Online Matching in Sparse Random Graphs: Non-Asymptotic Performances of Greedy Algorithm

1 Introduction . Finding matchings in bipartite graphs ( U [ V , E ) , where E ⇢ U⇥V is a set of edges , is a long-standing problem with different motivations and approaches [ Godsil , 1981 , Zdeborová and Mézard , 2006 , Lovász and Plummer , 2009 , Bordenave et al. , 2013 ] . If U is seen as a set of resources and V as demands , the objective is to allocate as many resources to demands ( an allocation - or a matching - between u and v is admissible if ( u , v ) 2 E ) with the constraint that a resource is allocated to only one demand and vice-versa . Motivated particularly by practical applications to Internet advertising , the online variant of this problem is receiving increasing attention ( we refer to the excellent survey [ Mehta , 2012 ] for more applications , specific settings , results and techniques ) . In this case , the set of vertices U is present at the beginning and the graph unveils sequentially : vertices v 2 V are observed sequentially , one after the other , along with the edges they belong to . An online algorithm must decide , right after observing vk and its associated set of edges Ek : = { ( u , vk ) 2 E } to match it to some other vertex u 2 U , at the conditions that ( u , vk ) 2 Ek and u 2 U has not been matched yet . The performance of an online algorithm is evaluated by its competitive ratio , which is the ratio between the size of the matching it has created and the highest possible matching in hindsight [ Feldman et al. , 2009 ] . This theoretical setting is particularly well suited for online advertising : U is the set of campaigns/ads that an advertiser can run and users v1 , v2 , . . . , vT arrive sequentially [ Mehta , 2012 , Manshadi et al. , 2012 ] . Some of them are eligible for a large subset of campaigns , others are not ( usually based 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . on their attributes/features , such as the geographic localization , the browsing history , or any other relevant information ) . The objective of an advertiser ( in this over-simplified model ) is to maximize the number of displayed ads . In practice , campaigns/ads are not displayed only once but have a maximal budget of impressions ( say , a specific ad can be displayed only 10.000 times each day ) . A possible trick consists of duplicating the vertices of U as many times as the budget . However , this results in strong and undesirable correlations between vertices . It is , therefore , more appropriate to consider a bipartite graph with capacities and admissible matchings as subsets of edges such that each vertex belongs to several different edges , but not more than their associated capacities ! 2 N ( a vertex v 2 V is matched once while u 2 U can be matched ! u times ) . This online matching problem with capacities has been quite extensively studied . It is known that GREEDY , which matches all incoming vertices to any available neighbor has a competitive ratio of 1/2 in the worst case , albeit it achieves 1 1/e as soon as the incoming vertices arrive in Random Order [ Goel and Mehta , 2008b ] . The worst-case optimal algorithm is the celebrated RANKING , which achieves 1 1/e on any instance [ Karp et al. , 1990 , Devanur et al. , 2013 , Birnbaum and Mathieu , 2008 ] , and also has better guarantees in the Random Order setting [ Mahdian and Yan , 2011 ] . Beyond the adversarial setting , the following stochastic setting has been considered : there exist a finite set of L “ base ” vertices v ( 1 ) , . . . , v ( L ) associated to base edge-sets E ( 1 ) , . . . , E ( L ) . When a vertex vk arrives , its type ✓k 2 { 1 , . . . , L } is drawn iid from some distribution ( either known beforehand or not ) and then its edge set is set as Ek = E ( ✓k ) . In the context where the distribution is known , algorithms with much better competitive ratios than GREEDY or RANKING were designed [ Manshadi et al. , 2012 , Jaillet and Lu , 2014 , Brubach et al. , 2019 ] , specifically with a competitive ratio of 1 2/e2 when the expected number of arrival of each type are integral and 0.706 without this assumption . Notably , those competitive ratios still hold with Poisson arrival rates rather than a fixed number of arrivals . On a side note , a vast line of work considers online matching in weighted graphs [ Devanur et al. , 2012 , Goel and Mehta , 2008a , Mehta , 2012 ] , which is outside the scope of this paper . However , it is still worth noting that the unweighted graph is a weighted graph with all weights equal . This model of the stochastic setting is quite interesting but rather strong : it lacks flexibility and can not be used to represent some challenging instances ( for example when the degrees of each vertex U increase linearly with the number of vertices in V , or when the set U of campaigns must be fixed so that the model is well specified , etc ... ) . Another tentative is to consider Erdős-Rényi graphs assuming that each possible edge is present in U ⇥V with some fixed probability and independently of the other edges ( see [ Mastin and Jaillet , 2013 ] ) . The most interesting and challenging setting corresponds to the so-called sparse regime where each vertex of U has an expected degree independent of the size n of V , which amounts to take a probability of connection equal to c/n . Interestingly enough , even the analysis of the simplest GREEDY algorithm is quite challenging and already insightful in those models [ Borodin et al. , 2018 , Arnosti , 2019 , Dyer et al. , 1993 , Mastin and Jaillet , 2013 ] . Unfortunately , although this Erdős-Rényi model is compatible with growing sets U and V , it also turns out to be quite restrictive . The main reason is that the approximate Poisson degree distribution of the vertices has light-tail and does not allow for the appearance of the so-called scale-free property satisfied by many real-world networks [ Barabási et al. , 2000 , Van Der Hofstad , 2016 ] . We , therefore , consider a more appropriate random graphs generation process called configuration model , introduced by [ Bender and Canfield , 1978 ] and [ Bollobás , 1980 ] . The optimal matching of this model has been computed in [ Bordenave et al. , 2013 ] . The configuration model is particularly well suited to handle different situations such as the following one . Assume that campaigns can either be “ intensive ” ( with many eligible users ) or “ selective/light ” ( few eligible users ) , with an empirical proportion of , say , 20 % /80 % . Then whether an advertiser handles 100 campaigns at the same time or 10.000 , it will always have roughly this proportion of intensive vs. light campaigns . Similarly , some users are more valuable than others , and are thus eligible for more campaigns than the others ; the proportion of each type being independent of the total population size . The configuration model accommodates these observations by basically drawing iid degrees for vertices U and V ( accordingly to some different unknown distributions for U and V ) and then by finding a graph such that those degrees distribution are satisfied ( up to negligible errors ) ; as a consequence , the graphs generated are sparse , in the sense that the number of edges grows linearly with the number of vertices . Additionally , the configuration model is a well-suited random graph model which mimics a number of properties of real-world complex networks , while being analytically tractable . For instance , choosing power-law distributions for the degrees allows to obtain the so-called scale-free property ( often observed in practice , as highlighted for the web by Faloutsos et al . [ 1999 ] ) . The configuration model also displays the so called “ small-world phenomenon ” ( observed for instance in the graph of Facebook by Backstrom et al . [ 2012 ] ) as its diameter is of logarithmic order . Main contribution We investigate the performances ( in terms of expected competitive ratio ) of the GREEDY matching algorithm in configuration models and we provide explicit quantitative results using stochastic approximation techniques [ Wormald , 1995 ] ; we prove that the increasing size of the random matching created is arbitrarily close to the solution of some explicit ODE . Solving the latter then gives in turn the solution to the original problem . The remaining of the paper is organized as follows . Section 2 describes precisely the problem and Theorem 1 is our first main result : it describes the performances of GREEDY in the capacity-less problem . The proof of Theorem 1 is delayed to Appendix D , but the main ideas and intuitions are provided in Section 3 . The online matching with capacities problem is treated in Appendix A . 2 Online Matching Problems ; Models and main result . Consider a bipartite graph with capacities G = ( U , V , E , ! ) where U = { 1 , . . . , N } and V = { 1 , . . . , T } are two finite set of vertices , E ⇢ ( u , v ) , u 2 U , v 2 V is the set of edges and ! : U ! N⇤ is a capacity function . A matching M on G is a subset of edges e 2 E such that any vertex v 2 V is the endpoint of at most one edge e 2 M and any vertex u 2 U is the endpoint of at most ! u edges in M . We will denote by M the set of matchings on G ; the optimal matching M⇤ 2M is the one ( or any one ) with the highest cardinality , denoted by |M⇤| . The batched matching problem consists in finding any optimal matching M⇤ given a graph with capacities G ; the online variant might be a bit more challenging , as the matching is constructed sequentially . Formally , the set of vertices U and their capacities ! are known from the start , and vertices v 2 V arrive sequentially ( with the edges they belong to ) and M0 = ; . At stage t 2 N – assuming a matching Mt 1 has been constructed – , a decision maker observes a new vertex1 vt and its associated set of edges { ( u , vt ) ; u 2 E } . If possible , one of these edges ( ut , vt ) is added to Mt 1 , with the constraint that Mt = Mt 1 [ { ( ut , vt ) } is still a matching . The objective is to maximize the size of the constructed matching MT . The classical way to evaluate the performances of an algorithm is the competitive ratio , defined as |MT |/|M⇤| 2 [ 0 , 1 ] ( the higher the better ) .

Summary: The authors investigate bipartite matching in an online setting were the graph edges are generated via the configuration model (i.e. from a chosen degree distribution). They bound the estimation error, measured via the expected ratio between the online matching obtained by Greedy and the hindsight-optimal matching. This allows them to compare the performance guarantee for Greedy versus the state-of-the-art Ranking algorithm. The authors prove that surprisingly, there exist problem instances (including simple and natural ones) where Greedy outperforms Ranking.