Online Stochastic Max-Weight Bipartite Matching: Beyond Prophet Inequalities
Christos Papadimitriou, Tristan Pollner, Amin Saberi, David Wajc
aa r X i v : . [ c s . D S ] F e b Online Stochastic Max-Weight Bipartite Matching:Beyond Prophet Inequalities ∗Christos Papadimitriou , Tristan Pollner , Amin Saberi , and David Wajc Columbia University Stanford University
Abstract
The rich literature on online Bayesian selection problems has long focused on so-called prophetinequalities, which compare the gain of an online algorithm to that of a “prophet” who knowsthe future. An equally-natural, though significantly less well-studied benchmark is the opti-mum online algorithm, which may be omnipotent (i.e., computationally-unbounded), but notomniscient. What is the computational complexity of the optimum online? How well can apolynomial-time algorithm approximate it?Motivated by applications in ride hailing, we study the above questions for the online stochas-tic maximum-weight matching problem under vertex arrivals. This problem was recently intro-duced by Ezra, Feldman, Gravin and Tang (EC’20), who gave a / -competitive algorithm forit. This is the best possible ratio, as this problem is a generalization of the original single-itemprophet inequality.We present a polynomial-time algorithm which approximates optimal online within a factorof . —beating the best-possible prophet inequality. At the core of our result are a new linearprogram formulation, an algorithm that tries to match the arriving vertices in two attempts,and an analysis that bounds the correlation resulting from the second attempts. In contrast, weshow that it is PSPACE -hard to approximate this problem within some constant α < . ∗ Research supported in part by NSF Awards CCF1763970, CCF191070, CCF1812919, ONR award N000141912550,and a gift from Cisco Research.
Introduction
Decision-making in an uncertain, dynamic environment influenced by one’s decisions has arguablyalways been the essence of life, and yet it appears to have been first confronted mathematically byHerbert Robbins and Richard Bellman, from different perspectives, in the late 1940s and early 1950s.Decision theory initially focused on instantaneous decisions, but later gave us stopping rules andthe gem of prophet inequalities [34]. Later, the Internet age brought us new business models relyingexclusively on stochastic decision making — online advertising, ride hailing, kidney exchanges —in which the changing environment affected by the agents’ decisions can often be abstracted as anevolving weighted bipartite graph.Here we study one such problem, the online Bayesian bipartite matching, or
RideHail , problem.The input to this problem is a random bipartite graph, revealed over time. Initially, the m nodeson one side of the graph, termed taxis or bins, are present. The n nodes on the other side, termedpassengers or balls, are revealed over time. Initially, we know for each ball t the probability p t ofit actually arriving, as well as the weight w i,t of the edge connecting it to any bin i — if it arrives.If ball t does not arrive, we do nothing at time t ; if it does arrive, we can choose to match it,irrevocably, to some unmatched neighbor i before time t + 1 , yielding a profit w i,t . Our goal is tomaximize the overall expected profit. RideHail essentially generalizes the classic single-item online Bayesian stopping rule problem— the so-called prophet inequality problem. In particular, our problem with a single offline nodealready captures the worst-case examples of the prophet inequality, for which no online algorithm isbetter than / -competitive. On the other hand, RideHail is a special bipartite case of the onlinestochastic max-weight matching problem in general graphs studied by Gravin and Wang [23] andEzra et al. [16]. In the latter work, Ezra et al. present a / -competitive algorithm for this problem,which is worst-case optimal for our problem too.There is an extensive literature on numerous variations of online Bayesian selection problems,that relate the performance of online algorithms with the omniscient prophet of inequality fame —that is to say, with the offline optimum (see Section 1.2). In particular, these works study achievablecompetitive ratios: the worst-case ratio over all inputs between the online algorithm and the bestoffline algorithm. While this may be the right thing to do when the input is adversarial, when theinput is generated stochastically one perhaps could do better. In particular, in the stochastic case,the optimum online algorithm for the given input is a well-defined benchmark that can be computedin exponential time. Suddenly we are in the realm of approximation algorithms, rather than ofcompetitive analysis.In approximation algorithms, typically one explores two interesting questions: First, is approxi-mation hard? And second, what is the best approximation ratio achievable in polynomial time?In this paper we address both questions. First, we show that for some α < it is PSPACE -hardto approximate the
RideHail problem within a factor of α . Theorem 1.1.
It is
PSPACE -hard to approximate the optimal online
RideHail algorithmwithin a factor of α , for some absolute constant α < . Here, − α is small, limited by the current status of expander constructions and approximationhardness of MAX-SSAT (see Section 2). To our knowledge, no past work on variants of onlinematching had demonstrated such level of hardness.We then develop an approximation algorithm, as well as a technique to bound the (online)optimum. The upper bounding technique is our main innovation. To our knowledge, all past workon approximating this large family of problems, with the exception of [5], has used the prophetinequality bound on the offline optimum, which necessarily limits the approximation ratio for manyvariations to be below / . e go for bounding the online optimum. We achieve this by identifying a new constraint whichseparates online from offline algorithms. In particular, we note that online algorithms cannot matchan edge ( i, t ) with probability greater than the probability of ball t arriving, times the probability ofbin i not being matched by the online algorithm beforehand, due to independence of these events.This constraint, which is not true of offline algorithms, poses restrictions on the marginal probabilitiesof edges to be matched by the optimal online algorithm. Combining this constraint with the naturalmatching constraints we obtain a new LP which bounds the optimal online algorithm’s gain. Usingthis new LP bound (and a number of further ideas, see Section 1.1), we design a new algorithmwhich recovers at least of the online optimum, i.e., a ratio strictly better than the optimalcompetitive ratio of / . Theorem 1.2.
There exists a polynomial-time online algorithm which is a . -approximationof the optimal online algorithm for the RideHail problem.
We further generalize our algorithm and achieve the same approximation bound for the moregeneral problem in which weights of any given ball’s edges can follow any joint distribution, butweights of different ball’s edges are independent. That is, we extend our positive results to the moregeneral problem studied by Ezra et al. [16], in bipartite graphs. (See Section 5.)
Here we give a very brief overview of the key ideas used to obtain our main results.
For our
PSPACE -hardness result, we refine the result of Condon et al. [11] for maximum satisfiabilityof stochastic SAT instances. In the stochastic SAT (SSAT) problem, introduced by Papadimitriou[39], a 3CNF formula is given, and variables x , x , . . . , x n are alternatingly set by an (online)algorithm and randomly set by nature. Condon et al. [11] proved that approximating the maximumexpected number of satisfiable clauses of an SSAT instance is PSPACE -hard. Using an expandergraph construction, we extend this result to SSAT instances in which each variable appears in atmost a constant number of clauses. We then give a polynomial-time reduction from approximatingmaximum satisfiability of a bounded-occurrence SSAT instance to approximating the optimal onlinealgorithm for the
RideHail problem, implying our claimed
PSPACE -hardness.
Our algorithmic results involve a number of ideas. We outline the key ones here.
A New LP.
We recall that we want to approximate the optimal online algorithm within a factorstrictly greater than the / which is best-possible against the optimal offline algorithm. Hence, ourfirst objective is to identify a property which separates online from offline algorithms. To this end,we note that for any online algorithm A , the probability of arrival of ball t is independent of theevent that bin i is not matched by Algorithm A prior to time t . Consequently, the probabilitythat edge ( i, t ) is matched by Algorithm A is at most the product of these two events’ probabilities.Combining this constraint with natural matching constraints, we obtain an LP which bounds theexpected gain of the optimal online algorithm (but not its offline counterpart) . Note that this constraint does not necessarily hold for the prophetic optimum offline algorithm A ∗ , which makesits matching choices based on both past and future balls’ arrivals. Second Chance Algorithm.
We present an efficient online algorithm for approximately round-ing a solution to the above LP. Let x i,t be the decision variables of this LP. Intuitively, these x i,t serve as proxies for the probability of ( i, t ) to be matched by the optimal online algorithm. Ouronline algorithm matches each edge ( i, t ) with probability at least x i,t · ( / + c ) for c = / . Ouralgorithm can be seen as a generalization and extension of the / -competitive algorithm of Ezra etal. [16] for our problem. Their algorithm can be thought of as approximately rounding the aboveLP (without the new constraint) as follows. After each arrival of ball t , pick a bin i with probabilityproportional to x i,t , and then, if bin i is unmatched, match edge ( i, t ) with some probability q i,t .These q i,t are set to guarantee that each edge ( i, t ) is matched with marginal probability x i,t · / . Toimprove on this, we first note that modifying these q i,t appropriately results in each edge ( i, t ) beingmatched with probability precisely x i,t · ( / + c ) if P t ′ To prove that conditioning on ball t not being matched after its first pick indeed doesnot decrease the probability of bin i being free by much, we show that (i) the bins’ matched statusesby time t have low correlation, and (ii) bin i is unlikely to be picked twice by ball t . To prove Property(i), we show that most of the probability of a bin to be matched by this algorithm is accounted forby variables which are negatively correlated, and even negatively associated (see Section 2). For ourproof of Property (ii), we finally reap the rewards from our new LP constraint. In particular, thisconstraint implies that for bins i with P t ′ Let Q, Q ′ ≥ be positive quantities, such that Q ′ /Q ≤ β , and let α ∈ (0 , . Then, an (cid:0) α + β β (cid:1) -approximation to Q + Q ′ yields an α -approximation to Q . We now turn to providing background on problems and tools used in this work. Stochastic SAT. The stochastic SAT (SSAT) problem was first defined by Papadimitriou [39].In this work, we will consider the maximization variant of this problem, defined below. Definition 2.2. The input to the MAX-SSAT problem is a 3CNF formula φ over an ordered listof variables ( x , x , . . . , x n ) . We choose a value of either True or False for x , nature chooses avalue of either True or False for x uniformly at random, we choose a value of either True or False for x , and so on. Our goal is to maximize the expected number of satisfied clauses in φ afterall the variables have been assigned a value. For convenience, we will refer to { x , x , . . . } as the“deterministic variables" and { x , x , . . . } as the “random variables."In his work introducing SSAT, Papadimitriou [39] proved PSPACE-hardness of determining theprobability of satisfiability of an SSAT instance. Over a decade later, this was improved to a hardnessof approximation result by Condon et al. [11], via extensions of the PCP theorem [7]. In particular,they prove the following hardness of approximation result. Lemma 2.3. ([11, Theorem 3.3]) There exist constants k ∈ N and α ∈ (0 , so that it is PSPACE-hard to compute an α -approximation to OP T on ( φ ) for a MAX-SSAT instance φ satisfying:1. no random variable appears negated in any clause of φ , and2. each random variables appears in at most k clauses of φ . It is worth noting that Theorem 3.3 in [11] only includes the statement about random variablesbeing non-negated. The second property is a direct consequence of the proof of the theorem. InAppendix A we explain the necessary modifications to the proof to add this guarantee. xpander Graphs. Define the expansion of a graph G as h ( G ) := min S ⊆ V, | S |≤| V | / | E ( S, V \ S ) || S | , where E ( X, Y ) := { ( x, y ) ∈ E | e ∈ X, y ∈ Y } denotes the edges with one endpoint in X andthe other in Y . We will utilize results providing explicit, deterministic constructions of graphs withconstant degree and constant expansion (e.g. [20, 35]). Lemma 2.4. There exists a deterministic, polynomial-time construction of a graph on n verticeswith expansion at least 1 and maximum degree at most some constant d . Negative Association. We briefly review some notions of negative dependence we need in thiswork, in particular, the notion of Negatively Associated random variables. Definition 2.5 ([30, 32]) . Random variables X , . . . , X n are negatively associated (NA) , if everytwo monotone non-decreasing functions f and g defined on disjoint subsets of the variables in ~X arenegatively correlated. That is, E [ f · g ] ≤ E [ f ] · E [ g ] . (1)A family of independent random variables are trivially negatively associated. A more interestingexample of negatively associated random variables is the following. Proposition 2.6 (0-1 Principle [15]) . Let X , . . . , X n ∈ { , } be binary random variables such that P i X i ≤ always. Then, the joint distribution ( X , . . . , X n ) is negatively associated. More elaborate NA distributions can be obtained via the following closure properties. Proposition 2.7 (NA Closure Properties [15, 30, 32]) . 1. Independent union. Let ( X , . . . , X n ) and ( Y , . . . , Y m ) be two mutually independent negativelyassociated joint distributions. Then, the joint distribution ( X , . . . , X n , Y , . . . , Y m ) is also NA.2. Function composition. Let X = ( X , . . . , X n ) be NA, and let f , . . . , f k be monotone non-decreasing functions defined on disjoint subsets of X . Then the joint distribution ( f , . . . , f k ) is also NA. Negative association implies many powerful concentration inequalities and other useful properties(see e.g., [8, 15, 30, 32]). For our purposes we will use the pairwise negative correlation of NAvariables, implied by Equation (1) with the disjoint functions f ( ~X ) = X i and g ( ~X ) = X j for i = j . Proposition 2.8. Let X , . . . , X n be NA random variables. Then, for all i = j , Cov( X i , X j ) ≤ . In this section, we prove our PSPACE-hardness result. Theorem 1.1. It is PSPACE -hard to approximate the optimal online RideHail algorithm withina factor of α , for some absolute constant α < . .1 Extending Stochastic SAT Hardness We first extend hardness of approximation for MAX-SSAT instances as in Lemma 2.3 to instanceswhich in addition satisfy that deterministic variables appear in at most k clauses. Lemma 3.1. There exist constants k ∈ N and α ∈ (0 , so that it is PSPACE-hard to compute an α -approximation to OP T on ( φ ) for a MAX-SSAT instance φ satisfying(1) no random variable appears negated in any clause of φ , and(2) each variable (both random and deterministic) appears in at most k clauses of φ . We give a polynomial-time reduction from α -approximating OP T on ( φ ) for a MAX-SSAT instance φ as in Lemma 2.3 to α ′ -approximating OP T on ( φ ′ ) on a MAX-SSAT instance φ ′ satisfying bothproperties (1) and (2) for some k ′ = O (1) and constant α ′ ∈ (0 , . The reduction. For odd (deterministic) i , if the variable x i appears in a ( i ) clauses in φ , we replacethe j th occurrence of x i with a new variable x i,j for ≤ j ≤ a ( i ) . Let φ ′ denote the new 3CNFformula after these replacements. We also add clauses to force the optimal online algorithm to setall of ( x i, , x i, , . . . x i,a ( i ) ) equal to each other, without increasing their number of occurrences bymore than a constant. Specifically, for each odd i , we construct via Lemma 2.4 an expander graph G i on a ( i ) vertices with maximum degree at most d = O (1) and expansion at least 1. Associate thevertices of G i with the literals ( x i, , x i, , . . . , x i,a ( i ) ) arbitrarily. For any edge in G i between x i,j and x i,j ′ , add the following two clauses to φ ′ : ( x i,j ∨ x i,j ′ ) ∧ ( x i,j ∨ x i,j ′ ) . (2)Note that if x i,j = x i,j ′ , we satisfy exactly one of these two clauses, while if x i,j = x i,j ′ we satisfy both.The order of variables x i,j and x i in φ is some arbitrary order such that variables in φ ′ correspondingto (copies of) variables x i and x j in φ appear in an order consistent with the variables x i and x j in φ . By adding dummy random variables, we further guarantee that copies of deterministic/randomvariables in φ are likewise deterministic/random in φ ′ .The following lemma relates the maximum expected number of satisfiable clauses in φ and φ ′ ,needed to complete our reduction’s analysis. Lemma 3.2. Let E n := P odd i ≤ n | E ( G i ) | . Then, the MAX-SSAT instances φ and φ ′ satisfy OP T on ( φ ′ ) = OP T on ( φ ) + E n . Proof. We first prove OP T on ( φ ′ ) ≥ OP T on ( φ ) + E n . Consider an online algorithm A which for odd i sets x i, = x i, = . . . = x i,a ( i ) = b i , where b i is the assignment for x i of OP T on on φ given theinduced history. This algorithm for φ ′ is clearly implementable. Moreover, this algorithm satisfieseach of the E n clauses of form (2), and satisfies OP T on ( φ ) of the original clauses in expectation.Hence OP T on ( φ ′ ) ≥ A ( φ ′ ) = OP T on ( φ ) + E n . We now prove that OP T on ( φ ′ ) ≤ OP T on ( φ ) + E n . Assume that for some odd i , and some fixedhistory for all variables before ( x i, , . . . , x i,a ( i ) ) , an SSAT algorithm A sets ( x i, , x i, , . . . , x i,a ( i ) ) suchthat they do not all take the same value (with some positive probability). Consider the minimumsize subset S ⊆ { , , . . . , a ( i ) } such that flipping all { x i,j } j ∈ S would result in all variables beingset to the same value (so, ≤ | S | ≤ a ( i ) / ). Since the expansion of G i is at least 1, we know that | E ( S, V \ S ) | ≥ | S | ; flipping all the { x i,j } j ∈ S would hence let us satisfy at least | S | additional clauses ofthe form (2), and possibly satisfy | S | fewer clauses corresponding to clauses in φ containing x i . Thus, A would satisfy at least as many clauses in expectation by flipping the sign of { x i,j } j ∈ S . Repeatedlyapplying this transformation results in an improved online algorithm A ′ as stated in the previousparagraph, from which we find that OP T on satisfies at most OP T on ( φ ) ≤ A ′ ( φ ′ ) ≤ OP T on ( φ ′ ) + E n clauses in expectation. The lemma follows.e now that E n is bounded from above by a constant times OP T on ( φ ) . Observation 3.3. E n ≤ d · OP T on ( φ ) . Proof. Since for each odd i , the expander graph G i contains at most d edges per each of the a ( i ) occurrences of i in φ , we have that E n = P odd i ≤ n | E ( G i ) | ≤ P odd i ≤ n d · a ( i ) . Next, for m thenumber of clauses in φ , since φ is a 3-CNF formula, P odd i ≤ n a ( i ) ≤ m . Finally, we note that, sincesetting each variable randomly satisfies at least half of the clauses in expectation, m/ ≤ OP T on ( φ ) .Combining these observations, we find that E n = X odd i ≤ n | E ( G i ) | ≤ X odd i ≤ n d · a ( i ) ≤ dm ≤ d · OP T on ( φ ) . Given the above, we are now ready to prove Lemma 3.1. Proof of Lemma 3.1. Let α ∈ (0 , and k be the constants in the statement of Lemma 2.3. Let φ be a MAX-SSAT instance as in the statement of that lemma and φ ′ be the obtained instance fromthe reduction of this section, which is polynomial-time, by Lemma 2.4. By construction, no randomvariable appears negated in any clause, and each variable appears in at most k ′ = max( d + 2 , k ) = O (1) clauses. By Lemma 3.2, OP T on ( φ ′ ) = OP T on ( φ ) + E n . Next, we let Q = OP T on ( φ ) , Q ′ = E n ,and β = 12 d , and note that Q ′ /Q ≤ β , by Observation 3.3. Thus, by Fact 2.1, for the constant α ′ := (cid:16) α +12 d d (cid:17) ∈ (0 , , an α ′ -approximation to OP T on ( φ ′ ) = OP T on ( φ ) + E n = Q + Q ′ yields an α -approximation of Q = OP T on ( φ ) , which is PSPACE-hard, by Lemma 2.3. RideHail We are now ready to prove our main theorem about the hardness of RideHail . Throughout thisproof, we will let k = O (1) be the constant in the statement of Lemma 3.1. Denote the variablesin an SSAT instance φ as in Lemma 3.1 by ( x , x , . . . , x n ) and the number of clauses of φ by m .Without loss of generality, suppose n is even. From φ , we construct a RideHail instance I φ , withweights w i,t = w t for each pair ( i, t ) ∈ E , where we refer to w t as the weight of ball t . The instancehas n bins, corresponding to the literals { x i , x i | i ∈ [ n ] } . The instance I φ has n + m balls; we willrefer to the first n balls as “literal balls" and the final m balls as “clause balls" (for reasons that willbecome clear shortly). For odd t ≤ n , ball t arrives with probability 1, has weight , and has anedge only to bins x t and x t . For even t ≤ n , ball t arrives with probability / , has weight , andhas an edge only to bin x t . The last m clause balls t = n + 1 , . . . , n + m each have weight m k andarrive with probability m − . The clause ball t = n + r corresponding to clause C r neighbors onlythe bins corresponding to literals in C r . (See Figure 1.)Figure 1: The RideHail instance I φ Bins are labeled by their corresponding literal, while balls are labeled by their weight. We shall see that OP T on ( I φ ) and OP T on ( φ ) are, up to a negligible error term, related by asimple linear relation. In particular, we will show that OP T on ( I φ ) = 0 . n + (1 − m − ) m − k · OP T on ( φ ) + o (1) . (3)We prove Equation (3) in the following two lemmas. The first proves that OP T on run on I φ matches all arriving literal balls. Lemma 3.4. Algorithm OP T on matches all arriving literal balls of I φ .roof. Suppose that there is some history h (occurring with probability q > ) after which OP T on does not match a literal ball t which arrives; let A ′ be the algorithm that follows exactly what OP T on does, with the exception that it will match t if t arrives after the history h . Then, A ′ ( I φ ) − OP T on ( I φ ) ≥ q · (cid:18) − k · m k · m − (cid:19) = q/ > . Indeed, if the history h occurs, A ′ gets a guaranteed profit of 1 from matching t that OP T on does notreceive. The expected profit OP T on gets from having the additional bin available to be potentiallymatched to clause balls is at most k · m k · m − , since each literal bin has at most k clause ballsadjacent to it, each of which has value m k and arrives with probability m − . As the above wouldimply A ( I φ ) > OP T on ( I φ ) , we conclude that OP T on must match each literal ball that arrives.A simple corollary of the above is that OP T on gets value of . n in expectation from the literalballs it matches. Moreover, this lemma gives a natural correspondence between OP T on on I φ andalgorithms for φ . The following lemma relies on Lemma 3.4 to bound the value OP T on obtains fromthe clause balls in terms of the expected number of clauses of φ satisfied by OP T on . Lemma 3.5. Let B be the gain of OP T on from clause balls of I φ . Then, for some δ ∈ [0 , m − ] , E [ B ] = (1 − m − ) m − k · OP T on ( φ ) + δ. Proof. By Lemma 3.4, OP T on matches each arriving literal ball. We consider the following naturalmapping between MAX-SSAT algorithms A on φ and families of algorithms F A which match eachliteral ball in I φ . For odd t ≤ n , an algorithm A ′ ∈ F A matches ball t to bin x t ( x t ) iff algorithm A sets x t to True ( False ). For even t ≤ n , if ball t arrives, an algorithm A ′ ∈ F A matches ball t to bin x t ; this corresponds to nature setting x t = False . Otherwise, bin x t is unmatched up to time m + 1 ,and we will think of this as nature setting x t = True . (Note that ball t arrives with probability 50%,so the variables are set to True / False with the correct probability.) Finally, algorithms A ′ ∈ F A match each arriving clause ball to some available neighboring bin when possible. A simple exchangeargument shows that OP T on ( I φ ) ∈ F A for some algorithm A .Let C be the number of clause balls of I φ that arrive. Then, with probability Pr[ C = 1] = m · m − · (1 − m − ) m − , exactly one such clause ball arrives, equally likely to correspond to anyof the m clauses in φ . On the other hand, a literal x t (respectively, x t ) is unmatched by A ′ ∈ F A immediately prior to time m + 1 iff A ( φ ) or nature set x t to True (respectively, False ). We concludethat Algorithm A ′ ∈ F A gains A ( φ ) m · m k expected value from conditioned on a single clause ballarriving. Thus, the expected gain E [ B ] of OP T on ( I φ ) from clause balls is at least E [ B ] ≥ E [ B | C = 1] · Pr[ C = 1] = (1 − m − ) m − k · OP T on ( φ ) . (4)Let A be the MAX-SSAT algorithm for which OP T on ( I φ ) ∈ F A . By the above argument yieldingEquation (4), the expected gain of OP T on ( I φ ) from clause balls conditioned on C = 1 is precisely Pr[ B | C = 1] = A ( φ ) m · m k ≤ OP T on ( φ ) m · m k . (5)Next, we note that the probability that multiple clause balls arrive is inverse polynomial in m . Pr[ C ≥ 2] = m X t =2 (cid:18) mt (cid:19) m − t (1 − m − ) m − t ≤ m X t =2 m t · m − t ≤ m − + m · m − ≤ m − . (6)On the other hand, conditioned on at multiple clause balls arriving, the expected profit of OP T on from clause balls is at most E [ B | C ≥ ≤ m · m k ≤ m . (7)ombining equations (5), (6) and (7), we find that the expected gain of OP T on ( I φ ) from matchingclause balls is at most E [ B ] = E [ B | C = 1] · Pr[ C = 1] + E [ B | C ≥ · Pr[ C ≥ ≤ OP T on ( φ ) m · m k · m · m − (1 − m − ) m − + m · m − = (1 − m − ) m − k · OP T on ( φ ) + 2 m − . We now conclude the reduction, and obtain the proof of our hardness result. Proof of Theorem 1.1. Let α ∈ (0 , be the constant from the statement of Lemma 3.1 and φ be aMAX-SSAT instance as in the statement of that lemma. Without loss of generality, we assume that φ has no pairs of consecutive variables x k − and x k which appear in no clauses. (Else, we removethese variable pairs and relabel the remaining variables while preserving parity of indices. Thisdoes not change the clauses, nor does it change the expected number of clauses satisfied by OP T on .)Next, let I φ be the obtained RideHail instance from the (clearly polynomial-time) reduction of thissection. From Lemma 3.4, the expected gain of OP T on ( I φ ) from literal balls is . n . Combiningthis with Lemma 3.5 we find that for γ := (1 − m − ) m − k and some δ ∈ [0 , m − ] , OP T on ( I φ ) = 0 . n + γ · OP T on ( φ ) + δ. Next, since φ is a 3-CNF formula with at least half its variables appear in at least one clause, thenumber of variables is at most n ≤ m . Moreover, since setting all variables randomly satisfies atleast half of the clauses in expectation, we have m/ ≤ OP T on ( φ ) . Combining these two observations,we get . n < n ≤ · OP T on ( φ ) , (8)Next, let Q = γ · OP T on ( φ ) + δ , Q ′ = 0 . n , and β = γ . Note that Q ′ /Q ≤ β by Equation (8),and that β = O (1) , since k = O (1) . Therefore, by Fact 2.1, for the constant α ′ := ( α · ( γ +2 m − ) /γ + β β ) ,which is in the range (0 , for sufficiently large m , an α ′ -approximation to OP T on ( I φ ) = OP T on ( φ )+0 . n = Q + Q ′ yields an α · ( γ +2 m − ) /γ -approximation of Q ∈ [ γ · OP T on ( φ ) , ( γ +2 m − ) · OP T on ( φ )] .By scaling appropriately, this yields an α -approximation to OP T on ( φ ) , which is PSPACE-hard toobtain, by Lemma 3.1. The theorem follows. In this section we give an algorithm to approximate the profit of OP T on , for any joint distributionsover edge weights of each ball t . Theorem 1.2. There exists a polynomial-time online algorithm which is a . -approximation ofthe optimal online algorithm for the RideHail problem. An LP Relaxation. Our starting point is a linear program (LP) called LP-Match, which we showupper bounds the gain of any online algorithm for RideHail . Below, the variables we optimize overare { x i,t } , which we think of as “the probability that the online algorithm matches ball t to bin i ”.Recall that ball t arrives with probability p t . P-Match: max X i,t w i,t · x i,t s.t. X t x i,t ≤ for all i (9) X i x i,t ≤ p t for all t (10) x i,t ≤ p t · − X t ′ For any RideHail instance I , we have that LP-Match ( I ) ≥ OP T on ( I ) . Proof. Let x ∗ i,t denote the probability that OP T on matches bin i to ball t . We note that x ∗ constitutesa feasible solution for LP-Match because (i) the probability OP T on matches a bin i is at most 1,(ii) the probability OP T on matches a ball t is at most p t (the probability that t arrives), (iii)the probability OP T on matches a bin i to a ball t is at most p t (the probability t arrives) times − P t ′ OP T on by Lemma 4.1, a natural ap-proach to approximate OP T on is to round this solution online. By simple “integrality gap” examples(see Appendix B), this is impossible to do perfectly. Instead, we show how to do so approximately,by rounding a solution to LP-Match while only incurring a / + c multiplicative loss in the rounding,for the constant c := 0 . .For notational simplicity, assume without loss of generality that an optimal solution to LP-Match to the input instance I satisfies all Constraints (10) at equality, i.e., P i x i,t = p t for all balls t . This can be guaranteed by adding a dummy bin i t for each ball t with w i,t = 0 , and setting x i t ,t ← p t − P i x i,t . These dummy edges do not affect the gain of OP T on , nor that of the onlinealgorithm.After computing a solution to LP-Match as above, our algorithm proceeds iteratively as follows.For each time t , if ball t arrives, we pick a single bin i with probability x i,t /p t , and if this is binis vacant (unmatched), we match ( i, t ) with some probability q i,t . (We sometimes refer to this as i accepts t .) If this did not result in t being matched, we repeat the process a second time, but thistime we match t to its picked bin i , provided i is vacant, and the edges until time t have nearlysaturated Constraint (9) for i . See Algorithm 1. Here, we use the fact that arrival of t is independent of the online algorithm’s previous choices. Note thatthis constraint is not valid for the probabilities induced by an offline algorithm, so our LP does not upper bound OP T off ( I ) . lgorithm 1 Rounding LP-Match Online solve LP-Match for { x i,t , y i,t,r } add dummy neighbor for each t so that P i y i,t,r = p t,r for all r M ← ∅ for all balls t = 1 , , . . . do let r be the index of the realization of { w i,t = w i,t,r | i } pick a single bin i with probability y i,t,r p t,r if i is unmatched in M then with probability q i,t := min (cid:16) , / + c − P t ′ Each edge ( i, t ) ∈ E is matched by Algorithm 1 with probability at least Pr[( i, t ) ∈ M ] ≥ x i,t · ( / + c ) . Theorem 4.2 implies that our algorithm is a polynomial-time . -approximation of the optimalonline algorithm, thus proving Theorem 1.2. Proof of Theorem 1.2. All steps of Algorithm 1, including solving the polynomially-sized LP inLine 1, can be implemented in polynomial time. The approximation ratio follows directly fromlinearity of expectation, together with Lemma 4.1 and Theorem 4.2.The remainder of this section is dedicated to proving Theorem 4.2. To this end, we consider twoevents for edge ( i, t ) being matched—depending on whether it was matched as a first pick or secondpick, in Line 9 or Line 13, respectively. We bound the probability of an edge being matched eitheras a first pick or as a second pick in the following sections. In this section we bound the probability of an edge being matched as a first pick. That is, theprobability that edge ( i, t ) is added to M in Line 9. We start with the following useful definition. Definition 4.3. Ball t is early for bin i if P t ′ Fix i . We prove by strong induction that these bounds hold for all edges ( i, t ′ ) with t ′ < t .The base case, for t = 1 , is vacuously true. Assume the claim holds for all t ′ < t ; we will prove itholds for t as well.The event ( i, t ) ∈ M requires that ball t arrives and bin i is picked in Line 6, that bin i is vacantat time t , and that bin i accepts the offer. Note that i being vacant at time t is independent fromthe arrival of t , and the first pick of t . Therefore, Pr[( i, t ) ∈ M ] = x i,t · Pr[ V i,t ] · q i,t . (13)For this reason, we turn our attention to bounding the probability of i being vacant at time t , Pr[ V i,t ] = 1 − X t ′ For any edge ( i, t ) , we have that Pr[ V i,t ] ≥ / − c . For any late ( i, t ) , we have that Pr[ V i,t ] ≤ / + c . For any early ( i, t ) , we have that Pr[ V i,t ] = 1 − P t ′ For any late edge ( i, t ) ∈ E , Pr[( i, t ) ∈ M ] ≥ x i,t · c. Before proving the above theorem, we provide some useful intuition and outline the challengesthe proof of Theorem 4.6 needs to overcome.By Lemma 4.4, the probability of a late edge ( i, t ) being matched as a first pick is at least Pr[( i, t ) ∈ M ] ≥ x i,t · ( / − c ) . (19)Moreover, by the same lemma, each edge ( i, t ) ∈ E (whether early or late) is matched as a first pickwith probability at most Pr[( i, t ) ∈ M ] ≤ x i,t · ( / + c ) . Denote by A t the event that t arrives anddenote by U ( t ) the event that t is unmatched after its first pick of i = j . Then, we have Pr[ U ( t ) | A t , i = j ] = 1 − Pr[ V j,t ] · q j,t . If ( j, t ) is late, then because Pr[ V j,t ] ≤ / + c by Corollary 4.5, the above quantity is at least / − c .If ( j, t ) is early, then because Pr[ V j,t ] = 1 − P t ′ Challenge 1: Re-drawing i . Unfortunately, conditioning on U ( t ) does not result in the prob-ability of ( i, t ) being matched in the second pick equalling that of it being matched in the firstpick. To see this, suppose a ball t was late for a single bin i , and x i,t /p t = 1 . In that case,conditioning on U ( t ) is equivalent to conditioning on i being occupied (matched) before time t .Consequently, for this late edge ( i, t ) , we have that Pr[( i, t ) ∈ M ] ≥ x i,t · ( / − c ) by Lemma 4.4,while Pr[( i, t ) ∈ M | U ( t )] = 0 , which implies that the second pick does not increase the probabilityof ( i, t ) to be matched at all , as Pr[( i, t ) ∈ M ] = 0 (!).This is where Constraint (11) of LP-Match comes in: This constraint implies that if t is late forbin i , then the probability that i was picked in Line 6 at time t conditioned on arrival of t is at most x i,t p t ≤ − X t ′ For any late edge ( i, t ) , for i the bin picked in Line 6 at time t , Pr[ i = i | A t ] ≥ − c. hallenge 2: Positive Correlation Between Bins. Lemma 4.7 alone does not resolve our prob-lems. Suppose that ball t is late for all bins for which x i,t = 0 , and all these bins have perfectlypositively correlated matched status, i.e., V i,t = V j,t for all bins i, j always. If this were the case,then we would have that Pr[ V i,t | U ( t )] = 0 , since if t is not matched to its first i , then i and i must both have been matched before. This again would result in Pr[( i, t ) ∈ M ] = 0 .To overcome the above, we show that the above scenario does not occur. In particular, weshow that while positive correlations between different bins’ matched statuses are possible, suchcorrelations cannot be too large. More formally, we show the following. Lemma 4.8. For any time t and bins i = j , we have that Cov( V i,t , V j,t ) ≤ c. The crux of our analysis is proving Lemma 4.8. Using it, we will be able to argue that for anylate edge ( i, t ) , the probability that i is free at time t , conditioned on U ( t ) and on the first picksatisfying i = i (a likely event, by Lemma 4.7), is not changed much compared to the unconditionalprobability of i being free at time t . In particular, this implies that the probability of ( i, t ) beingmatched as a second pick, conditioned on U ( t ) , is not too much smaller compared to its probabilityof being matched as a first pick. In particular, we will show that Pr[( i, t ) ∈ M ] ≥ x i,t · c , forsufficiently small c > , as stated in Theorem 4.6.We prove that lemmas 4.7 and 4.8 indeed imply Theorem 4.6, as outlined above, in Section 4.3.3.But first, we turn to proving our key technical lemma, namely Lemma 4.8. To bound the correlation of vacancy indicators, it is convenient to define the indicator randomvariable O i,t := 1 − V i,t , which indicate whether i is occupied (i.e., matched) at time t . We additionallydecompose the variables O i,t into two variables, based on whether i was matched (became occupied)along an early or late edge. In particular, we let O Ei,t ≤ O i,t be an indicator for the event that i ismatched along an early edge before t arrives. Similarly, we let O Li,t := O i,t − O Ei,t be an indicator forthe event that i is matched along a late edge before t arrives. To bound the pairwise correlations ofvariables O i,t , we will show that O Ei,t contributes most of the probability mass of O i,t , and that thevariables O Ei,t and O Ej,t are negatively correlated. To prove this negative correlation, we will provethe following, stronger statement. Lemma 4.9. For any time t , the variables { O Ei,t } i are negatively associated (NA).Proof. For every edge ( i, t ) , let X i,t be the indicator random variable for the event that ball t arrivesand picks bin i as its first pick. Let Y i,t ∼ Ber ( q i,t ) be an indicator for the event that bin i accepts,i.e., it will be matched to ball t if it arrives and picks i as its first pick and i is free.For fixed t , the variables { X i,t } are 0/1 random variables whose sum is at most 1 always, sothey are NA by the 0-1 Principle (Proposition 2.6). On the other hand, the variables { Y i,t } i areindependent, and hence NA. Moreover, { X i,t } i , { Y i,t } i are mutually independent distributions, andso by closure of NA under independent union (Proposition 2.7), we also have that { X i,t , Y i,t } i isNA. Likewise, the lists { X i,t , Y i,t } i are mutually independent as we vary t ; again using closure of NAunder independent union we find that { X i,t , Y i,t } i,t are also NA.Fix t . For each bin i , let t i denote the largest t ′ < t so that ( i, t ′ ) is early. We note that bin i cannot be matched as a second pick to any t ′ ≤ t i . So, it is matched along an early edge before t arrives if and only if there are some t ′ ≤ t i and r such that ball t ′ arrives and picks bin i , and bin i accepts the proposal (for the smallest such t ′ , bin i is guaranteed to be free). Therefore, we havethat O Ei,t = _ t ′ ≤ t i ( X i,t ′ ∧ Y i,t ′ ) . ote that we have written { O Ei,t } i as the output of monotone non-decreasing functions defined ondisjoint subsets of the variables in { X i,t , Y i,t } i,t . Hence, by closure of NA under monotone functioncomposition (Proposition 2.7), we have that { O Ei,t } i are NA.By Proposition 2.8, the above lemma implies that any O Ei,t and O Ej,t are negatively correlated. Corollary 4.10. For any time t and bins i = j , we have that Cov( O Ei,t , O Ej,t ) ≤ . We are now ready to prove Lemma 4.8. Proof of Lemma 4.8. First, we show that the probability of a bin i being matched along a late edgebefore time t is small, which we later use to bound the covariance of O Li,t and other binary variables.Indeed, as Pr[ V i,t ] ≥ / − c (Corollary 4.5), we have that Pr[ O i,t ] ≤ / + 3 c . Additionally, Pr[ O Ei,t ] ≥ / − c / + c · ( / + c ) = / − c by Lemma 4.4. Therefore, Pr[ O Li,t ] = Pr[ O i,t ] − Pr[ O Ei,t ] ≤ ( / + 3 c ) − ( / − c ) = 4 c. (21)Thus, using the additive law of covariance for Cov( O i,t , O j,t ) = Cov(1 − O i,t , − O j,t ) = Cov( V i,t , V j,t ) ,we obtain the desired bound, Cov( V i,t , V j,t ) = Cov( O Ei,t + O Li,t , O Ej,t + O Lj,t )= Cov( O Ei,t , O Ej,t ) + Cov( O Ei,t , O Lj,t ) + Cov( O Li,t , O Ej,t ) + Cov( O Li,t , O Lj,t ) ≤ O Ei,t , O Lj,t ] + Pr[ O Li,t , O Ej,t ] + Pr[ O Li,t , O Lj,t ] ( Cor. 4.10 ) ≤ O Lj,t ] + Pr[ O Li,t ] + Pr[ O Li,t ] ≤ c. ( Eq. (21 )) We are now ready to use weak positive correlation (if any) between vacancy indicators V i,t and V j,t .In particular, we will show that the probability of bin i to be occupied a time t is not changed muchwhen conditioning on A t (arrival of t ), the first picked bin at time t being i = i , and U ( t ) (ball t bot being matched to its first pick). Lemma 4.11. For any late edge ( i, t ) , we have that Pr[ O i,t | A t , i = i, U ( t )] ≤ Pr[ O i,t ] · (cid:18) c ( / − c ) (cid:19) . Proof. To analyze the conditional probability above, we first look at Pr[ O i,t , A t , i = j, U ( t )] . Thisis the probability of bin i being occupied at time t , ball t arriving and picking j as its first pick,and not being matched due to this first pick. Note that A t and the first pick is independent ofbins’ occupancy statuses at time t . Additionally, we notice that with probability − q j,t bin j willdeterministically reject. With probability q j,t , it rejects if and only if j is occupied. So, for any j = i , Pr[ O i,t , A t , i = j, U ( t )] = Pr[ O i,t ] · Pr[ A t , i = j ] · ((1 − q j,t ) + q j,t · Pr[ O j,t | O i,t ]) . (22)We now turn to relating the last term in the above product, namely (1 − q j,t ) + q j,t · Pr[ O j,t | O i,t ] ,to its "unconditional" counterpart, Pr[ U ( t ) | A t , i = j ] = (1 − q j,t ) + q j,t · Pr[ O j,t ] . For notationalconvenience, we which we abbreviate by z i,j,t := (1 − q j,t ) + q j,t · Pr[ O j,t | O i,t ] . ecalling that Cov( O i,t , O j,t ) = Cov( V i,t , V j,t ) ≤ c , by Lemma 4.8, we have Pr[ O j,t | O i,t ] = Pr[ O j,t , O i,t ]Pr[ O i,t ] = Pr[ O j,t ] · Pr[ O i,t ] + Cov ( O j,t , O i,t )Pr[ O i,t ] ≤ Pr[ O j,t ] + 12 c Pr[ O i,t ] . (23)Hence, z i,j,t ≤ (1 − q j,t ) + q j,t · (cid:18) Pr[ O j,t ] + 12 c Pr[ O i,t ] (cid:19) (Eq. (23)) ≤ (1 − q j,t ) + q j,t · (cid:18) Pr[ O j,t ] + 12 c / − c (cid:19) (Cor. 4.5, c < / )= Pr[ U ( t ) | A t , i = j ] + q j,t · c / − c ≤ Pr[ U ( t ) | A t , i = j ] · (cid:18) c ( / − c ) (cid:19) (Eq. (20), q j,t ≤ (24)Using this bound in Equation (22) and summing over all j = i , we have Pr[ O i,t , A t , i = i, U ( t )] ≤ Pr[ O i,t ] · Pr[ A t , i = i, U ( t )] · (cid:18) c ( / − c ) (cid:19) . The desired inequality therefore follows by Bayes’ theorem.With this lemma in place, we are ready to conclude this section by proving Theorem 4.6, i.e.that Pr[( i, t ) ∈ M ] ≥ x i,t · c for any late edge ( i, t ) . Proof of Theorem 4.6. We start by bounding Pr[( i, t ) ∈ M ] ≥ Pr[( i, t ) ∈ M | A t , i = i, U ( t )] · Pr[ A t , i = i, U ( t )] . (25)In words, the probability ( i, t ) is matched as a second pick is at least the probability of the sameevent and i = i . By Lemma 4.7 we know that Pr[ A t , i = i ] ≥ p t · (1 − c ) ; by Equation (20), weknow that Pr[ U ( t ) | A t , i = j ] ≥ / − c for any j = i . As a consequence, by Bayes’ theorem andour choice of c < / , we have that Pr[ A t , i = i, U ( t )] = Pr[ A t , i = i ] · Pr[ U ( t ) | A t , i = i ] ≥ p t · (1 − c ) · ( / − c ) . (26)Next, we note that Pr[( i, t ) ∈ M | A t , i = i, U ( t )] = x i,t p t · Pr[ V i,t | A t , i = i, U ( t )] (27)because conditioned on A t , picking someone other than i first, and being rejected, we will match ( i, t ) exactly when t ’s second pick is i and i is vacant.Lemma 4.11 yields the following lower bound on the probability of [ V i,t | A t , i = i, U ( t )] : Pr[ V i,t | A t , i = i, U ( t )] = 1 − Pr[ O i,t | A t , i = i, U ( t )] ≥ − Pr[ O i,t ] · (cid:18) c ( / − c ) (cid:19) = Pr[ V i,t ] − c ( / − c ) · (1 − Pr[ V i,t ]) ≥ / − c − c ( / − c ) · ( / + 3 c ) ( Cor. 4.5 ) (28)Combining equations 27 and 28 we thus have Pr[( i, t ) ∈ M | A t , i = i, U ( t )] ≥ x i,t p t · (cid:18) / − c − c ( / − c ) · ( / + 3 c ) (cid:19) . (29)utting it all together, equations (25), (26), and (29) and our choice of (sufficiently small) c = 0 . imply the desired inequality, Pr[( i, t ) ∈ M ] ≥ x i,t p t · (cid:18) / − c − c ( / − c ) · ( / + 3 c ) (cid:19) · p t · (1 − c ) · ( / − c ) ≥ x i,t · c. Our algorithm and its analysis of Section 4 generalize seamlessly to a setting in which weights ofeach online node t are drawn from discrete joint distributions. For brevity, we only outline the smallchanges in the LP, algorithm and analysis here. Problem Statement. We are given a complete bipartite graph, with vertices of one side (bins) giveup front, and vertices of the other side (balls) arriving sequentially, with ball t arriving at time t (withprobability one). The vector of edge weights of any ball t , denoted by w t := ( w ,t , w ,t , . . . ) , is drawnfrom some discrete joint distribution, w t ∼ D t . The vector of all edge weights, w := ( w , w , . . . ) ,is drawn from the product distribution, w ∼ D := Q t D t . That is, the weights of any ball’s edgesmay be arbitrarily correlated, but weights of different balls’ edges are independent. We assumethat these discrete distributions are given explicitly, e.g., via a list of tuples of the form ( v t,j , p t,j ) with p t,j := Pr D t [ w t = v t,j ] . We note that the problem considered in previous sections is a specialinstance of this problem with each D t consisting of two-point distributions, with one of the possiblerealizations of w t ∼ D t being the all-zeros vector. Generalizing LP-Match. The generalization of LP-Match now has decision variables y i,t,j , whichwe think of as proxies for the probability of edge ( i, t ) being matched by the optimal online algorithmwhen ball t ’s edge weights are w t = v t,j . Generalizing the argument behind Constraint (11), we notethat w t is independent of bin i not being matched by the optimal online algorithm by time t . Fromthis we obtain Constraint (30) below. The remaining constraints of the obtained LP (below) arematching constraints. LP-Match: max X i,t,j w i,t,j · y i,t,j s.t. X t X j y i,t,j ≤ for all i X i y i,t,j ≤ p t,j for all t, jy i,t,j ≤ p t,j · − X t ′ Our general algorithm will match each edge ( i, j ) when w t = v t,j with marginal probability at least probability Pr[( i, t ) ∈ M , w t = v t,j ] ≥ y i,t,j · ( / + c ) . To do so, when ball t arrives, we first observe the realization of the edge weight vector w t = v t,j .Then, When picking a bin i (either as first or second pick) at time t , we now do so with probability i,t,r p t,j . Moreover, we take q i,t := min (cid:16) , / + c − P t ′ Extending the analysis of Algorithm 1 to this more general problemis a rather simple syntactic generalization. We therefore only outline the changes in the analysis.Broadly, all changes needed for the analysis require us to refine our claims as follows. Denoteby R t a random variable denoting the random index of the weight vector of edges of t . Thatis, R t = j ⇐⇒ w t = v t,j . Then, all our bounds for the probability of ( i, t ) being matched(as a first or second pick, or either) now need to refer to R t = j , and relate to y i,t,j . So, forexample, Lemma 4.4 will be restated to show that for each early edge ( i, t ) and index j , we havethat Pr[( i, t ) ∈ M , R t = j ] = y i,t,r · ( / + c ) , and for any edge ( i, t ) , we have that y i,t,r · ( / − c ) ≤ Pr[( i, t ) ∈ M , R t = j ] ≤ y i,t,r · ( / + c ) . Lemma 4.9 requires some care in setting up the NA variablesto prove that O Ei,t are NA, by also accounting for the realization of R t , with indicators [ R t = j ] ,which are NA by the 0-1 Principle (Proposition 2.6). Apart from that, the proofs are essentiallyunchanged, except for replacing occurrences of A t by R t = j in every probability conditioned onarrival of t , and appropriately replacing x i,t p t by y i,t,j p t,j . We studied the online stochastic max-weight bipartite matching problem through the lens of approx-imation algorithms, rather than that of competitive analysis. In particular, we study the efficientapproximability of the optimal online algorithm on any given input. On the one hand, we showthat the optimal online algorithm cannot be approximated beyond some constant (barring shock-ing developments in complexity theory). On the other hand, we present a polynomial-time onlinealgorithm which yields a . approximation of the optimal online algorithm’s gain—surpassing theapproximability threshold of / of the optimal offline algorithm. Many intriguing research questionsremain.First, it is natural to further study the efficient approximability of our problem. We suspect thatmuch better approximation guarantees are achievable. One might also ask if our general algorithmicapproach can be extended to implicitly represented weight distribution D . For example, what canone show if D t is itself a product distribution, D t = Q i D i,t , with w i,t ∼ D i,t ? A related interestingquestion is to obtain better approximation for the widely-studied special case of balls drawn fromsome i.i.d distribution (see, e.g., [24, 29, 31, 37, 38]).More broadly, one might ask how well one can approximate the optimal online algorithm of onlineBayesian selection problems under the numerous constraints studied in the literature, includingmatroid and matroid intersections, knapsack constraints, etc. For which of these problems is theonline optimum easy to compute? Which admit a PTAS? Which admit constant approximations?Which are hard to approximate? We are hopeful that the ideas developed here, both algorithmic, aswell as our new hardness gadgets, will prove useful when exploring this promising research direction. A Omitted Proofs of Section 2 In this section we provide proofs deferred from Section 2, restated below for ease of reference. Fact 2.1. Let Q, Q ′ ≥ be positive quantities, such that Q ′ /Q ≤ β , and let α ∈ (0 , . Then, an (cid:0) α + β β (cid:1) -approximation to Q + Q ′ yields an α -approximation to Q .Proof. As f ( x ) = α + x x = 1 − − α x is monotone increasing in x ≥ − for α ∈ (0 , , we have that α + β β ) ≥ α + Q ′ /Q Q ′ /Q = α · Q + Q ′ Q + Q ′ . Thus, An ( α + β β ) -approximation to Q + Q ′ yields a number in the range T ∈ h α + β β · (cid:0) Q + Q ′ (cid:1) , Q + Q ′ i ⊆ [ α · Q + Q ′ , Q + Q ′ ] . Subtracting Q ′ from T then yields a number T − Q ′ in the range T − Q ′ ∈ [ α · Q, Q ] . Next, we provide a proof of the underlying PSPACE-hardness result of Condon et al. [11] usedin our reductions. Lemma 2.3. ([11, Theorem 3.3]) There exist constants k ∈ N and α ∈ (0 , so that it is PSPACE-hard to compute an α -approximation to OP T on ( φ ) for a MAX-SSAT instance φ satisfying:1. no random variable appears negated in any clause of φ , and2. each random variables appears in at most k clauses of φ .Proof. This lemma follows from the proof in [11]; here, we briefly explain why.In that paper, the authors prove their main result that RPCD (log n, 1) = PSPACE in Theorem2.4. Using this theorem, they prove that it is PSPACE-hard to approximate MAX-SSAT in Theorem3.1. In their proof, they start with a language L in PSPACE and an input x , and construct anRPCDS for L flipping O (log n ) coins and reading O (1) bits of the debate. From this, they constructa MAX-SSAT instance φ such that if x ∈ L , all clauses of φ can be satisfied with probability 1,while if x / ∈ L there is no way to satisfy more than an α < fraction of the clauses of φ . Theirconstruction of φ builds a constant-size 3CNF for each possible realization of the O (log n ) coin flips,and takes the conjunction of these 3CNFs. Each constant-size 3CNF has variables corresponding tothe bits of the debate that V queries for a specific realization of the coin-flips. Hence, to show that φ only has each random variable appear in O (1) clauses, it suffices to show that each random-bit inthe RPCDS constructed is queried for only O (1) realizations of the coin flips.To show this, we turn to the construction of the RPCDS used to prove Theorem 2.4. Via Lemma2.1, the authors first show that it is sufficient to consider RPCDSs where the verifier can read aconstant number of rounds of Player 1 (and not just a constant number of bits).In Lemma 2.3, the authors describe their protocol for a verifier V which can read O (1) rounds ofPlayer 1. Note that the random coins in this protocol are used to select a “random odd-numberedround k > " and a “random bit of round k − of Player 0." In fact, this is the only time that theverifier reads a random bit of Player 0. So, in this construction, each random bit is only queried in O (1) realizations of the coin flips. With Lemma 2.1, the authors transform this RPCDS to one thatonly reads a constant number of bits. We note that this transformation only impacts the stringsthat player 1 writes, and does not affect the coin flips or the bits of player read.From this, it holds that the MAX-SSAT instance φ constructed in Theorem 3.1 has each randomvariable appear in O (1) clauses. That instance does not yet satisfy the property that randomvariables only appear non-negated. Condon et al. give a fix for this in the proof of Theorem 3.3; webriefly note that after the modification provided in this proof, it will still hold that random variablesappear in O (1) clauses. B LP-Match: Additional Observations Here we make a few additional observations concerning the usefulness of Constraint (11) and LP-Match in general, as well as some natural limits to this LP.First, we note that LP-Match captures the optimal online algorithm precisely for the classicsingle-item prophet inequality problem. That is, for RideHail instances with a single bin i , solutionsto this LP can be rounded online losslessly. bservation B.1. LP-Match ( I ) = OP T on ( I ) for any RideHail instance I with a single bin i .Proof. Consider the following online algorithm, which starts by computing a solution ~y to LP-Match.Next, upon arrival of ball t with with w i,t = w i,t,r (i.e., R t = r ), match ( i, t ) with probability y i,t,r p t,r · (cid:0) − P t ′ OP T on ( I ) ≥ LP-Match ( I ) . The opposite inequality follows from Lemma 4.1.On the other hand, for general RideHail instances, there is a limit to the approximation guar-antees obtainable using LP-Match. In particular, simple examples show that there is a gap betweenthe upper bound given by LP-Match and the expected profit of OP T on , appropriately restrictingthe approximation guarantees provable using this LP. This is to be expected, given our work inSection 3. We present a simple example of such a gap instance below. Observation B.2. There exists a RideHail instance I with w i,t ∈ { , } for all ( i, t ) ∈ E forwhich LP-Match ( I ) ≥ / · OP T on ( I ) .Proof. We consider an instance I with three balls and two bins. For k = 1 , , ball t = k has withprobability p k, = 1 / edge weights w i,t = 0 for all i . With the remaining probability p k, = 1 / , itsedges have weights w k,k = 1 and w k, − k = 0 . The last ball has weights w ,k = 1 for all bins k = 1 , with probability one. An optimal solution to LP-Match on this Instance I assigns y k,k, = 1 / for k = 1 , , and y ,k, = 1 / for k = 1 , , achieving an objective value of P i,t,r y i,t,r = 2 . However, withprobability / , both of the first two balls have all their edge weights zero, and so an online algorithmcan at most achieve an expected value of / . That is, OP T on ( I ) ≤ / · LP-Match ( I ) . References [1] Melika Abolhassani, Soheil Ehsani, Hossein Esfandiari, MohammadTaghi Hajiaghayi, RobertKleinberg, and Brendan Lucier. Beating 1-1/e for ordered prophets. 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