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: The paper considers a version of the stochastic matching problem. We have a graph with known weights on the edges but where each edge is present with probability p_e (also known). Sequentially, a decision-maker has to probe edges. When DM probes an edge, she can see whether the edge was present or not, and in the former case, she must include it in the matching. The main result in the paper is 0.382 approximation-algorithm for this problem.