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 paper studies the online bipartite matching problem with capacities in random graphs that are generated by a stochastic process called the configuration model. The configuration model generates random graphs whose degree distributions are sub-Gaussian. The main contribution of the paper is an analysis of the competitive ratio achieved by the well-studied Greedy algorithm for online bipartite matching. The main result relates the competitive ratio to the solution of a certain ODE. The ODE can either be solved in closed form for some special cases of interest or it can be solved numerically to obtain estimates of the competitive ratio. As a corollary of the main result, the paper derives competitive ratios for several special cases, including random d-regular graphs and Erdos-Renyi graphs.