of a set system — for our purposes, a matroid — are each equipped with a single-bit stochastic datapoint indicating whether that element is active or inactive
. The joint distribution of these datapoints, henceforth referred to as theprior distribution, is assumed to be known and given. An algorithm for this problem — which we often refer to as a contention resolution scheme (CRS) — is tasked with accepting an independent set of active elements with the goal of maximizing the balance ratio
: the minimum, over all elements, of the ratio of the probability the element is accepted to the probability the element is active. When a CRS achieves a constant balance ratio for a distribution or class of distributions, we simply call itbalanced. In the original offline setting of contention resolution, the algorithm observes all datapoints before choosing which elements to accept. Most pertinent for us is the online random order setting: elements and their datapoints arrive in a uniformly random order, and the algorithm must must decide whether to accept or reject each element, subject to independence, knowing only the activity status of elements which have arrived thus far.
Most work on contention resolution has restricted attention to product prior distributions: elements are active independently, with given probabilities. Sweeping positive results hold for product priors, for both offline and online contention resolution on matroids (see [6, 15]), and those results tend to extend to negative correlation between elements. In contrast, it is easy to see that not much is possible in the presence of unrestrained positive correlation, even offline. We build on our recent work in , which observed that some forms of positive correlation are relatively “benign” for contention resolution, at least in the offline setting. We characterized uncontentious distributions — those permitting a balanced offline CRS — and delineated some of their basic properties. Leveraging this characterization, we then related the matroid secretary conjecture to online contention resolution for these uncontentious distributions, via a pair of complementary reductions.
One of the reductions in  is of unambiguous significance, and follows from unsurprising duality arguments: given a competitive algorithm for the secretary problem on a matroid, one can derive an (online) random-order CRS which is balanced for every uncontentious distribution on that matroid. We refer to such an online CRS, which is balanced for all uncontentious (correlated) distributions, as universal.
The second reduction in  is from the matroid secretary problem to a more restrictive model for online contention resolution, and therefore falls short of establishing an equivalence between the two problems. At the center of this reduction is the (random) set of improving elements for a weighted matroid, as originally defined by Karger : a random sample consisting of a constant fraction of the elements is set aside, and an element outside the sample is deemed improving if it increases the weighted rank of the sample. It is shown in  that improving elements, though they may exhibit nontrivial positive correlation, are nonetheless uncontentious — i.e., they admit a balanced offline CRS. Achieving such balance online as well, in the random-order model, is then shown to imply the matroid secretary conjecture. The major caveat to this reduction is the following: the prior distribution of improving elements is only partially known when the online CRS is invoked by this reduction. In essence, the reduction requires online contention resolution in a nontraditional, and more restrictive, model of a partially-described prior distribution.
Results and Technical Approach
This is where the present paper picks up. We restrict attention to matroids, and derive a reduction from the secretary problem to random-order contention resolution with a fully known and given uncontentious distribution. In doing so, we establish equivalence of the matroid secretary conjecture and universal random-order contention resolution on matroids. A conceptual take-away from our result is that the key challenge of the matroid secretary problem lies in resolving contention for random sets exhibiting positive correlation, in particular when such correlation is “benign” for offline contention resolution.
We face a series of technical obstacles, which we isolate by expressing our reduction as the composition of three component reductions. This takes us through two “bridge problems” along the way. The first of these bridge problems is the correlated version of the familiar matroid prophet secretary problem of Ehsani et al. , which relaxes the matroid secretary problem by assuming that weights are drawn from a known distribution rather than adversarially.111An alternate, equivalent, description of the prophet secretary problem is as a relaxation of the prophet inequality problem to random order arrivals. The second bridge problem is a generalization of contention resolution — in particular on matroids, in the online random-order model — which we define and term labeled contention resolution. Here, each active element comes with a stochastic label, and balance is evaluated with respect to element/label pairs rather than merely with respect to elements. Figure 1 summarizes the cycle of reductions between all four problems, which we conclude are all equivalent up to constant factors in their competitive and balance ratios.
Our first component reduction, motivated by the aforementioned caveat to the results of 
, is from the secretary problem to the prophet secretary problem on matroids. Fairly standard duality arguments allow us to replace the adversarial weight vector in the secretary problem with a stochastic one of known distribution. Modulo some simple normalization and discretization of the weights, at the expense of a constant in the competitive ratio, this yields an instance of the prophet secretary problem. With a stochastic weight vector drawn from a known distribution, we now face aknown mixture of improving element distributions. Moreover, since it is shown in  that uncontentious distributions are closed under mixing, this mixture is still uncontentious. At first glance, it would appear that we have now resolved the caveat of ,
Unfortunately, shifting to a stochastic weight vector introduces a new obstacle. With the set of improving elements now correlated with the vector of element weights, balanced contention resolution no longer guarantees extracting a constant fraction of the expected weight of the set. This is because a contention resolution scheme may preferentially accept an improving element when it has low weight, and reject it when it has high weight, while still satisfying the balance requirement in the aggregate. In fact, we show by way of a simple example that egregious instantiations of this phenomenon are not difficult to come by. This motivates our reduction from the matroid prophet secretary problem to labeled contention resolution, also in the online random-order model. By labeling each element with its weight, and requiring balance with respect to element/label pairs, we exclude contention resolution policies which favor low-weight elements.
Our final, and most technically involved, component reduction is from labeled to unlabeled contention resolution, for matroids in the random-order model. Such a reduction would be trivial in the offline setting: by thinking of each (element,label) pair as a distinct parallel copy of the element, we obtain an equivalent instance of unlabeled contention resolution, albeit on a larger matroid. One might hope for an online version of this reduction, which interleaves inactive element/label pairs amidst the active element/label pairs from the labeled instance. However, we argue at length that such an approach appears unlikely to succeed, for two fundamental reasons. First, we present evidence that not any interleaving will do: we show formally that an arbitrary interleaving produces a contention resolution problem which does not admit a constant balance ratio, ruling out such a reduction if the matroid secretary conjecture were true. In other words, it really is important that both active and inactive elements are ordered randomly in random-order contention resolution, since the semi-random generalization which provides no guarantees on the positions of inactive elements is strictly more difficult (assuming the matroid secretary conjecture). Second, we argue that natural interleaving approaches fail to produce a uniformly-random sequence of element/label pairs (both active and inactive), even in an approximate sense. Roughly speaking, the difficulty is thus: natural online reductions from the labeled problem to its unlabeled counterpart must randomly interleave many (inactive) labeled copies of an element early into the sequence, well before the active copy (if any) arrives online. Without knowing the identity of this active copy (if any) in advance, there simply is not enough information, in a statistical distance sense, to approximately simulate a uniformly-random interleaving. We overcome these obstacles by “blowing up” the matroid even further, creating a large number of identical duplicates of each label. As the number of duplicates grows large, a random order of element/label pairs converges in distribution to a deterministic order (modulo the equivalence relation between duplicates). The required interleaving of inactive element/label pairs is now essentially deterministic, and in particular approximately invariant — in a statistical distance sense — to the identities of active elements and their labels.
Additional Discussion of Related Work
Contention resolution in the offline setting was formalized by Chekuri et al. , motivated by applications to approximation algorithm design via randomized rounding. For product priors and a given packing set system,  shows that the optimal offline balance ratio equals the worst-case correlation gap, as defined by Agrawal et al. , of the set system’s weighted rank function. Starting with the work of Feldman et al. , contention resolution was extended to online settings and applied to a variety of problems in mechanism design and adaptive stochastic optimization (see also [1, 26]). Regardless of the set system, balanced contention resolution is obviously only possible for priors that are (approximately) ex-ante feasible: the random set is feasible on average, in the sense that the per-element marginal probabilities lie in the polytope associated with the set system. One message of the aforementioned prior work is that — for product priors and many natural set systems such as matroids, knapsacks, and their intersections — approximate ex-ante feasibility is also sufficient for balanced contention resolution, whether offline or online in any natural arrival model. Beyond product priors, the difficulty lies with resolving contention in the presence of positive correlation. Without any assumptions on the kind or degree of positive correlation, there exist simple examples of highly contentious yet ex-ante feasible distributions.222Consider a -uniform matroids with elements, all of which are active simultaneously with probability . Motivated by the existence of relatively “benign” forms of positive correlation, and the connection thereof to the secretary problem, our work in  characterized uncontentious distributions regardless of correlation, and established some of their basic properties.
The (single-choice) secretary problem is due to Dynkin . It was subsequently generalized to a uniform matroid constraint by Kleinberg , and to a general matroid constraint by Babaioff et al. . A long line of work has designed constant-competitive algorithms for special cases of the matroid secretary problem, and we refer the reader to the semi-recent survey by Dinitz . The current state-of-the art for general matroids is an -competitive algorithm due to Lachish , which was since simplified by Feldman et al. . Beyond matroids, the secretary problem with general packing constraints was recently studied by Rubinstein .
Closely related to the secretary problem are the prophet inequality problem and the prophet secretary problem, which analogously admit combinatorial generalizations to matroids and other packing set systems. Whereas a secretary problem features adversarial data (i.e., element weights) arriving online in a random order, a prophet inequality problem features stochastic data (typically assumed to be independent) arriving online in an adversarial order. A prophet secretary problem is a relaxation of both, featuring stochastic data arriving online in a random order. The original (single-choice) prophet inequality is due to Krengel, Sucheston, and Garling [23, 24], and was generalized to matroids by Kleinberg and Weinberg . Generalizations beyond matroids have also received much study; see for example [9, 10, 29]. The (single-choice) prophet secretary problem was introduced by Esfandiari et al. , and further studied in . Generalizations to combinatorial constraints, including matroids, were studied by Ehsani et al. .
One take-away from this paper is that stochastic decision making in the presence of correlation, and in particular positive correlation, is deserving of more attention. Most prior work on aforementioned stochastic decision-making models restricts attention to product priors, with a few exceptions which we now mention. For contention resolution, the only exception we are aware of is our aforementioned work . The classical (single-choice) prophet inequality was extended to negatively correlated variables by Rinott and Samuel Cahn [28, 30], whereas no nontrivial prophet inequality holds in the presence of unrestrained positive correlation . The only nontrivial prophet inequalities we are aware of in the presence of positive correlation are from the recent work of Immorlica et al. , who pose a particular linear model of correlated distributions.
2.1 Miscellaneous Notation and Terminology
We denote the natural numbers by , the real numbers by , and the nonnegative real numbers by . We also use as shorthand for the set of integers .
For a set , we use to denote the family of distributions over , use to denote the family of subsets of , and use to denote finite strings with alphabet . When is finite, we use to denote uniformly sampling from . When is equipped with weights , and , we use the shorthand . For a distribution supported on , we often refer to the vector of marginals of , where is the marginal probability of in .
For a finite set , we use to denote the family of permutations of . We think of as a bijection from the integers to , where is the th element of the permutation.
2.2 Matroid Theory Basics
We use standard definitions from matroid theory; for details see [27, 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. For , we denote the restriction of to as , deletion of as , and contraction by as .
We use to denote the rank — i.e. the maximum cardinality of an independent set — of a matroid , and to denote the weighted rank — i.e. the maximum weight of an independent set — of a weighted matroid . Overloading notation, we use to denote the rank of , and to denote the weighted rank of with weights , though we omit the superscript when the matroid is clear from context. We note that both and are submodular set functions on the ground set of the matroid.
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 .
In parts of this paper, we restrict attention to weighted matroids where all non-zero weights are distinct. 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. We also use as shorthand for the maximum-weight independent set of of minimum cardinality.
We also assume without loss of generality that each matroid we consider has no loops (i.e., each singleton is independent), since we can restrict the matroid to non-loop elements at the outset.
2.3 The Matroid Secretary Problem
In the matroid secretary problem, originally defined by Babaioff et al. , there is matroid with nonnegative weights on the elements. Elements arrive online in a uniformly random order , and an online algorithm must irrevocably accept or reject an element when it arrives, subject to accepting an independent set of . Only the matroid is given to the algorithm at the outset — say, as an independence oracle. The weights , on the other hand, are chosen adversarially, and without knowledge of the random order . The elements then arrive online, along with their weights, in the random order .
The goal of the online algorithm is to maximize the expected weight of the accepted set of elements. Given , we say that an algorithm for the secretary problem is -competitive for a class of matroids, in the worst-case, if for every matroid in that class and every adversarial choice of , the expected weight of the accepted set (over the random order and any internal randomness of the algorithm) is at least a fraction of the offline optimal — i.e., at least .
The matroid secretary conjecture, posed by Babaioff et al. , can be stated as follows
Conjecture 1 ().
There exists an absolute constant such that the matroid secretary problem admits an (online) algorithm which is -competitive for all matroids.
We note that we are considering the known matroid model of the secretary problem, which is the original model defined by Babaioff et al. . A potentially more challenging variant, where only the size of the ground set is known at the outset, but the structure of the matroid is revealed online, has also been considered (see e.g. ). We are unaware of any evidence of a separation between the two models, and in fact most algorithms in the matroid secretary literature work for both models. Nonetheless, the known setting lends itself best to our reduction.
2.4 The Matroid Prophet Secretary Problem
The matroid prophet secretary problem relaxes the matroid secretary problem by assuming that the weights are drawn from a known prior distribution , independent of the random order , rather than being chosen adversarially. Both and are given at the outset, whereas the random order and the realized weight vector are revealed online as elements arrive. The single-choice prophet secretary problem was introduced by Esfandiari et al. , and later studied for matroids and other set systems by Ehsani et al. . To our knowledge, all prior work on the prophet secretary problem has considered independent weights — i.e., is a product distribution. We make no assumption here, allowing the weights to be correlated arbitrarily.
Given , we say that an algorithm for the secretary problem is -competitive for a class of matroids and prior distributions if for every matroid and distribution in that class, the expected weight of the accepted set (over the random order , the weight vector , and any internal randomness of the algorithm) is at least a fraction of the expected offline optimal — i.e., at least .
The matroid prophet secretary problem also relaxes the matroid prophet inequality problem of Kleinberg and Weinberg , in particular by assuming that the arrival order is uniformly random rather than adversarial. It follows that the competitive ratio of for the matroid prophet inequality from  generalizes to the matroid prophet secretary problem when weights are independent. This was improved to by . No constant is known for the matroid prophet secretary problem with general correlated priors, though one would immediately follow from the matroid secretary conjecture. In fact, along the way to our results we show that the existence of a constant competitive algorithm for the matroid prophet secretary problem, with arbitrary matroids and arbitrary correlated priors, is equivalent to the matroid secretary conjecture.
2.5 Contention Resolution
For classical contention resolution, we roughly follow the notation and terminology from . Let be a set system. A contention resolution map (CRM) for is a randomized function from to with the property that for all . Such a map is -balanced for a distribution if, for , we have for all . Every CRM can be implemented by some algorithm in the offline model, where the set is provided to the algorithm at the outset; when we emphasize this we sometimes say it is an offline CRM. If a distribution admits an (offline) -balanced CRM for , we say is -uncontentious for . When and is -uncontentious, we often abuse terminology and also say that the random set is -uncontentious. We omit reference to in these definitions when the set system is clear from context.
The following Theorem characterizes uncontentious distributions for matroids, and the subsequent proposition is an immediate consequence; both are shown in .
Theorem 2.1 ().
Fix a matroid , and let . The following are equivalent for every .
is -uncontentious (i.e., admits an -balanced offline contention resolution map).
For every weight vector , the following holds for :
For every , the following holds for :
Proposition 2.2 ().
Fix a matroid. A mixture of -uncontentious distributions is -uncontentious.
An online random-order contention resolution map (henceforth RO-CRM for short) is a CRM which can be implemented as an algorithm in the online random-order model. In the online random-order model, is presented to the algorithm in a uniformly random order , and at the th step the algorithm learns whether is active — i.e., whether — and if so must make an irrevocable decision on whether to accept — i.e., include it in the set — or otherwise reject it.
A contention resolution scheme (CRS) for a set system and class of distributions is an algorithm which takes as input a description of a distribution and a sample , and outputs satisfying . In effect, is a collection of contention resolution maps , one for each . If each is -balanced for , we say that the is an -balanced CRS for . If each is an RO-CRM, we say that is an online random order CRS (RO-CRS). Every CRS can be implemented offline, and we say offline CRS if we wish to emphasize this.
In much of the prior work on contention resolution schemes, was taken to be the class of product distributions with marginals in , and each is described completely via its marginals . Here, we consider more elaborate classes , most notably -uncontentious distributions for various . We refer to a balanced CRS for such a class as universal.
Fix a set system. For , a -universal CRS is a CRS which is -balanced for the class of -uncontentious distributions.
The above definition is only interesting in restricted input models: there always exists an (offline) -universal CRS for every and every set system, by definition. Moreover, it is only interesting for , since the identity CRS is -balanced otherwise. We will be concerned with the existence of -universal RO-CRS’s, for constants , and matroid set systems.
3 High Level Approach
Our main result is the following.
The matroid secretary conjecture holds if and only if there exists constants such that every matroid admits a -universal CRS in the online random order model.
We note that the forward direction of this theorem, namely that the matroid secretary conjecture implies the existence of a universal RO-CRS, was already shown in [8, Theorem 4.1]. In fact, it was shown there that the matroid secretary conjecture implies that for every there exists a non-zero such that every matroid admits a -universal RO-CRS. Therefore, one can equivalently state Theorem 3.1 by quantifying universally over .333A notable, and perhaps surprising, consequence is that the existence of an -universal RO-CRS on matroids for some implies the same for all other .
In this paper, we prove the backward direction of Theorem 3.1 by reducing the matroid secretary conjecture to universal random-order contention resolution. We emphasize that unlike in [8, Section 5], we reduce the matroid secretary problem to random-order contention resolution in the traditional setting of a known and given prior distribution over sets.
First, we introduce a “bridge problem” which we term labeled contention resolution, generalizing classical contention resolution.
3.1 Labeled Contention Resolution
Labeled contention resolution generalizes (classical) contention resolution to a setting where each active element arrives with a label, and a scheme is -balanced if each (element,label) pair is accepted with probability at least -times the probability that the element is active with that label. More formally, let be a set system, and let be a finite set of labels. A labeled set for is a pair where and is an labeling of with . A labeled contention resolution map (LCRM) for takes as input such a labeled set , where is again referred to as the set of active elements, and outputs with the property . Such an LCRM is -balanced for a distribution over labeled sets for if, when the input is drawn from , we have for every and . When an (offline) -balanced LCRM exists for a distribution over labeled sets, we again say that is -uncontentious for . When and is -uncontentious, we often abuse terminology and also say that the random labeled set is -uncontentious. We omit reference to and/or when they are clear from context.
In the online random order setting, elements of arrive in a uniformly random order , and at the th step the algorithm learns whether is active — i.e., whether — and if so the algorithm also learns its label . The algorithm must then make an irrevocable decision on whether to accept .
Remaining notions and terms from unlabeled contention resolution generalize naturally to the labeled setting: A labeled contention resolutions scheme (LCRS) for set system takes as input a description of a distribution over labeled sets for and some finite set of labels, and implements an LCRM for . As before, an LCRS may offline or online, and is -balanced for a class of distributions if, for in that class, is -balanced for . We focus on -universal RO-LCRSs: those which are -balanced for all -uncontentious distributions over labeled sets (for every finite set of labels), in the online random order model.
Note that classical contention resolution is the special case of labeled contention resolution in which each element of the ground set is associated with a single label. We also note that labeled contention resolution offers little beyond classical contention resolution in the offline model for matroids: if we think of labeled copies of an element as parallel elements in a new matroid, we obtain an equivalent unlabeled contention resolution problem.444More generally, this is also the case for any family of set systems closed under duplication of elements. Formally, for a matroid and set
of labels, we define their “tensor product”, where includes for each and each . It is easy to verify that is a matroid: each element of was just replaced with parallel elements, one for each label. In the offline setting, a labeled contention resolution problem on and is equivalent to an unlabeled one on . In particular, we can think of a labeled set for and as an (unlabeled) set for . It follows that a random labeled set is -ucontentious (in the labeled sense, for and ) if and only if the corresponding unlabeled set is -uncontentious (in the unlabeled sense, for ). Given this equivalence, the following labeled analogue of Proposition 2.2, which will be useful in Section 5, is immediate.
Fix a matroid and a set of labels. A mixture of -uncontentious distributions over labeled sets is -uncontentious.
Our main concern will be labeled contention resolution in the online random order model. Unlike in the offline model, the reduction from the labeled to the unlabeled problem is nontrivial, as will be shown in Section 6.555Though not a concern of this paper, the relationship between the labeled and unlabeled problems is interesting to contemplate in other online order models. In the adversarial order model, it is not too hard to see that the two problems are again equivalent. In the free order model, however, no such equivalence is immediately obvious.
3.2 Proof Outline
Our proof is the composition of three reductions, one from the matroid secretary problem to the (correlated) matroid prophet secretary problem, one from the matroid prophet secretary problem to universal labeled contention resolution, and finally one from labeled to unlabeled contention resolution, all in the online random order model. Theorem 3.1 is a consequence of the following three lemmas, combined with the reverse reduction in [8, Theorem 4.1].
Fix a constant . If there is a -competitive algorithm for the matroid prophet secretary problem with finitely-supported arbitrarily-correlated priors, then there is a -competitive algorithm for the matroid secretary problem.
Fix constants . If there is a -universal RO-LCRS for a matroid , then there is a -competitive algorithm for the matroid prophet secretary problem on with finitely-supported arbitrarily-correlated priors.
Fix constants . If every matroid admits a -universal RO-CRS, then for each , every matroid admits a -universal RO-LCRS.
4 Reducing Secretary to Prophet Secretary
We now reduce the matroid secretary problem to the matroid prophet secretary problem with a finitely-supported, arbitrarily-correlated prior distribution on weight vectors. Our reduction loses a constant factor in the competitive ratio.
First, we observe that we can restrict attention to instances of the matroid secretary problem which are normalized, in that the offline optimal value is roughly , and discretized, in that weights are contained in a known finite set. The following Sublemma is shown using standard arguments, and its proof is therefore deferred to Appendix A. We note that we make no attempt to optimize the constants here.
The matroid secretary problem reduces, at a cost of a factor of in the competitive ratio, to its special case where the matroid and weights are guaranteed to satisfy the following:
Discretized: The weight of each element is either zero, or is an integer power of contained in .
We now fix the matroid , and reduce the normalized and discretized matroid secretary problem on , in the sense of Sublemma 4.1, to the prophet secretary problem on the same matroid , losing a constant factor in the reduction. To keep the proof generic, we use to denote the (known) constant such that offline optimal value is guaranteed to lie in , and use to denote the (known) finite set of permissible weights for . We also use to denote the (known) finite set of permissible weight vectors for , yielding a normalized and discretized instance.
Our reduction invokes minimax duality to replace the adversarially-chosen weight vector , as in the secretary problem, with a weight vector drawn from a known and arbitrarily-correlated distribution , as in the prophet secretary problem. Discretization is needed so that we can invoke the minimax theorem for finite games. However, straightforward application the minimax theorem produces a variant of the prophet secretary problem where the goal is to maximize the expected ratio between the online and offline optimal values, rather than the (usual) goal of maximizing the ratio of the two expectations. Normalization serves to obviate the distinction between these two goals.
An algorithm for normalized and discretized secretary problem on maps a permissible weight vector and an order on the elements to an independent set . When is deterministic, we can think of it as a function from to . Since , , and are all finite sets, there are finitely many such functions that are computable online. A randomized algorithm can be thought of as simply a distribution over these functions. For an algorithm , be it deterministic or randomized, we use to denote the expected weight of the independent set chosen by algorithm for weight vector , where expectation is over the uniformly random order . Note that
is a random variable whenis randomized.
Consider the following finite two-player zero-sum game played between an algorithm player and an adversary. The pure strategies of the algorithm player are deterministic algorithms for the secretary problem on , which we think of as functions from to , and mixed strategies are naturally randomized algorithms. The pure strategies for the adversary are the permissible weight vectors . The algorithm player’s utility if he plays a deterministic algorithm and the adversary plays is simply the competitive ratio of on , given by .
For a randomized algorithm for the secretary problem, its competitive ratio on a weight vector is given by, where expectation is over any internal randomness in . The worst-case competitive ratio of is at least if
Inequality (1) can be equivalently interpreted as follows: if the algorithm player moves first by playing mixed strategy , he guarantees an expected utility of at least regardless of the response
of the adversary. By the minimax theorem for finite two-player zero-sum games, and through the associated dual pair of linear programs, the design of an algorithmsatisfying Equation (1) reduces to the following (dual) problem faced by an algorithm player who moves second: for each (a mixed strategy of the adversary), design an algorithm for the secretary problem on which satisfies:
We note that our minimax reduction is not necessarily efficient, as both players in our zero-sum game have exponentially many strategies in the size of the ground set of the matroid. An efficient reduction is not necessary, however, for our (information theoretic) result. We also note that there is no benefit to randomization in when computational efficiency is not a concern: a randomized algorithm satisfying inequality (2) can be derandomized, albeit perhaps inefficiently, by appropriately choosing a deterministic algorithm in its support. Nevertheless, we permit randomization in for our reduction to be as general as possible.666This is convenient since the reduction from the prophet secretary problem to contention resolution in Sections 5 and 6 will, in general, produce a randomized algorithm, as contention resolution schemes are typically randomized.
Finally, we claim that a -competitive algorithm for the prophet secretary problem on and satisfies inequality (2) with . By definition, the assumption that is a -competitive prophet secretary algorithm for and can be written as
It follows that
|( for all )|
|( for all )|
Since our reduction lost a factor of in the normalization and discretization step (Sublemma 4.1), and a factor of due to the discrepancy between the objective of the matroid prophet secretary problem and the dual of the matroid secretary problem, this completes the proof of Lemma 3.3 with the claimed loss in the competitive ratio of .
5 Reducing Prophet Secretary to Labeled Contention Resolution
Recall that in , the matroid secretary problem is “reduced”, with a major caveat, to random-order contention resolution for the set of improving elements. The random set of improving elements, adapted from the original definition of Karger , is defined next. In our definition for improving elements, and in this section generally, we assume the non-zero entries of a matroid weight vector are distinct; this is without loss of generality by standard tie-breaking arguments, and serves to simplify our definitions and proofs.
Let be a matroid, let be a parameter, and let be a weight vector. The random set of improving elements for is sampled as follows: Let include each element independently with probability , and let .777Equivalently, is the set of elements in which are not spanned by higher weight elements in . Another equivalent definition is . We say improves , in the sense that adding to improves its weighted rank. We use to denote the distribution of .
Key to the “reduction” in  are the following two properties of the set of improving elements.
Fact 5.2 ().
Let be a matroid, let , and let be a weight vector. The set of improving elements for holds a fraction of the weighted rank of in expectation. Formally:
Theorem 5.3 ().
Let be a matroid, let , and let be a weight vector. The distribution is -uncontentious for .
Consider the following “reduction”, outlined in , from the secretary problem on an -element matroid , and (a-priori unknown) weights , to online contention resolution: Observe the weights of the first elements arriving online, then resolve contention for the set of elements which improve as they arrive online.888Note that membership in can be determined online, as needed. When is a constant, follows an -uncontentious distribution (Theorem 5.3), and holds a constant fraction of the optimal value (Fact 5.2). Therefore, it suffices to resolve contention online for almost as well (up to a constant in the balance ratio) as is possible offline. Since arrive in uniformly random order after , and we can “interleave” among them to create a uniformly random order on , universal contention resolution in the random order model suffices. The important caveat to this “reduction” of  is that the distribution of , being a function of the unknown and adversarial weight vector , is unknown to the contention resolution scheme. This is a departure from the traditional notion of contention resolution, involving a known and given prior distribution.
In this section, we overcome this caveat by instead reducing from the prophet secretary problem, where is drawn from a known prior distribution . Proposition 2.2 implies that set of improving elements is still -uncontentious when is random. This, however, introduces additional difficulties: contention resolution with a constant balance ratio no longer recovers a constant fraction of the weighted rank when and are correlated, as illustrated by the following example.
Consider the truncated graphical matroid in Figure 2, with the weights labeling the edges and . We can guarantee that weights are distinct by introducing small perturbations. The graph on the left is the classical “hat example” often employed in the literature on the matroid secretary problem. We take the disjoint union of the hat example with the free matroid on elements (represented by the isolated edges on the right), and truncate the resulting matroid to a rank of . We fix the sampling parameter , and examine the set of improving elements for two settings of the weights and .
For the first scenario, let and . With high probability as grows large, the set of improving elements does not include any of the “hat” edges with weights or . Moreover, . The following simple scheme is -balanced: Discard the edges with probability (and otherwise discard nothing),999Discarding these edges serves solely to guarantee balance for the “hat edges”, in the low probability event that any of the hat edges are improving. then run greedy random-order contention resolution on the remaining edges.
For the second scenario we let (or a very large constant) and . Setting effectively takes the edges “out of the running”, leaving only the hat example. The set of improving elements, though -uncontentious, is now less amenable to greedy contention resolution: when , there are typically many “hats” in as well: for a constant fraction of , both edges and are in . It follows that the above-described discard-then-greedy scheme is no longer -balanced. In particular, it selects with probability , despite the fact that . A slightly more involved contention resolution scheme is needed for a constant balance ratio.
Suppose we randomize between the above scenarios, with each scenario equally likely. Let be the resulting set of improving elements, and note that is -uncontentious. It is easy to verify that the discard-then-greedy scheme is -balanced here. However, is accepted with probability when it is active with weight (in the first scenario), but with probability when it is active with weight (in the second scenario). Therefore, the discard-then-greedy scheme does not recover a constant fraction of the expected weighted rank of , despite being -balanced.
A similar situation arises for any nontrivial randomization between the two scenarios, even if we make the second scenario exceedingly unlikely.
This example suggests that we must constrain contention resolution to not “favor” improving elements that have low weight. We accomplish this by labeling each improving element with its weight, and requiring contention resolution in the (stronger) labeled sense. We use the following labeled notion of improving elements:
Let be a matroid, let be a parameter, and let be a weight vector. The random labeled set of improving elements for is the pair , where is the (random) set of improving elements, and is the labeling with for all . We use to denote the distribution of the labeled set .
When is fixed, each element is associated with a single label , so labeled contention resolution for is equivalent to unlabeled contention resolution for , and by Theorem 5.3 it follows that is -uncontentious in the labeled sense. When is a drawn from a known prior with finite support, the labeled set of improving elements is drawn from a mixture of the -uncontentious distributions , for the finitely-many realizations of . When and , we refer to as the labeled set of improving elements for , and denote its distribution by . The following is then a direct consequence of Proposition 3.2.
Let be a matroid, let , and let be a distribution over weight vectors with finite support. The distribution is -uncontentious (in the labeled sense) for .
Fixing matroid and , we reduce the prophet secretary problem on to -universal random-order labeled contention resolution with . The reduction is shown in Algorithm 1 for the prophet secretary problem on , which takes as an offline input a prior on weight vectors, and as its online inputs a sequence of weighted elements of . We assume that the online inputs to the Algorithm are distributed as specified in the prophet secretary problem, namely with and drawn independently, and analyze the algorithm’s competitive ratio. In particular, we will see that the algorithm achieves its competitive ratio by resolving contention, in the random order model, for a labeled set drawn from the -uncontentious distribution .
Let denote the elements improving , as determined in Step (11), and let be the label of determined in Step (12). We also let denote the list of elements (whether active or inactive) fed to by Algorithm 1, in that order. First, we show that the inputs to are as stipulated in random-order contention resolution for , and that is -balanced for that distribution.
The labeled set follows the distribution . Moreover, is a uniformly random order on independent of .
Since is a uniformly random permutation of , and , it follows that includes each element of independently with probability . The set consists of all elements improving with respect to weight vector , so by Definition 5.1. Since and , it follows that .
We now condition on and , which in turn fixes , and show that is a uniformly random permutation of . Since each iteration of the while loop feeds one of or to , and increments the corresponding counter ( or ), it follows that is a permutation of . Now consider the th iteration of the while loop, let and , and notice that is the set of elements not yet fed to . It is easy to see inductively that and , where and are as in iteration . Since is uniformly random, is a uniformly random element of , and is a uniformly random element of . The bias of the coin in Step (8) is such that with probability , and with probability . Therefore, is a uniformly-random sample, without replacement, from . It follows inductively that is a uniformly random permutation of . ∎
The RO-LCRM instantiated in Step (2) is -balanced for .
Follows directly from the fact that is -universal, and the fact that is -uncontentious as shown in Sublemma 5.6. ∎
Let denote the set of elements accepted by Algorithm 1, as determined in Step (12). We can bound the expected weight of these elements as follows, where expectations are with respect to , , the internal randomness in Algorithm 1, and any randomness in the instantiated contention resolution map .
6 Reducing Labeled to Unlabeled Contention Resolution, Online
Consider labeled contention resolution for matroid and labels in the online random-arrival model, and denote and . Here, a labeled set drawn from a known distribution is presented online to an LCRM for and as the string
where is a uniformly random permutation of , and is the label if (i.e. is active) and is otherwise. Entries of are revealed online, with iteration revealing , at which point the LCRM must immediately decide whether to accept in the event it is active.
Recall from Section 3.1 that, in the offline setting, labeled contention resolution on and reduces to unlabeled contention resolution on , via the map . It is therefore tempting to attempt a similar reduction in the online random order model as well. When the unlabeled problem on is considered in the online random order model, the (unlabeled) active set is presented online to an (unlabeled) CRM for as the string
where is a uniformly random permutation of , and designates whether . The string is revealed online, with iteration revealing , at which point the CRM must immediately decide whether to accept in the event that . We emphasize that the string is longer than : whereas an element appears exactly once in , it appears times in (once for each possible label, with at most one of these appearances active).
In attempting an online reduction from the labeled problem to its unlabeled counterpart, the problem we face at this point, intuitively, is the following: Given , how do we “interleave” the “missing” element/label pairs to form the string . This interleaving must be done online, before we know exactly which elements are active and what their labels are. Moreover, it must be such that the resulting order of element/label pairs in
is uniformly distributed, at least approximately, in order to make use of any gaurantee on the balance ratio of the (unlabeled) RO-CRM. This, it so happens, is nontrivial.
The reader might understandably furrow their brow at this point: Surely, any “reasonable” random-order contention resolution algorithm need only exploit the relative ordering of active elements. This is already true in , so an arbitrary interleaving of the missing element/label pairs should suffice! Certainly, this additional difficulty is an artifact of the precise technical definition of the random order model, rather than a conceptually interesting distinction! The reader would be justified in expressing such skepticism. However, intuitive as it may seem, this knee-jerk reaction is flawed in a formal sense. Specifically, we show in Appendix B that there does not exist a constant-competitive universal CRS in the online model where active elements arrive in a uniformly random order, but inactive elements are ordered arbitrarily. This impossibility result holds even for a -uniform matroid. Therefore, for online contention resolution to plausibly encode the matroid secretary problem, it needs to exploit randomness in the arrival order of both active and inactive elements!
6.1 Difficulties with Direct Approaches
We begin by explaining how the direct approach, namely reducing the online labeled contention resolution for to online unlabeled contention resolution for , appears unlikely to succeed. Let be the online input string to the labeled problem, as in Equation (3). All online reductions to the corresponding unlabeled problem which are conceivable to us fit the following template, which has oracle access to an online CRM for , and produces an online LCRM for .
While not all element/label pairs have been fed to , do one of the following:
Read the next active element/label pair in (if any), skipping inactive elements as needed. If has not previously been fed to , then feed to , and accept iff accepts .
“Hallucinate” an element/label pair which has not yet been fed to , and feed to .
Notice that, in each iteration, the choice to do (i) or (ii), and the choice of “hallucination” in (ii), can depend on previously observed entries of , on previous acceptance/rejection decisions of , and on previous “hallucinations”. These choices may also be randomized. Let denote the string fed to through the course of the reduction, and let denote the sequence of element/label pairs appearing in .
For an instantiation of the above template to serve as an approximation preserving reduction (up to a constant) from the labeled problem to its unlabeled counterpart in the online random order model, the following properties appear needed.
Condition on the labeled set , and assume that the order of elements in is uniformly distributed (as is guaranteed by the random order model for the labeled problem). The order of element/label pairs in should be uniformly distributed (as is required by the random order model for the unlabeled problem) or approximately so (say, in terms of total variation distance).
In the event that is an entry of (i.e., is active with label ), it should hold with constant probability that is an entry of (i.e., is active in the corresponding unlabeled instance). This requires that is not “hallucinated” before it arrives in .
Trivial insantiations of our template satisfy one of (a) or (b), but satisfying both (a) and (b) simultaneously appears impossible. To illustrate the difficulty, consider the special case where the number of active elements is known in advance. Arguably the most natural instantiation of our template in this special case, and one which at first glance appears promising, is as follows. In each iteration, with active entries of remaining and element/label pairs not yet fed to , we choose (i) with probability and choose (ii) otherwise. When (ii) is chosen, we let be a uniformly random draw from the remaining element/label pairs. The probability is chosen to reflect the proportion of active to inactive element/label pairs.
It is not too difficult to verify that (b) is satisfied for this reduction. However, it can be shown that the permutation is not uniformly distributed after conditioning on . To see this, consider an element with . The probability that is the first element/label pair appearing in is given by
where the first term corresponds to the event that (i) is chosen and is the first active element/label pair in , and the second term corresponds to the event that (ii) is chosen and is hallucinated. Since , this expression is at least . When the number of labels is large, this is almost twice the probability that would appear first in a uniformly random permutation on element/label pairs! In other words, an active element/label pair is almost twice as likely to appear early in than an inactive element/label pair, rendering far from uniformly distributed. In fact, we can show that the total variation distance between and the uniform distribution tends to as grows large, violating (a).
One might hope that different choices of , coupled with a different rule for choosing the hallucinated element/label pair in (ii), might remedy this failure. However, some examination suggests that such approaches are unlikely to succeed. The difficulty, intuitively, is the following: when hallucinating inactive element/label pairs early in the sequence , we must do so without knowledge of which active elements/label pairs appear later in , and this is due to the online nature of the reduction. This gives active element/label pairs in a “greater than fair” shot at appearing early in the sequence (violating (a)), unless one is content with “ignoring” entries of with high probability (which results in violating (b)). Therefore, there is a tension between requirements (a) and (b).
These difficulties appear intrinsic to online reductions from the labeled problem on to the unlabeled problem on , leaving little hope for preserving the balance ratio with such a direct approach. A new idea appears to be needed.
6.2 An Indirect Approach: Duplicating the labels
We overcome these difficulties by reducing labeled contention resolution on and to unlabeled contention resolution on a much larger matroid than . Specifically, we “duplicate” each label a large number of times, creating many “identical copies” of each element/label pair. We associate an active element/label pair in with one of its copies uniformly at random, leaving all other copies inactive. Roughly speaking, a random permutation of the duplicated element/label pairs converges in probability to a limiting permutation as the number of copies grows large, modulo the symmetry between copies. An active element/label pair from is now merely a drop in a sea of its inactive brethren, and therefore interleaving into
has little influence on the probability distribution of.
Formally, we duplicate each label in a large number of times to form an expanded set of labels , where is an abstract set for indexing copies with . We then reduce labeled contention resolution on and to unlabeled contention resolution on the matroid . For and , we say the pair is a copy of label . We also say that an element of is a copy of .
An offline version of our reduction maps an active set in with labels in to an (unlabeled) active set of by selecting a copy of each label uniformly at random. Specifically, a labeled set of active elements is mapped to the (unlabeled) set