Improved Guarantees for Offline Stochastic Matching via new Ordered Contention Resolution Schemes

1 Introduction . The offline stochastic matching problem is about finding a maximum matching on a weighted graph . However , each edge e is active independently according to a known probability pe , and only active edges can be matched . The set of active edges is initially unknown . An edge whose endpoints are unmatched can be probed to determine whether it is active , and if so , it is irrevocably inserted into the matching . The objective is to sequentially probe the edges in a way to maximize the expected weighted matching at the end . Matching problems arise in numerous deployed AI systems , especially those dealing with allocation and scheduling . See , for example , the works of Ahmadi et al . ( 2020 ) ; Ahmed et al . ( 2017 ) ; Brubach et al . ( 2020 , 2021b ) ; Baveja et al . ( 2018 ) for applications to advertising , e-commerce , organ exchange , online dating , peer review , school matching , and hiring ; and Hikima et al . ( 2021 ) ; Nanda et al . ( 2020 ) ; Xu et al . ( 2019 ) for applications to ride sharing , crowdsourcing ( worker-task assignment ) , and 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . recommendation systems . The work of Esfandiari et al . ( 2016 ) ; Antoniadis et al . ( 2020 ) gives further discussion on applications to e-commerce and internet advertising , and illustrates the importance of designing algorithms with good theoretical guarantees in deployed AI systems . Stochastic edges play a major role in properly capturing many of these applications when we account for hidden information about whether a proposed match will be accepted . Is a kidney donor a good match ? Will a worker accept an offered task ? Does a user want to click on this ad ? The uncertainty inherent in the presence or absence of various edges naturally leads to the need for stochastic optimization . Achieving good results when the edge probabilities are known demonstrates the value of learning distributions in AI . We contribute improved algorithms for a range of fundamental problems , as well as useful-specialcase problems , in the realm of stochastic matching . Although our results are phrased for the offline setting , some of our algorithms also translate to the random-order online setting in which the edges arrive in random order . This online arrival model is well-motivated in these AI settings due to the inability to control the order in which the agents arrive . Below , we further describe three typical features that may be present in stochastic matching problems . Patience constraints . The offline stochastic matching problem is typically considered with the input including an integral patience parameter tv for each vertex v known as a patience constraint or timeout . This adds the constraint that at most tv of the edges incident to v can be probed , which is motivated by applications in which the vertices represent users who are only willing to view a finite number of potential matches . A vertex v which has been probed tv times is said to have timed out . To capture the absence of patience constraints in some models , we allow patience parameters to be∞ . Probing orders allowed . Most generally , we allow algorithms to probe the edges in any order based on the realizations of past probes . Sometimes we impose that the algorithm must make a single pass through the edges , according to a uniform random permutation . This can also represent an “ online ” setting in which the edges arrive in a uniformly random order , and upon arrival , an edge must either be probed ( only possible if both endpoints are unmatched and have remaining patience ) or irrevocably discarded . We say that such algorithms are random-order . Special case of graphs considered . We introduce a new class of graphs , bipartite graphs with a unit-patience side , for which we derive further improved guarantees . Such a graph can be divided into two sides such that : ( 1 ) all edges are between vertices on different sides ; and ( 2 ) for one of the sides , all patiences are 1 . This subclass of graphs captures a problem of interest in AI with applications such as crowdsourcing and ride-hailing , described in Hikima et al . ( 2021 ) . 1.1 Summary of Results . To derive computationally-efficient probing algorithms , we address the standard approach of first solving a fractional relaxation , which prescribes a probability ye ∈ [ 0 , 1 ] with which each edge e should be probed . These relaxed values ye only have to satisfy the matching and patience constraints in expectation , while a probing algorithm must satisfy these constraints with probability ( w.p . ) 1 . However , we can still hope to probe every edge e with probability at least c · ye , for some c ≤ 1 . It is well-known that such an algorithm would then be c-approximate , i.e . its expected weight matched would be at least c times that of an optimal probing algorithm . Result 1 . A 0.382-approximate random-order probing algorithm for general graphs . This makes a significant improvement to the line of work on patience-constrained offline stochastic matching for general graphs , which includes the 0.25-approximate algorithm of Bansal et al . ( 2012 ) , 0.269-approximate algorithm of Adamczyk et al . ( 2015 ) , and 0.31-approximate algorithm of Baveja et al . ( 2018 ) . We emphasize that although our algorithm considers the edges in a random order , no better guarantee is known even without imposing this property . Our general guarantee can be parametrized by the patiences in the graph and improves in certain special cases . Corollary 1 . A ( 1 − e−2 ) /2 ≈ 0.432-approximate random-order probing algorithm for general graphs without patience constraints . In the special case where every patience is∞ , the guarantee of our random-order probing algorithm improves to 0.432 . We note that without patience constraints , the Greedy algorithm which probes edges in decreasing order of weights ( ignoring the probabilities pe ) guarantees at least 1/2 the offline maximum weighted matching knowing the set of active edges in advance . However , our algorithm has the benefit of providing a guarantee relative to the relaxed values ye , and moreover only needs to consider the edges in a uniformly random order ( instead of needing to choose the order ) . This means that the special case of our algorithm where patiences are∞ implies new results for Random-order Contention Resolution Schemes and Prophet Secretary under Edge Arrivals , as we discuss below . Corollary 2 . A ( 1− e−2 ) /2-selectable Random-order Contention Resolution Scheme for the matching polytope of general graphs . For c ≤ 1 , the definition of a c-selectable Contention Resolution Scheme can be reduced to our goal of probing every edge e with probability at least c · ye , as we explain in Section 2 . Previous work in this area has derived a ( 1− e−2 ) /2-selectable “ offline ” Contention Resolution Scheme , which needs to know the set of active edges in advance , for the matching polytope of general graphs ( Guruganesh and Lee , 2017 ) . The selectability has been recently improved to be strictly greater than ( 1− e−2 ) /2 , while also satisfying a monotonicity property ( Bruggmann and Zenklusen , 2020 ) . However , both of these results require knowing the set of active edges in advance , while our algorithm observes the activeness of edges in an ordered fashion and must immediately decide whether to insert any active edges into the matching . In fact , our algorithm satisfies the definition of a Random-order Contention Resolution Scheme introduced in the papers by Adamczyk and Włodarczyk ( 2018 ) ; Lee and Singla ( 2018 ) . To this end , our specific setting of the matching polytope is not captured1 by these papers . Corollary 3 . A 0.432-guarantee for the Prophet Secretary problem under Edge Arrivals , in general graphs . Contention Resolution Schemes for the matching polytope also imply Prophet Inequalities under the Edge Arrival model introduced in Gravin and Wang ( 2019 ) . The best-known result here is a 0.337-selectable adversarial-order Contention Resolution Scheme for the matching polytope of general graphs due to Ezra et al . ( 2020 ) , which implies a 0.337-guarantee for Prophet Inequalities under Edge Arrivals in any order . When the edges arrive in a uniformly random order , this can be called the Prophet Secretary problem ( Esfandiari et al. , 2017 ) under Edge Arrivals , for which our Random-order Contention Resolution Scheme implies an improved 0.432-guarantee . Corollary 4 . A new class of non-adaptive ( 1−1/e ) -selectable Random-order Contention Resolution Schemes for rank-1 matroids . In the further special case of a star graph with infinite patiences , the selectability of our probing algorithm improves to 1− 1/e . The constraint that at most one edge in a star graph can be matched corresponds to a rank-1 matroid , for which a ( 1− 1/e ) -selectable Random-order Contention Resolution Scheme is already known ( e.g . Lee and Singla , 2018 ) . However , our analysis yields a wide range of new such schemes , which are simpler than existing ones in that they satisfy a non-adaptiveness property , which we we elaborate on in Subsection 1.2 . Result 2 . A ( 1 − 1/e ) ≈ 0.632-approximate probing algorithm for bipartite graphs with a unitpatience side . This result improves and generalizes the 1/3-guarantee of Hikima et al . ( 2021 ) which holds for a special case of bipartite graphs with a unit-patience side . Despite the ubiquity of ( 1−1/e ) -guarantees in online matching , to the best of our understanding , our guarantee requires the novel technical ingredient of a ( 1 − 1/e ) -selectable Ordered Contention Resolution Scheme for rank-1 matroids under negative correlation . The aforementioned ( 1− 1/e ) -selectability results for rank-1 matroids assume elements to be active independently and do not establish this guarantee , as we discuss in Subsection 1.2 . We note that our probing algorithm here must be able to choose the order , though . We note that concurrent and independent work of Borodin et al . ( 2021 ) study an online problem which implies the same ( 1− 1/e ) -approximation for the offline problem in the particular case where one side has unit patience and the other side has unlimited patience . 1.2 Description of Techniques . Random-order probing : finding an attenuation function which improves the worst case . As described before , our probing algorithm for general graphs considers the edges in a uniformly random 1If the graph is bipartite , then its matching polytope can be captured by the intersection of two matroids ; however , in this case the selectability guaranteed by Adamczyk and Włodarczyk ( 2018 ) is 1/3 , which is worse than our selectability of 0.432. order . This can be implemented by each edge e drawing an “ arrival time ” xe independently and uniformly from [ 0,1 ] . Baveja et al . ( 2018 ) have previously analyzed a similar algorithm , which probes each incoming edge e according to the fractionally-feasible probability ye as long as e is safe . They show that every edge e ends up being probed with probability at least 0.31 · ye , yielding a 0.31-approximate algorithm . The worst case occurs for an edge e′ whose values of ye′ , pe′ are close to 0 , with both endpoints of e′ being incident to other edges e′′ whose values of ye′′ , pe′′ are 1 . To improve this worst case , we attach an “ attenuation factor ” a ( e ) ∈ [ 0 , 1 ] to each edge e such that the probability of an incoming safe edge e being probed is scaled down by a factor of a ( e ) . We make a ( e ) decreasing in ye and pe , to dissuade the aforementioned edges e′′ with large values of ye′′ , pe′′ from being probed and blocking edge e′ . However , given an arbitrary function a defining the attenuation factors , computing the new worst case could be difficult . Therefore , our approach is instead to derive properties on a which cause the worst case to only involve edges e with a ( e ) = 1 . More specifically , we show that for functions a defined by a ( e ) = ã ( yepe ) for some univariate function ã with ã ( 0 ) = 1 , it is possible to design the derivatives of ã so that in the worst case , edge e′ is only incident to edges e′′ with ye′′pe′′ ≈ 0 ( which implies that a ( e′′ ) ≈ ã ( 0 ) = 1 ) . Our final bound is then derived from the same expression as in Baveja et al . ( 2018 ) , except the worst case ratio has improved to 0.382 . The specific attenuation function we compute is inconsequential to this improved ratio—the key is showing the existence of an attenuation function which eliminates the previous worst case . However , by deriving such necessary properties , we are able to see not only which attenuation functions work , but also which don ’ t work ( suggesting alternative attenuation functions can ’ t do better than ours ) . Further , having a family of valid attenuation functions allows for a choice of different attenuation functions for different application domains while keeping the same approximation guarantee . Using our attenuation for Random-order Contention Resolution Schemes . As stated in Corollaries 2 and 3 , our attenuation analysis in the special case of infinite patiences implies a previouslyunknown ( 1 − e−2 ) /2-selectable Random-order Contention Resolution Scheme for the matching polytope of general graphs . We now discuss the further special case in Corollary 4 of star graphs , for which we can contrast our technique with that used in the known Random-order Contention Resolution Schemes for rank-1 matroids . Here , Lee and Singla ( 2018 ) show that ( 1− 1/e ) -selectability can be achieved using what we would call the attenuation function a ( e ) = exp ( −xepe ) , which depends on the arrival time xe of each edge e. This function is designed ( see Ehsani et al. , 2018 ) to yield a closed-form expression for the probability of availability at any particular time x ∈ [ 0 , 1 ] , which allows them to elegantly compute that the selectability is 1− 1/e . In this special case , our analysis also yields a ( 1 − 1/e ) -selectable Random-order Contention Resolution Scheme . However , instead of designing a specific function , our analysis implies a class of functions which sufficiently2 attenuate large values of pe to prevent them from blocking smaller values of pe . To elaborate , we show that either of the functions a ( e ) = exp ( −αpe ) or a ( e ) = 1−αpe , with α ∈ [ 1/2 , 1 ] , ensures that in the worst case , all edges e have pe ≈ 0 . And in this worst case , a ( e ) ≈ 1 for all e , from which we can conclude that any edge has an ≈ 1 − 1/e chance of being selected . We note that our functions do not depend on the arrival time xe and can be seen as non-adaptive ( 1− 1/e ) -selectable Random-order Contention Resolution Schemes for a rank-1 matroid , which we believe could be applied elsewhere . Bipartite graphs with a unit-patience side . For offline stochastic matching on bipartite graphs , a standard technique ( Bansal et al. , 2012 ) is to randomly round the fractionally-feasible values ye to binary values Ye using the dependent rounding procedure of Gandhi et al . ( 2006 ) , which ensures the patience constraints on both sides to be satisfied w.p . 1 . Under our additional assumption that one of the sides V1 has unit-patience , the vertices in V1 must have rounded degree at most 1 , resulting in the rounded graph being a disjoint collection of stars . One could then separately handle the edges in each star using a Random-order Contention Resolution Scheme for rank-1 matroids , since its edges e will be disjoint from other stars and active independently w.p . pe . 2This intuition can be illustrated as follows . If there is no attenuation , then the worst case involves two edges e′ , e′′ with probabilities pe′ = 0 , pe′′ = 1 , which when shown in a random order implies that e′′ will block e′ w.p . 1/2 , whereas e′ will not block e′′ . The goal of “ attenuation ” is to scale down the probability of selecting e′′ , to increase the probability of selecting e′ . However , this does not lead to ( 1− 1/e ) -selectability . To elaborate , for any vertex v /∈ V1 , let δ ( v ) denote the set of edges incident to v. Fractional feasibility of the ye values ensures∑ e∈δ ( v ) peye ≤ 1 , ( 1 ) but the rounded star graph formed by edges { e ∈ δ ( v ) : Ye = 1 } could have ∑ e∈δ ( v ) peYe > 1 whenever3 some particular edge e′ has Ye′ = 1 , making it difficult to ensure that edge e′ gets selected with sufficient probability when Ye′ is rounded up . Reinterpretation as contention resolution under negative correlation . To resolve this issue , we instead imagine each edge e ∈ δ ( v ) as being active with probability ze : = peye , which satisfies∑ e∈δ ( v ) ze ≤ 1 , due to ( 1 ) . The active edges in δ ( v ) are correlated in a way such that they can not conflict with active edges in other stars ; however , due to this correlation , any of the aforementioned Random-order Contention Resolution Schemes which are agnostic to the correlation will only be 1/2-selectable , as we will show in Section 4 . Despite this apparent lack of a correlation-agnostic ( 1 − 1/e ) -selectable Contention Resolution Scheme , what we do show is that the optimal online algorithm , which trivially sorts the edges e ∈ δ ( v ) in decreasing order of weights we , obtains in expectation at least 1−1/e times the fractional value ∑ e∈δ ( v ) weze . This is only possible due to the following negative correlation property enjoyed by the rounding procedure of Gandhi et al . ( 2006 ) , with Ze ∈ { 0 , 1 } denoting the activeness of an edge e : Pr [ ⋂ e∈S ( Ze = b ) ] ≤ ∏ e∈S Pr [ Ze = b ] ∀S ⊆ δ ( v ) , b ∈ { 0 , 1 } . ( 2 ) Our analysis applies this negative correlation property with b = 0 to show that the expected overall weight obtained by the online algorithm is minimized when the Ze ’ s are independent , despite the fact that for a particular edge e′ , negative correlation among other edges in δ ( v ) could make it more likely that e′ is blocked than in the independent case . Through the equivalence derived in Lee and Singla ( 2018 ) , our analysis also implies the existence of a ( 1− 1/e ) -selectable Ordered Contention Resolution Scheme for rank-1 matroids under negative correlation , assuming the order can be chosen . We leave it as an open question whether a ( 1−1/e ) -selectable Random-order4 Contention Resolution Scheme is possible under property ( 2 ) .

Summary: They give an improved 0.38 approximation algorithm for a basic stochastic matching problem. They also get improvements for a few other variants. One notable result is improving a recent 0.33 approximation in [Hikima et al] to 0.63.