Simple and Optimal Greedy Online Contention Resolution Schemes
Anonymous Author(s)
AffiliationAddress email
Abstract
Real-world problems such as ad allocation and matching have been extensivelystudied under the lens of combinatorial optimization.
In several applications,uncertainty in the input appears naturally and this has led to the study of onlinestochastic optimization models for such problems.
For the offline case, theseconstrained combinatorial optimization problems have been extensively studied,and Contention Resolution Schemes (CRSs), introduced by Chekuri, Vondrák,and Zenklusen, have emerged in recent years as a general framework to obtaininga solution.
The idea behind a CRS is to first obtain a fractional solution to a(continuous) relaxation of the objective and then round the fractional solution toan integral one.
When the order of rounding is controlled by an adversary, Online
Contention Resolution Schemes (OCRSs) can be used instead, and have beensuccessfully applied in settings such as prophet inequalities and stochastic probing.
In this work, we focus on greedy OCRSs, which provide guarantees against thestrongest possible adversary, an almighty adversary.
Intuitively, a greedy OCRShas to make all its decisions before the online process starts.
We present simple1 / e - selectable greedy OCRSs for the single-item setting, partition matroids andtransversal matroids, which improve upon the previous state-of-the-art greedy
OCRSs of [FSZ16] for these constraints.
We also show that our greedy OCRSs areoptimal, even for the simple single-item case.
Introduction
In recent years, problems in Bayesian and stochastic online optimization have attracted significantinterest, especially in the field of machine learning.
In this setting, we are usually asked to makedecisions in an online manner, based on the information available to us so far and our objective is tominimize our “regret”, which is captured by a loss function and describes how much better we couldhave done if we had all the information available a priori.
In several applications, in which our decision relies also on hidden information, such as decidingwhether a user will click an ad they are presented with [AAD + 20, XSC + 19, BSSX20, BGMS21]or whether a kidney donor is a good match for another patient [BFT13, DS15], our online decisionproblem naturally becomes stochastic.
The inherent uncertainty of such applications makes the studyof stochastic optimization all the more significant, and demonstrating algorithms with reasonablecompetitive ratios when one has knowledge of the underlying distributions illustrates the importance of
One can consider a simpler setting in which our decision at each time step is a binary choice.
Inone such formulation, elements arrive in an online manner and we have to select a subset of them,subject to certain combinatorial constraints.
The simplest example of such a constraint is when we only want to select a single element. Under this formulation, the elements reveal one after the other in an online manner whether they are “active” or not, and our binary decision is whether to select anactive element, with the constraint of being able to select at most one active element. In this paper, westudy this problem and its generalizations.
Famous examples of such settings are prophet inequalityproblems [KS77, KW12, EFGT20] and secretary problems [Dyn63], which have found applicationin the design of posted-price mechanisms and auctions, among others.
One approach that has seen plenty of success for these problems is to use the known distributionalinformation to obtain a continuous relaxation of the objective.
One can then solve this relaxationand get an optimal fractional solution x ∗ , which corresponds to the marginals of the elements underthe optimal distribution.
Thus, the optimal value of the relaxation constitutes an upper bound to theperformance of any online (or even offline) algorithm.
Afterwards, x ∗ is used to devise an onlinealgorithm in order to maximize the value of the subset of elements selected.
It is easy to see that thisalgorithm essentially corresponds to an online rounding procedure for x ∗ .
In this paper, we contribute provably optimal algorithms for the problem of rounding an optimalsolution to a linear program in an online manner, for a range of fundamental constraints.
The onlinenature of our rounding is well-motivated in the field of AI due to the fact that we are not able tocontrol the arrival order of the agents.
Our algorithms apply for the single-item setting, for partitionmatroids, as well as transversal matroids which have been used extensively to model matchingmarkets [DRS09, CFMP, BIK07].
1.1 Online Contention Resolution Schemes
Such rounding algorithms have recently been used to obtain several optimal and interesting results[CVZ11, FSZ16, AW18, EFGT20, BZ20, RS17, CL21], and have more applications in online mechanism design and posted pricing mechanisms [CHMS10, HKS07].
General rounding algorithms foroffline problems are called Contention Resolution Schemes (CRSs) and were introduced by Chekuri,
Vondrák and Zenklusen [CVZ11] with the purpose of maximizing a submodular function.
A CRSis defined with respect to a constraint family. Examples of such combinatorial constraints includeselecting an independent set in a given matroid, selecting a feasible matching in a given a graph inwhich the elements correspond to edges, or selecting a feasible set of elements subject to a knapsackconstraint, where each element is associated with a size. Chekuri, Vondrák and Zenklusen [CVZ11]gave the first CRSs for all aforementioned constraint families as well as other constraints.
For a givenfractional point x ∗ , the main idea behind CRSs is to first obtain a random set R , drawn from theproduct distribution with marginals x ∗ , hence called the active elements .
Since R may be infeasiblewith respect to the constraints, the CRS proceeds to “drop” specific elements from R and obtain anew, feasible, set R ′ ⊆
R .
While the general applicability of the CRS approach is remarkable, they are unfortunately not usefulfor Bayesian and stochastic online optimization problems.
In particular, one can utilize CRSs whenthey have the ability to choose the order in which they obtain information about the underlying groundset of elements, as CRSs round a fractional point x ∗ in a particular order to obtain a feasible solution.
To overcome the inherently offline nature of CRSs, Feldman, Svensson and Zenklusen [FSZ16]introduced the notion of Online Contention Resolution Schemes (OCRSs) , applicable in a varietyof online settings in Bayesian and stochastic online optimization, such as prophet inequalities[LS18, RS17, CL21], stochastic probing [ASW16, GN13, GNS16, GNS17], and posted pricingmechanisms [HKS07].
Surprisingly, OCRSs yield constant-factor competitive ratios for severalinteresting feasibility constraints.
All of the results presented in [FSZ16] are based on a special subclass of OCRSs called greedy
OCRSs .
Intuitively, a greedy OCRS fixes a downward-closed subfamily of feasible sets F beforethe online process starts.
During the online process, the greedy OCRS maintains a subset S of theelements which is feasible in F , and then greedily accepts any active element i if S ∪ { i } is alsofeasible in F , i.e. if i does not violate feasibility, with respect to F , of the set maintained by thegreedy OCRS.
One can easily see that the final set at the end of the online process is feasible byconstruction.
Even though greedy OCRSs offer suboptimal performance guarantees with respect to (non-greedy) OCRSs, as we will see, their study remains interesting for two important reasons. First, greedy OCRSs are inherently simpler than their non-greedy counterparts.
Usually, to obtain an optimalnon-greedy OCRS for a non-trivial constraint, one has to use linear programming duality, as in the approach of [LS18]. This leads to a non-intuitive algorithm which, in many situations, can be difficultto implement 1 . In short, greedy OCRSs are simpler to implement and more intuitive. Furthermore, greedy OCRSs provide guarantees against an almighty adversary who has knowledge ofthe future as well as any random coins used by the algorithm.
This property is crucial for applicationsthat require the algorithm to compare against an almighty adversary.
One such example is [BDP22]in which the authors study the "delegation gap" of the generalized Pandora’s box problem and infact reduce the problem to the design of an OCRS which is necessarily greedy.
To the best of ourknowledge, this result is the first example of an application in which non-greedy OCRSs cannot beapplied and a greedy OCRS is needed.
1.2 Our contributions
In this paper, we analyze the performance of greedy OCRSs and provide the first provably optimalgreedy OCRS for the single-item setting, partition matroids and transversal matroids.
We have four main contributions:
• We design a 1 / e -selectable greedy OCRS for the single-item setting (Theorem 1.1). • We show that our greedy OCRS extends naturally to partition matroids 2 (Corollary 1.2).
• We proceed to show that no greedy OCRS can be ( 1 / e + ε ) -selectable, for any ε > 0 ,even for the single item setting.
This, combined with our first contribution, shows that our1 / e -selectable greedy OCRS is the best possible (Theorem 1.3).
• We extend our greedy OCRS to transversal matroids 3 as well, and show that the selectability can be increased to 1 − 1 / e for special cases of transversal matroids (Theorem 1.4).
Our results improve upon the 1 / 4 -selectable OCRSs of [FSZ16] for all the constraints discussed here.
As a corollary, our work presents the first instance of a dichotomy between the best possible guaranteesby greedy OCRSs and (non-greedy) OCRSs, since a 1 / 2 (non-greedy) OCRS is known for the singleitem setting [Ala14].
We proceed with our four main results.
The proof of the following theorem is found in Section 3.
Theorem 1.1.
There exists a 1 / e -selectable (randomized) greedy OCRS for the single-item setting.
Next, we extend the single-item greedy OCRS to a partition matroid constraint, by decomposingthe partition matroid into single-item instances, running the greedy OCRS above and acceptingan active element if and only if it is independent in the corresponding single-item instance of thedecomposition.
Corollary 1.2.
There exists a 1 / e -selectable (randomized) greedy OCRS for partition matroids.
We complement the results above by also showing that it is tight.
The proof of the following theoremcan be found in Section 4.
Theorem 1.3.
For every ε > 0 , there exists no greedy OCRS for the single-item setting that selectsan active element
i with probability at least 1 / e + ε for all i ∈ N .
Finally, we extend Theorem 1.1 to a more general class of matroids, transversal matroids, andstrengthen it for the special case in which every element’s neighborhood has size at least 3 .
The proofof the following theorem is found in Section 5.
Theorem 1.4.
Let M =
( U, I ) be a transversal matroid represented by a bipartite graph
G =(
U ∪ V, E ) .
Then, there exists a 1 / e -selectable (randomized) greedy OCRS π for M .
Furthermore, iffor every element u ∈ U we have | N ( u ) | ≥ 3 , where N ( u ) is the set of neighbours of u in G , then πis a (1 − 1 / e ) -selectable (randomized) greedy OCRS for M .
1.3 Related work
Since their introduction [CVZ11], Contention Resolution Schemes (CRSs) have found severalapplications.
Applications of CRSs in Bayesian mechanism design and posted price mechanisms[CHMS10] can be found in [CVZ11].
Later, Yan [Yan11] connected mechanism design with thenotion of correlation gap [ADSY12].
OCRSs were developed [FSZ16] with applications to Bayesianmechanism design as one of the main motivations as they directly translate to competitive ratiosfor the prophet inequality problem [FSZ16, Rub16, RS17, CL21].
In fact, Alaei’s work on uniformmatroids [Ala14] precedes [FSZ16] and can be seen as an OCRS, even though it is formulateddifferently.
Random order CRSs (ROCRSs) were introduced in [AW18] and yield improved boundswhen the arrival order is random.
As stated previously, Feldman, Svensson and Zenklusen [FSZ16] gave the first greedy OCRS formatroids, which is 1 / 4 -selectable.
Lee and Singla [LS18] showed a reverse connection between
OCRSs and prophet inequalities, obtaining a 1 / 2 -selectable (non-greedy) OCRS for matroids and a(1 − 1 / e ) -selectable
ROCRS for the single item setting.
Adamczyk and Wlodarczyk [AW18] obtainedseveral results, including a 1 / k +1 -selectable ROCRS for the intersection of k matroids.
For matchings,
Ezra et al [EFGT20] designed a 0 .
337 -selectable OCRS for bipartite graphs, while Bruggmann and
Zenklusen [BZ20] developed optimal monotone CRSs via a novel polyhedral approach.
This work is connected to stochastic optimization, online algorithms, mechanism design and submodular optimization, all of which have extensive literature.
There have been several surveys onthe topic [Gup17, Luc17, CFH + 19, Din13, HK92], as well as a survey on random-order modelsin general [GS20].
Singla’s thesis [Sin18] has connections to several of the topics discussed here.
On the application side, prophet inequality and secretary problems have received significant attention in the last years, due to their connections with Bayesian mechanism design and posted pricemechanisms [GM66, HK82, Ker86, Dyn63, KS77, KW12, ACK17, CFH + 17, CSZ20, AEE + 17,
EHLM15, HKS07, EHKS18, FZ18], while ROCRSs have found several applications to stochasticprobing [ASW16, GN13, GNS16, GNS17, BSZ19, AN16].
Recently, Dughmi [Dug20, Dug21]showed the equivalence between the existence of constant-factor universal OCRSs and a constantfactor approximation to the famous matroid secretary problem [BIKK18].
Independently, [FLT + 21] study the problem of designing an oblivious OCRS 4 for the same settingand obtain a similar result, showing that there exists a 1 / e -selectable oblivious OCRS and no oblivious
OCRS can be ( 1 / e + ε ) -selectable for any ε > 0 .
We note that the two results (and schemes) are verydifferent.
In fact, their OCRS is not greedy, while ours is not oblivious.
Whether one can achievesimilar guarantees with greedy and oblivious OCRSs for more general settings is an interesting openproblem.
1.4 Roadmap
We begin in Section 2 with some background.
Then, in Section 3, we present our first main result, the 168/ e -selectable greedy OCRS for the single-item setting and partition matroids.
Then, in Section 4, weshow that our greedy OCRS is optimal.
We proceed with our greedy OCRS for transversal matroidsin Section 5, which also achieves the optimal 1 / e selectability and show it performs even better undermild assumptions on the structure of the transversal matroid.
Finally, we present our experimentalresults in Section 6, demonstrating how our schemes outperform the best previously known schemesfor randomly generated as well as specific worst-case instances.
All omitted proofs can be foundin the Appendix which, along the code used for the experiments, is provided in the supplementarymaterial.
Preliminaries
Before we proceed, we present the formal definitions of CRSs, OCRSs and greedy OCRSs and brieflydescribe a 1 / 4 -selectable single item OCRS by [FSZ16].
Let N be a finite ground set.
A constraint family over N is a subset I ⊆ 2 N ; a set S ∈
I is calledfeasible, while a set S ̸∈ I is called infeasible.
We say P I ⊆
[0 , 1] N is a polyhedral relaxation of( N , I ) if P I is a polyhedron and 1 S ∈ P I for all S ∈
I (here 1 S is the characteristic vector of S ).
Given a polyhedral relaxation P I of a constraint ( N , I ) and a point x ∈ P I , a natural questionis whether we can round x in order to obtain a feasible set S ∈ I .
One way to achieve this is via
Contention Resolution Schemes, which we define below.
Definition 2.1 (Contention Resolution Scheme [CVZ11]) .
Let b, c ∈ [0 , 1] .
A ( b, c ) -balanced
Contention Resolution Scheme π for P I is a procedure that for every x ∈ b · P I and A ⊆ N , returnsa random set π x ( A ) ⊆
A ∩ support ( x ) and satisfies the following properties:
π x ( A ) ∈
I with probability 1 , A ⊆ N , x ∈ b · P I , and2.
for all i ∈ support ( x ) , Pr [ i ∈ π x ( R ( x )) | i ∈ R ( x )] ≥ c, [Equation] x ∈ b · P I , where R ( x ) ⊆
N denotes a random set in which every element i ∈ N appears independently withprobability x i .
The scheme is said to be monotone if Pr [ i ∈
π x ( A 1 )] ≥ Pr [ i ∈
π x ( A 2 )] whenever i ∈ A 1 ⊆
A 2 .
For the remainder of this paper, we drop the subscript in P I and simply write P whenever theconstraint is clear from context.
CRSs are offline rounding schemes.
In the case where the arrival order of the elements is selected byan adversary, we can use the following notion of Online Contention Resolution Schemes (OCRS) toround x .
Definition 2.2 (Online Contention Resolution Scheme (OCRS) [FSZ16]) .
For an online selectionsetting where a point x ∈ P is given, we draw a random subset of the elements R ( x ) , in whicheach element i appears independently with probability x i . We call R ( x ) the set of active elements.
Afterwards, we observe whether the element e ∈ N are active ( e ∈ R ( x ) ), one by one, and haveto immediately and irrevocably decide whether to select an element or not before the next elementis revealed.
An Online Contention Resolution Scheme for P is an online algorithm which selects asubset I
⊆ R ( x ) such that 1 I ∈ P .
A scheme is called a Random Order Contention Resolution Schemes (ROCRS) if, instead of beingchosen by an adversary, the arrival order of the elements is chosen uniformly at random.
Adamczykand Wlodarczyk present several interesting results on ROCRSs in [AW18].
In the case of adversarialarrival order, however, one can distinguish between three different adversaries in terms of theinformation they have at their disposal.
An offline adversary , which is the weakest of the three, hasto fix an ordering of the elements before any of the elements are revealed.
An almighty adversary ,the most powerful one, has access to the realizations of all random events; both the set of activeelements and any potential random bits the algorithm may use.
Therefore, an almighty adversarycan predict the algorithm’s behaviour and choose a truly worst-case ordering of the elements for theparticular algorithm.
In between the two extremes is the online adversary .
An online adversary’schoices can only depend on the realizations of the elements that have appeared so far.
In other words,the adversary has, at any step, exactly the same information as the algorithm, and their decision as towhich element to reveal at step i can only depend on the realizations of the elements revealed in steps1 through i − 1 .
We also define the notion of a greedy OCRS, which provide guarantees with respect to an almightyadversary.
Definition 2.3 (Greedy OCRS [FSZ16]) .
Let P ⊆
[0 , 1] n be a relaxation of the feasible sets F ⊆ 2 N .
An OCRS π for P is called a greedy OCRS if, for any x ∈ P , π defines a down-closed subfamily offeasible sets F π,x ⊆ F , and it selects an active element e when it arrives if, together with the set ofelements already selected, the resulting set is in F π,x .
We say that π is a randomized greedy OCRS if,given x , the choice of F π,x is randomized.
Otherwise, we say that π is a deterministic greedy OCRS.
For the remainder of this paper, we drop the subscript in F π,x and simply write F x or F , whenever πand x are clear from context.
Intuitively, we say a greedy OCRS is c -selectable if and only if an active element e ∈ R ( x ) can beincluded in the currently selected elements I ⊆
R ( x ) and maintain feasibility with probability atleast c .
Definition 2.4 ( c -selectability) .
Let c ∈ [0 , 1] .
A greedy OCRS for P is c -selectable if and only if forany x ∈ P we have Pr [ [Equation]
[Equation]
c ∀ e ∈ N . Notice that a c -selectable greedy OCRS guarantees that each active element e is selected withprobability at least c , even against the almighty adversary.
We should note that the randomness in theabove definition is with respect to both the randomness of R ( x ) and also any potential randomnessthe greedy OCRS might use to decide upon F x .
Next, we briefly describe the 1 / 4 -selectable single item greedy OCRS by [FSZ16].
Given a fractionalpoint x such that (cid:80) ni =1 x i ≤ 1 , the greedy OCRS will, at step i , observe whether element i is activeor not.
If it is active, the greedy OCRS will choose to select with probability 1 / 2 or discard it andmove on to the next element.
Since each element is active with probability x i and is selected withprobability x i / 2 , the expected number of selected elements is at most half, and thus, by Markov’sinequality, the probability the greedy OCRS selects no elements is at least 1 / 2 .
Therefore, for everyelement i , we reach i without having selected an element with probability at least 1 /
and we select i ,given that it is active, with probability 1 / 2 , for an overall selectability of 1 / 4 .
We should note that for the single item setting there exists a 1 / 2 -selectable OCRS [Ala14] but,crucially, it is not greedy.
In fact, we show in the Section 4 that there is no 1 / 2 -selectable greedy
OCRS for the single item setting.
An 1 / e -selectable greedy OCRS for the single-item setting
This section is dedicated to proving Theorem 1.1.
Before we begin, we need the following lemma.
Lemma 3.1.
Let a 1 , . .
. , a k ∈ [0 , 1] .
Then
[Equation]
Next, consider a ground set N = { e 1 , e 2 , . .
e n } , and let M =
( N , I ) be the uniform matroid ofrank 1 with respect to N , i.e. I = {{ e i } | e i ∈ N} .
Let P be the following polyhedral relaxation of
M :
[Equation]
For a given x ∈ P , let π
= π x denote the OCRS we will create.
π will draw a random set R ( q ) whereeach element e
i appears in R ( q ) independently with some probability q i .
The family of feasiblesubsets is
[Equation]
We set q i = 1 − x i / 2 for all e i ∈ N .
Afterwards, π selects the first element e i that is active and that{ e i } ∈ F .
Lemma 3.2.
π is a randomized greedy OCRS.
Next, we quantify the probability that each element is selected by π , given that it is active.
Lemma 3.3.
π selects every element e i ∈ N , given that it is active, with probability at least 1 / e .
Proof.
We relabel the elements of N so that each e i arrives in the i -th step.
Consider an elemente i ∈ N .
Given that e i is active, since π is a greedy OCRS, π will select e i if and only if it hasnot selected any elements before e i and also { e i } ∈ F π,x .
Recall that we have { e i } ∈ F π,x withprobability exactly q i =
− x i / 2 .
Furthermore, for every element e j where j < i , it needs to bethe case that we avoid having both { e j } ∈ F π,x and also e j coming up active.
This happens withprobability 1 − x j · (1 − x j / 2 ) =
1 − x j + x 2 j / 2 for every e j where j < i .
Overall, if we denote by r i the probability that e i is selected by π , given that it is active, we have
[Equation]
where the first inequality follows from Lemma 3.1 and the second inequality follows from (cid:80) i x i ≤ 1 .
Therefore r i ≥ 1 / e , for all
i ∈ N .
From Lemmas 3.2 and 3.3, it follows that π is a 1 / e -selectable (randomized) greedy OCRS for P .
Remark 3.4.
In a personal communication, Jan Vondrák devised an alternate scheme for the problem,after we notified him of our scheme.
With his consent [Von], we have included this alternate schemein the Appendix, which can be found in the supplementary material.
1 / e is tight
In this section, we present the proof of Theorem 1.3.
Consider the instance where x i = 1 / n for alle i ∈ N , where n = |N| , and let A denote the set of active elements.
Any greedy OCRS π will selecta subset S of F = { e i | e i ∈ N} with some probability α S , and then accept the first element in Sthat comes up active.
What is the worst-case probability that an element from S will be selected?
This is minimized for the element in S which is last in the arrival order, which has a probability ofbeing selected exactly equal to (1 − 1 / n ) | S |− 1 , because the OCRS is greedy, and it would select anelement from S which arrived earlier, if it came up active.
Therefore, no greedy OCRS can guarantee,for any S ⊆ F , that an element e ∈ N will be selected, when e ∈ A , with probability greater than(1 − 1 / n ) | S |− 1 .
Thus, for any e ∈ N and any greedy OCRS π , we have
[Equation]
Next, for a greedy OCRS π to be c -selectable, it needs to guarantee thatmin e ∈N Pr [ e ∈ π ( A ) | e ∈ A ] ≥ c .
Therefore, if we show that
[Equation]
by (1) it follows that π cannot be ( c + ε ) -selectable for any ε > 0 .
Lemma 4.1.
[Equation]
By Lemma 4.1, since lim n →∞ (1 − 1 / n ) n − 1 = 1 / e , it follows that there exists no greedy OCRS for
P that selects an element e , when active, with probability at least 1 / e + ε for all e ∈ N .
Extension to Transversal Matroids
In this section, we prove Theorem 1.4.
Let M =
( U, I ) be a transversal matroid and G =
( U ∪ V, E )denote the underlying bipartite graph, where | U | = n .
We know that a subset S ⊆
U is independentif and only if there exists a matching in G that covers S .
Let P be the natural polyhedral relaxationof M .
For a given x ∈ P , let π
=
π
x be the greedy OCRS we will create.
For each v ∈ V , π willdraw a random set R v ⊆ N ( v ) , in which each element u ∈ U appears with probability q u .
For everyu ∈ U , let N ( u ) denote the set of neighbors of u in G .
Then, we set
[Equation]
It is easy to see that q u ∈ [0 , 1] for every | N ( u ) | ≥ 1 , and thus q u is well-defined.
Next, we create a down-closed subfamily of feasible sets by taking all possible combinations of setscreated by taking at most one element from each R v and then taking the union of all such elements.
Specifically,
[Equation]
Any set S in F is clearly an independent set of M , as the constraints guarantee that there alwaysexists a matching in G that covers S .
During the online process, π starts with a set of selectedelements S = ∅ , and greedily selects an active element u if S + u ∈ F .
The proof of the following lemma is identical to the proof of Lemma 3.2 and follows from thediscussion above.
Lemma 5.1.
π is a randomized greedy OCRS.
Next, we again lower bound the selection probability of an active element.
Lemma 5.2.
π selects every element u ∈ U , given that it is active, with probability at least 1 / e .
Furthermore, if | N ( u ) | ≥ 3 for all u ∈ U , π selects every element u ∈ U , given that it is active, withprobability at least 1 − 1 / e .
We conclude that π is a 1 / e -selectable greedy OCRS for M and that if | N ( u ) | ≥ 3 for every u ∈ U ,π is a (1 − 1 / e ) -selectable greedy OCRS for M .
Experimental Results
In this section, we present our experimental results which give a quantitative view into how ourgreedy OCRSs outperform the former state-of-the-art greedy OCRSs by [FSZ16].
For the constraintsystem, we consider both the single-item setting as well as randomly-generated transversal matroidconstraints.
The code used to run these experiments can be found in the supplementary material.
For our experiments, we have the following parameters:
• N : The number of elements.
For the single item setting, this varies from 10 to 100 .
For thetransversal matroid constraint, this is set to 50 .•
ITERATIONS :
For a fixed x , the number of times we simulate the realization of the elements.
This is used to approximate the selectability of each element within a reasonable estimateand is set to 200 , 000 .•
REPETITIONS :
The number of times we run the entire experiment from scratch.
This has noeffect on the uniform instance of the single-item setting or the transversal matroid constraint,but affects the random vector x in the randomly-generated instance of the single-item setting,and is set to 10 .
We focus on two metrics: the minimum selectability and the average selectability of each element.
For a fixed x , the minimum (resp. average) selectability records the minimum (resp.
average) over allelements of the fraction of the times an element was selected by our greedy OCRS over the number oftimes it was active.
While the average gives an idea of what selectability to expect for
For both types of constraints, we can see that our greedy OCRSs outperform the 1 / 4 -selectablesingle-item greedy OCRS by [FSZ16] in the minimum selectability metric and performs substantiallybetter in the average selectability metric.
6.1 Single-Item Setting
We consider two different types of instances for the single-item setting: the uniform instance , wherex i = 1 / n for every element i ∈ N , as well as a randomly-generated instance , in which x is a randomvector of length n with the constraint that (cid:80) i x i = 1 .
6.2 Transversal Matroid
For the transversal matroid constraint, we consider three different types of instances: a transversalmatroid with | V | = 2 , | N ( u ) |
= 2 and x u = 2 / n for all vertices u ∈ U , a transversal matroid with| V | = 3 , | N ( u ) | = 3 and x u = 3 / n for all vertices u ∈ U , as well as a transversal matroid with| V | = 4 , | N ( u ) |
= 4 and x u = 4 / n for all vertices u ∈ U .
While there are many options for the choice of what transversal matroid to pick, we selected these“uniform” instances of transversal matroids to illustrate how the selectability changes as | N ( u ) |increases.
Since a transversal matroid with | V | = 1 , | N ( u ) | = 1 and x u = 1 / n for all vertices u ∈ Uis equivalent to a uniform instance of the single-item setting, we do not consider it explicitly, but weplot it together with the previous three instances. The x-axis in the figures below corresponds to the size of | N ( u ) | .
Checklist
For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
contributions and scope?
[Yes]
(b) Did you describe the limitations of your work?
[Yes] See Section 1.
(c) Did you discuss any potential negative societal impacts of your work?
[N/A]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to
them?
[Yes]
If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results?
[Yes]
(b) Did you include complete proofs of all theoretical results?
[Yes] All proofs can be
found in the Appendix, which is part of the supplemental material.
If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi-
mental results (either in the supplemental material or as a URL)?
[Yes] The code used
to run the experiments can be found in the supplemental material.
(b) Did you specify all the training details (e g., data splits, hyperparameters,
how they
were chosen)?
[Yes] See Section 6.
(c) Did you report error bars (e g., with respect to the random seed after running experiments multiple times)?
[Yes]
(d) Did you include the total amount of compute and the type of resources used (e
g.,type of GPUs, internal cluster, or cloud provider)?
[No] The experiments were simple
enough that they were all completed within a few hours in our personal computers.
If you are using existing assets (e
g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators?
[N/A]
(b) Did you mention the license of the assets?
[N/A]
(c) Did you include any new assets either in the supplemental material or as a URL?
[N/A]
(d) Did you discuss whether and how consent was obtained from people whose data you’re
using/curating?
[N/A]
(e) Did you discuss whether the data you are using/curating contains personally identifiable
information or offensive content?
[N/A]
If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if applicable?
[N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review
Board (IRB) approvals, if applicable?
[N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount
spent on participant compensation?
[N/A]