The topic of prophet inequality broadly refers to the study of trade offs between online and offline selection algorithms in Bayesian settings. The first fundamental result is due to Krengel and Suncheon (krengelS77, ), who considered the following problem in the optimal stopping theory. Let
be a sequence of independent, non-negative random variables with, then there is an online stopping policy such that
I.e., if a gambler plays a sequence of games with rewards and who can stop at any time and take the most recent reward, then a prophet who can foretell which reward to take in the whole sequence cannot gain more than two times the reward of the gambler. In fact, it was later shown by Samuel-Cahn (Cahn84, ) that a simple stopping policy with a uniform threshold gives the same approximation guarantee of .
In computer science terms, inequality (1) compares the performance of online algorithm (the gambler) and offline optimal solution (the prophet) in the competitive analysis framework. The inequality (1) became one of many results in the Bayesian settings where the algorithm has to select a feasible subset of elements from a random sequence of samples with the knowledge about the distribution from which this sequence is generated and where the value of the selected set is compared to the offline optimum solution with the complete information about the sequence of samples. For example, a general result of this type is due to Kleinberg and Weinberg (KW12, ), who gave a competitive algorithm for the Bayesian selection problem where feasibility constraint on the element sets that can be selected is given by the intersection of matroids.
The interest in prophet inequalities as a tool for the design and analysis of algorithms has been largely driven by important applications in algorithmic mechanism design. In particular, the work in Bayesian mechanism design showed a remarkable power of simple sequential posted price mechanisms (SPM) in a variety of allocation problems such as the settings with unit-demand bidders (Chawla10, ),(Alaei14, ) and combinatorial auctions (FeldmanGL15, ),(DuettingFKL17, ). An underlying environment for all these applications is matching in bipartite graphs.
In this paper, we adopt the general stochastic model of Kleinberg and Weinberg (KW12, ) to study an important basic setting of Bayesian (weighted) matching. Specifically, we assume that edges
of a bipartite graph are the elements of our online selection problem which arrive in an arbitrary order with random values (often called weights) sampled independently from different probability distributions on. An online algorithm knows the distribution of each edge’s value but could only select a subset of edges that must form a matching. Upon arrival of every new edge the algorithm observes edge’s value and must immediately decide whether to add this edge if possible to the current matching or not. The algorithm’s objective is to maximize the total value of the selected edges.
This model is more general than the matching (unit-demand) models or sequential posted prices from the mechanism design literature. Indeed, the previous work assumes that one part of the bipartite graph is already present and when a new vertex of the other side arrives all the values of its incident edges are revealed to the algorithm111To be fair, the independence assumption (for all edges) of Kleinberg and Weinberg (KW12, ) is a bit stronger than in some mechanism design models with vertex arrivals. However, without this assumption, it is impossible to recover constant fraction of the prophet’s value if the algorithm only sees only one edge at a time.. On the practical side, the model with edge arrivals (as opposed to vertex arrivals) gives much finer control over preferences of the agents which may dynamically evolve over time. On the negative side, the online policy proposed in (KW12, ) is not as simple as in (Cahn84, ) and is highly adaptive222Although, adaptivity does not preclude Kleinberg and Weinberg’s result to translate into a posted price mechanism for unit-demand multi-parameter bidders who report all their preferences at once, it makes it hard to apply to dynamic settings where valuations of the bidders may dynamically evolve over time. to the online information revealed to the algorithm, i.e., the value thresholds for accepting the elements are constantly changing throughout the algorithm’s execution.
Non adaptivity and simplicity.
Indeed, all application of prophet inequality in mechanism design starting with the work of Hajiaghayi (HajiaghayiKS07, ) and Chawla et al. (Chawla10, ) rely on non adaptive versions of prophet inequality rather than the comparison between optimal online and offline solutions. E.g., (Chawla10, ) refers to the result of Samuel-Cahn (Cahn84, ) that uses simple and robust constant threshold policy but not the earlier version of Krengel and Suncheon (krengelS77, ) which compares the optimal online and offline policies, while Feldman et al. (FeldmanGL15, ) use static item prices that never change throughout the execution of their algorithm and still obtain tight approximation guarantee of in Bayesian combinatorial auctions. Therefore, it is much desirable to have prophet inequalities with non adaptive and simple threshold policy.
In this paper, we prove a non adaptive prophet inequality for the Bayesian bipartite matching setting. Our online algorithm belongs to a natural class of simple vertex-additive threshold policies: (i) vertices on either side of the graph receive threshold values for the left hand side and for the right hand side vertices (ii) the online algorithm accepts every edge that can be added to the current matching if . We find a vertex-additive policy with the expected payoff of at least one third of the prophet’s payoff (the expected value of the optimal matching), i.e., we give a approximation guarantee. We give a complementary lower bound of that shows clear separation from the setting where vertices from one side arrive online and where the upper bound is (see, e.g., (FeldmanGL15, )).
The design of our algorithm and its analysis consist of two stages. First, we combine the ideas from (KW12, ) and (FeldmanGL15, ) of the additive threshold structure and their accounting of online algorithm expected payoff with respect to the offline optimum. Generally, we follow the plan of (KW12, ) but instead of the element’s (edges) contribution to the optimum we work in a “dual space” of vertices similar to item prices in (FeldmanGL15, ), but applied to the both sides of the bipartite graph. Surprisingly, our non adaptive pricing scheme actually works in Bayesian matching setting and would immediately yield a simpler algorithm with the same performance guarantee of 333In fact, one can give a slightly better approximation guarantee of as in (KW12, ) for the intersection of matroids. as in (KW12, ) for the intersection of two matroids. However, a simpler form of the thresholds allows us to improve the approximation guarantee.
In the second stage, we turn the optimization problem of vertex prices into a bilinear algebraic form instead of using existing explicit solutions for the additive thresholds from the prior literature. We further guess that certain vectors of parameters given implicitly as a solution to a semi-linear system of equations should yield significantly better approximation guarantee of . We then present a gradient-decent type algorithm that quickly finds a solution to this system of equations.
1.1. Related Work
Starting with the work of Krengel and Sucheston (krengelS77, ) there has been a lot of work studying different restrictions on stopping rules, distributions, independence assumption etc., which is too broad to discuss in this amount of space. We recommend (Hill_Kertz_survey92, ) for a detailed survey. The line of earlier work (kennedy85, ; kennedy87, ; Kertz86, ) on multiple-choice prophet inequalities, where the online algorithm and the prophet can choose more than one element, is particularly relevant to our paper.
In computer science literature, the research on prophet inequalities and their applications to posted price mechanisms was initiated by Hajiaghayi et al. (HajiaghayiKS07, ). Prophet inequalities were obtained for a variety of multiple choice combinatorial settings: for matroid and intersection of matroids (KW12, ; AzarKW14, ; FeldmanSZ16, ), polymatroids (DuttingK15, ), the generalized assignment problem (AlaeiHL12, ; AlaeiHL13, ), and general downward closed feasibility constraints (Rubinstein16, ). In algorithmic mechanism design, Chawla et al. (ChawlaHK07, ) developed approximately optimal in terms of revenue posted price mechanisms for unit-demand buyers. Other results on Bayesian auctions with the objectives of revenue and welfare maximization include (Chawla10, ; Alaei14, ; Cohen-AddadEFF16, ) for unit-demand buyers, and (Alaei14, ; FeldmanGL15, ; DuettingFKL17, ; RubinsteinV17, ) for combinatorial auctions. A direct connection between pricing mechanisms and prophet inequalities is shown in (CorreaFPV19, ). The success story of prophet inequality in mechanism design applications have instigated further studies of dynamic posted prices in other online optimization settings such as k-server (CohenEFJ15, ), and makespan minimization for scheduling problems (FeldmanFR17, ).
Our matching setting is closely related to online bipartite matching problem introduced by Karp et al. (KarpVV90, ) which is a central topic in the area of online algorithms with a wide range of applications. In this model one side of the bipartite graph is given in advance, while the vertices of the other side arrive online in an arbitrary order. The online algorithm observes all edges incident to the arriving vertex, however the algorithm does not have any prior information about distribution of the edges (typically in this line of work edges have only weights). Karp et al. gave tight approximation guarantee for the adversarial order of vertex arrivals. The result was extended to the settings with weighted vertices (AggarwalGKM11, ) and Adword problem (MehtaSVV07, ). The latter problem considers generalized matching where each edge has a weight and each vertex in the given side of the graph has a budget (capacity constraint). Earlier work (MehtaSVV07, ; BuchbinderJN07, ) on Adword problem focused on worst-case performance guarantees for the adversarial or random arrival order (GoelM08, ; DevenurH09, ). More recent papers consider Bayesian setting in which distribution of weights (bids) usually are known in advance, e.g., (FeldmanMMM09, ; MahdianY11, ; KarandeMT11, ; ManshadiGS11, ; HaeuplerMZ11, ; HaeuplerMZ11, ; MirrokniGZ12, ; DevanurJSW19, ). A few papers consider matching models closer to our edge arrival setting. E.g., McGregor (McGregor05, ) gave a -competitive online algorithm for (weighted) edge arrivals with preemption, i.e., where the online algorithm can discard any previously matched edges. Later, improved approximation guarantees were obtained for randomized algorithms (EpsteinLSW13, ) and for special cases of growing tree and forests (ChiplunkarTV15, ; BuchbinderST17, ). A very recent line of work (KTWZZ18, ; Ashlagi_arxiv18, ; HPTTWZ19, ) considers fully online matching model in (unweighted) bipartite graphs where vertices of both sides arrive online revealing edges to all previously arrived vertices. In this model (HPTTWZ19, ) gave tight -competitive algorithm.
Closest to our work is the result of Kleinberg and Weinberg (KW12, ) who obtained -approximation prophet inequality for the online Bayesian selection problem from the intersection of matroids. In comparison, our work can be seen as specialization of their result to an important setting of matching in bipartite graphs that corresponds to the intersection of matroids. In this setting, which is arguably considered to be central for the area of online algorithms, we design much simpler non-adaptive policy with significantly better approximation guarantees than the guarantee of (5.82) from (KW12, ). In terms of techniques, we first combine approaches from (KW12, ) and (FeldmanGL15, ) to derive a clean mathematical formulation, which we then efficiently tackle using novel techniques that may be useful in other stochastic settings.
Let be a bipartite multi-graph between the left and the right parts with and vertices. The graph has a set of multi-edges between and with different values for each edge , where denotes the set of non-negative real numbers. The edges arrive online in an unknown order , where edges are enumerated by their arrival time from to . For a given profile of values , we let to denote the value of the maximal (offline) matching in the graph with edge values .
Bayesian online selection problem.
We consider a Bayesian setting, where edge values are drawn independently from the distributions . Write , so that joint valuation profile
of all edges is drawn from the joint distribution. Both the graph and the distribution are known in advance. An input to the Bayesian online selection problem (BOSP) is a sequence of pairs revealed one by one from time to . An online selection algorithm (also called online policy) 444The algorithm can be randomized, but it must be independent of the randomness (if any) in the choice of . Any randomized algorithm is a distribution over deterministic policies. upon receiving a new piece of input at time must irrevocably decide whether to take the edge subject to the feasibility constraint that the selected set of edges must form a matching. The goal of the online algorithm is to maximize the total value of the selected edges in expectation over .
Every online policy at each time may be described as a function of the past selection decisions (not the edge values), since the posterior distribution of values and feasibility constraint for the edges coming after time (for a fixed sequence of past decisions) do not depend on the realized values before time . We use to denote the set of all edges accepted by an online selection algorithm . When it is clear from the context, we sometimes will drop the dependency on the arrival order . Without loss of generality, one may restrict attention to monotone selection policies, i.e., policies that for any given history before time select the edge with higher probability for larger values of . It means that any monotone deterministic selection policy can be described by a sequence of thresholds , where each threshold only depends on the prior history , such that edge is accepted if and only if and neither of vertices incident to was covered by previously selected edges.
The algorithm is agnostic to the order of edge arrivals, that is should perform well regardless of the arrival order of the edges. We study performance of in the worst-case for a fixed-order adversary which is the standard assumption in the prior work on BOSP. Formally, we assume that an adversary selects the order of edge arrivals (or distribution of different orders) that does not change with the choices of algorithm and/or realized edge values. As the optimal solution is usually quite complex even in the most basic settings, the performance of the online policy is compared against a stronger benchmark of the expected prophet’s performance who knows all the realized values in advance before selecting any edges, i.e., the benchmark is the offline optimum solution. Prophet inequality refers to the approximation guarantee of that online algorithm can achieve compared to the prophet, i.e., for any arrival order
3. Online Algorithm
Our algorithm belongs to a natural class of simple non-adaptive threshold policies that we call vertex-additive policies. Any vertex-additive policy (denoted as V-add) is described by two positive real vectors and that accepts a new arriving edge with value if and only if
vertices are available, i.e., both , are not covered by previously accepted edges;
edge’s value exceeds the sum of its vertex thresholds .
If at least one of the previous two conditions fails, then the edge is rejected. The class of vertex-additive policies bears certain resemblance to the solution proposed in (KW12, ) for the intersection of matroids set systems, but has some important distinctions.
3.1. Analysis of vertex-additive policies
We assume that incoming edges of graph arrive in a fixed order , which is unknown to us. Similar to the previous literature on the prophet inequality in the strategic settings (e.g. (Chawla10, ; FeldmanGL15, )), we treat the total value of the accepted edges as the social welfare that is comprised of two components: (i) the revenue part that is equal to the sum of accepted edges’ thresholds, i.e., the total payment received by the algorithm if it takes from each accepted edge a price equal to its threshold; (ii) the surplus part, which is equal to the sum of extra values that every accepted edge contributes to the total value on top of its guaranteed threshold payment. In particular, we will use price terminology interchangeably with thresholds when referring to our vertex-additive policy. In the following we will analyze separately the revenue and surplus parts of any vertex-additive policy . To simplify notations we use to denote the set of accepted edges for a particular run of on the valuation profile .
The important property of the vertex-additive prices is that we can conveniently attribute the expected revenue of our policy (denoted as Rev) to the average set of covered vertices instead of the set of accepted edges. We denote the final set of covered vertices of for a given valuation profile as that consists of the sets vertices in the part of and vertices in the part of . The expected revenue can be written as follows.
We can give a lower bound on the expected surplus using the edges between uncovered vertices of . The expected surplus (denoted as Surplus) by definition is equal to
where denotes . Since our goal is to compare the surplus with the optimal matching selected by the prophet, we focus on the edges in the optimal matching , i.e., the optimal matching for another independently drawn valuation profile . Specifically, our lower bound will include all the edges in a random optimal matching between uncovered vertices for the valuation profiles . To deal with specific edges of the optimal matching
we employ indicator random variablefor the event that there is an edge between vertices and random variable :
Lemma 3.0 ().
The expected surplus is at least
where can be also written as .
To analyze performance of a vertex-additive policy , we consider another indicator random variable of that indicates whether vertices are free to take (not yet covered) on the valuation profile right before the arrival of the edge at time . The surplus of V-add is given by (3) which we further rewrite and bound as follows
where the second equality holds, since the indicator function does not depend on and thus is independent of the value of which we substituted with another independent and identically distributed random variable ; and the inequality holds simply because is a decreasing function in time 555Time denotes the time after arrival of the last edge.. We note that the right hand side (RHS) of (6) can be directly related to the sets of covered vertices .
where to get the last inequality we ignored some non negative terms in the previous summation and counted only the edges between and that appear in the optimal matching . To conclude the proof we will use the following simple fact about function .
Fact 3.1 ().
for any real random variables .
Before we continue with the lower bound of (RHS) (7), we observe that for any fixed vertices , , and valuation profile . Indeed, by definition and, therefore, whenever indicator . Finally, we obtain the required lower bound on the (RHS) of (7).
where the last inequality follows from Fact 3.1 applied to random variable , where and so that . ∎
3.2. Optimization of parameters.
The V-add policy has two variable parameters representing the prices on the vertices in the left and right parts of . One needs carefully choose these parameters to compete with the prophet. The standard approach in the prior work on prophet inequalities is to explicitly set the thresholds based on certain statistics of each element’s marginal contribution to the optimal solution (usually a constant fraction of the expectation, or sometimes the median of its distribution). We take a different approach by casting our problem into a linear algebraic form and then implicitly choosing parameters that would maximize the bound in this formulation. In this section we derive this formulation and show how an implicitly described (via certain semi-linear equation) solution to this problem yields a -approximation to the prophet’s expected value. Furthermore, we give a gradient-decent type algorithm that efficiently calculates vectors up to any given precision error in the following section.
Our linear algebraic formulation is based on the lower bounds (2) and (5) for the revenue and surplus terms of the vertex-additive policy . Namely, the expected value of the is at least the sum of the right hand sides in (2) and in (5)
We note that RHS of (8) is calculated in expectation over all valuation profiles , which only determines the sets and . We further relax the bound in RHS of (8) by letting the sets to be selected by an adversary who wants to minimize our performance guarantee. Let the worst-case sets be for and for . Hence,
We rewrite RHS of (9) by introducing two matrices and that respectively comprise expected contributions and selection probabilities to the optimal matching of the edges between all pairs of vertices and . Formally,
Thus lower bound (9) can be written as
where are and dimensional vectors with all coordinates equal to .
Now RHS of (11) can be rewritten using linear algebraic notations. To this end we change vectors to diagonal and matrices , and represent sets , and respectively as vectors , with
With these notations at hand, RHS of (11) is equal to
where is a matrix operator that changes each entry of the matrix to . Our optimization problem is to find vertex prices so as to maximize expression (14) with diagonal matrices . Now we can state our main approximation guarantee.
Theorem 3.2 ().
There are vertex prices such that vertex-additive policy .
We will find non negative price vectors , s.t. for . Let . We view the linear algebraic expression under minimization in (14) as a bilinear function of and rewrite it as a separate sum of constant, linear, and bilinear terms.
where the last inequality holds, since for any choice of vectors (recall that all entries in the matrix and vectors are non negative). The latter relaxation (15) of the bound (14) allows us to simplify the problem so that it can be solved optimally. It turns out that the optimal solution that maximizes RHS of (15) satisfies the following system of semi-linear equations (it is not linear, since matrix has a non linear dependency, e.g., on due to operator).
We note that if (16) holds, then RHS of (15) is independent of and is equal to . For the remainder of the Theorem 3.2’s proof, we shall assume that the system of equations (16) has a solution and prove the required bound on the expected value of the prophet, opt. We give an algorithmic proof for the existence of the solution to (16) in the next section666We do no provide a proof for the optimality of that satisfy (16), as it is not needed for any of our results..
Fact 3.2 ().
and , where stands for coordinate-wise inequality .
Finally, we show that expression (14) is at least of the prophet’s expected value.
where the first inequality and the first equality directly follow from (16) and (15); to get the second inequality we observe that all entries in , and are non negative and also use the Fact 3.2; to get the last inequality, we notice that matrices and vectors are comprised of non negative entries, and observe that since , matrix dominates matrix in every entry; the last equality is (12). ∎
3.3. Algorithmic solution to the system of semi-linear equations (16)
In this section, we describe a gradient decent type algorithm that solves system of equations (16) up to an arbitrary precision error in polynomial in , and time. We start by rewriting (16) as a system of equations in and variables.
Our algorithm iteratively updates the solution of to (17) reducing the total additive approximation error in every such iteration. Specifically, we look at each separate equation in (17) and write their approximation errors in two vectors .
The update rule for in our iterative procedure is quite simple: we choose one of or vectors with the larger -norm error in (18) and then correct this vector by subtracting half of its error vector. The description of our update rule for is summarized in Algorithm 1 below. The initialization of and vectors can be arbitrary, e.g., one can set both to be -vectors.
Theorem 3.3 ().
First, we show that our algorithm decreases the total -norm of the combined error in all iterations and thus converges to the solution of (17). Specifically, we show a constant fraction decrement in -norm of the total error in each iteration.
Without loss of generality, let us assume that in a given iteration and thus we update vector to a new value in this iteration with the respective new errors given by formula (18) for the updated vectors. Let us bound first the decrease in the -norm of the error compared to its previous value .
where to get the last equality we simply plugged in the definition of from (18). We let
Hence, for every . We notice that where as per our definition of matrix . Now, the function satisfies the following two simple properties: (i) has the same sign as ; (ii) for any real numbers . Thus all for every have the same sign as (property (i) for and ). Moreover, by the second property
where the last inequality holds since as the sum of the row entries in the matching probability matrix . Therefore, we have
We obtain a bound on the decrease of -norm of by adding previous equation over all .
Next we bound the increase in the -norm of error compared to . We note that
for every . Thus and we get
where the last inequality holds because we assumed . That means that the -norm of the combined error decreases by a factor of in every round and after steps the -norm of the combined error will be where . Thus Algorithm 1 terminates in steps. Furthermore, if we let the Algorithm 1 to continue indefinitely, it will produce a sequence of states for each iteration that converges to an exact solution of (17) as . Indeed, we can show that is a Cauchy sequence, i.e., for any we have
since for any and for any . As is a complete space, the Cauchy sequence has a limit which must be an exact solution to (17) as the combined error