Prophet Matching Meets Probing with Commitment
aa r X i v : . [ c s . D M ] F e b Prophet Inequality Matching Meets Probing with Commitment
Allan Borodin ∗ Calum MacRury † Akash Rakheja ‡ Abstract
Within the context of stochastic probing with commitment, we consider the online stochasticmatching problem for bipartite graphs where edges adjacent to an online node must be probedto determine if they exist, based on known edge probabilities. If a probed edge exists, it must beused in the matching (if possible). In addition to improving upon existing stochastic bipartitematching results, our results can also be seen as extensions to multi-item prophet inequalities.We study this matching problem for given constraints on the allowable sequences of probesadjacent to an online node. Our setting generalizes the patience (or time-out) constraint whichlimits the number of probes that can be made to edges. The generality of our setting leads tosome modelling and computational efficiency issues that are not encountered in previous works.We establish new competitive bounds all of which generalize the standard non-stochastic settingwhen edges do not need to be probed (i.e., exist with certainty). Specifically, we establish thefollowing competitive ratio results for a general formulation of edge constraints, arbitrary edgeweights, and arbitrary edge probabilities:1. A tight ratio when the stochastic graph is generated from a known stochastic type graphwhere the π ( i ) th online node is drawn independently from a known distribution D π ( i ) and π is chosen adversarially. We refer to this setting as the known i.d. stochastic matchingproblem with adversarial arrivals.2. A 1 − /e ratio when the stochastic graph is generated from a known stochastic type graphwhere the π ( i ) th online node is drawn independently from a known distribution D π ( i ) and π is a random permutation. This is referred to as the known i.d. stochastic matchingproblem with random order arrivals.We note that the known i.d. model generalizes the online stochastic matching model wherethe stochastic graph (but not the edge probabilities nor the order of online arrivals) is knownto the algorithm. Our i.d. model also generalizes the prophet inequality and prophet secretarymodels to the probing setting.In deriving our results, we clarify and expand upon previous offline benchmarks, relative towhich one defines an appropriate definition of the competitive ratio. In particular, we introducea new LP relaxation which upper bounds the performance of “an optimum offline probing al-gorithm”. This new LP allows us to overcome some previous negative results (i.e. stochasticitygaps). While this LP has exponentially many variables, it has polynomially many constraintsand we show how it can be solved efficiently (i.e., in polynomial time) under some mild assump-tions on the edge probing constraints. ∗ Department of Computer Science, University of Toronto, Toronto, ON, Canada [email protected] † Department of Computer Science, University of Toronto, Toronto, ON, Canada [email protected] ‡ Department of Computer Science, University of Toronto, Toronto, ON, Canada [email protected] Introduction
Stochastic probing problems are part of the larger area of decision making under uncertainty andmore specifically, stochastic optimization. Unlike more standard forms of stochastic optimization,it is not just that there is some stochastic uncertainty in the set of inputs, stochastic probingproblems involve inputs that cannot be determined without probing (at some cost and/or withinsome constraint). Applications of stochastic probing occur naturally in many settings, such as inmatching problems where compatibility cannot be determined without some trial or investigation(for example, in online dating and kidney exchange applications). There is by now an extensiveliterature for stochastic matching problems. For space efficiency, we will give an extended overviewof related work in Appendix C. Research most directly relating to this paper will appear as weproceed.The stochastic matching problem was introduced by Chen et al. [20]. In this problem, we aregiven an adversarially generated stochastic graph G = ( V, E ) with a probability p e associated witheach edge e and a patience (or timeout) parameter ℓ v associated with each vertex v . An algorithmprobes edges in E within the constraint that at most ℓ v edges are probed incident to any particularvertex v ∈ V . The patience constraint can be viewed as a simple budgetary constraint, whereeach probe has unit cost and the patience constraint is the budget. When an edge e is probed,it is guaranteed to exist with probability exactly p e . If an edge ( u, v ) is found to exist, then thealgorithm must commit to the edge – that is, it must be added to the current matching (if possible).The goal is to maximize the expected size of a matching constructed in this way. This problem canbe generalized to offline vertices or edges having weights and then the objective is to maximize theexpected weight of the matching. Notably, in Chen et al., the algorithm knows the entire stochasticgraph in advance.In addition to generalizing the setting of the results of Chen et al., Bansal et al. [9] introducedan i.i.d. (one-sided) bipartite version of the problem where nodes on one side of the partition arriveonline and edges adjacent to that node are then probed. In their model, each online vertex (andits adjacent edges) is drawn independently and identically from a known distribution. That is, thepossible “type” of each online node (i.e., the adjacent edge probabilities and edge weights) is knownand the input sequence is then determined i.i.d. from this known distribution, where the type of anode is presented to the algorithm upon arrival. In the Bansal et al. model, each offline node hasunlimited patience, whereas each online node specifies its patience upon arrival. The match for anonline node must be made before the next online arrival. As in the Chen et al. model, if an edge isprobed and confirmed to exist, then it must be included in the current matching (if possible). Thisproblem is referred to as the online stochastic matching problem (with patience) and also referredto as the stochastic rewards problem .In more general settings, we will study the (one-sided) online bipartite stochastic matchingproblem. More specifically, we generalize the patience constraint to apply to any downward-closedset of constraints including a budget constraint . We first consider the original stochastic matchingsetting where the algorithm knows the stochastic graph, and the online vertices arrive either in Unfortunately, the term “stochastic matching” is also used to refer to more standard optimization where theinput (i.e., edges or vertices) are drawn from some known or unknown distributions but no probing is involved. The online stochastic matching problem is sometimes meant to imply unit patience but we will be interested inresults which hold for more general probing constraints. A budget constraint involves placing a budget B v ≥ v ∈ V and costs ( c e ) e ∈ ∂ ( v ) on its adjacent edges. Thecost of the probed edges then cannot exceed B v . We now informally point ahead to our main results. Our results will be stated more formally inSection 2 following the relevant definitions. Proofs of the main results will be given in Sections 3and 4. All of our results apply to general constraints on allowable probing sequences (generalizingpatience constraints) and the competitive ratios are with respect to an optimum offline probingalgorithm (the adaptive benchmark) which we define precisely in Section 2. Our results for thecase of a known stochastic graph are subsumed by the analogous results for the known i.d. modelbut we state them here so as to relate these results to previous work.1. Theorem 2.1 shows that Algorithm 4 is a non-adaptive algorithm with competitive ratio1 − /e in the following setting: • There is a known edge-weighted stochastic graph. • Online vertices are presented in random order.Algorithm 4 is non-adaptive in the sense that its probes are a randomized function of thestochastic graph it executes on. In particular, the probes of an arriving online node donot depend on the probed edge states of the previous online nodes. Non-adaptive probingalgorithms were first studied in Dean et al. [23] for the stochastic knapsack problem, as they3re conceptually simpler than adaptive probing algorithms, and require far less space to bespecified.The work of Gamlath et al. [32] showed that a 1 − /e competitive ratio is possible when thereare no constraints on the online vertices. Our result can therefore be viewed as a extensionof their result to general probing constraints. Our result also shows that the Brubach et al.[15] 0 .
544 inapproximation bound against the Bansal et al. [9] LP does not hold with respectto our new LP relaxation.In Theorem 2.2, we provide a negative result which shows that Theorem 2.1 is an optimumresult for non-adaptive online probing algorithms . Roughly speaking, in our setting, wedefine the adaptivity gap to be the worst case ratio of performance between the optimum non-adaptive online probing algorithm, and the adaptive benchmark. We conclude that theonline stochastic matching has an adaptivity gap of precisely 1 − /e . Adaptivity gaps wereinitially studied by Dean et al., and have since received lots of attention (see [35, 7, 36, 37, 14]for some of the latest results).2. Theorem 2.6 shows that Algorithm 6 is a non-adaptive online algorithm with competitiveratio 1 − e in the following stochastic i.d. setting (improving upon the previously best i.i.d.ratio of 0 .
46 in [15]). • There is a known edge weighted stochastic (type) graph. • Online vertices are drawn independently from distributions on the online vertices therebyproducing an edge-weighted stochastic graph. • The instantiation of the stochastic graph from the type graph is not known to thealgorithm. Online vertices are in random order.In the classical i.i.d. setting with non-integral arrival rates, Manshadi et al. [49] present anexample that shows that 1 − /e is optimal for classically non-adaptive online algorithms.Manshadi et al. [49] use the terminology non-adaptive to mean that a (classical) onlinealgorithm in the known i.i.d. setting uses only the type of the arriving node to determineits matching decisions. Our definition of non-adaptivity for an online probing algorithm inthe known i.d. setting generalizes this classical definition (and well as our definition from theknown stochastic graph setting) as we discuss in Subsection 2.1. Since our probing algorithmfits this classical definition, it has an optimal competitive ratio amongst this restricted classof probing algorithms.3. Theorem 2.7 shows that Algorithm 8 is an online algorithm with competitive ratio in thefollowing stochastic i.d. setting: • There is a known edge weighted stochastic (type) graph. • Online vertices are drawn independently from distributions on the online vertices therebyproducing an edge-weighted stochastic graph. • The instantiation of the stochastic graph from the type graph is not known to thealgorithm or the adversary who chooses the order of the online vertices. Our negative result on the adaptivity gap in fact applies to all offline non-adaptive probing algorithms.
In Section 2 we provide the required definitions and precise statements of our main results. Inparticular, we define the (offline) adaptive benchmark relative to which we state our competitivebounds. We also define the oracles we use to ensure that our algorithms can efficiently abide bythe probing constraints.Section 3 defines a new LP which we use to guide our algorithm for the known stochastic setting.We provide a non-adaptive algorithm that establishes the 1 − /e competitive ratio in the randomorder model assuming that the new LP is a relaxation of the adaptive benchmark. Our algorithmmakes use of the random order contention scheme (RCRS) for rank 1 matroids utilized by Lee andSingla [46].Section 4 extends our LP and results to the known i.d. setting. We note that the knowni.d. setting subsumes the known stochastic graph setting when each distribution D i is a singletondescribing the i th online vertex v i . In this setting we establish a 1 − /e competitive ratio forthe random order model and a ratio for adversarial input sequences. We again use the Lee andSingla [46] RCRS for the ROM setting, and we use the recently studied online contention resolutionscheme (OCRS) of Ezra et al. [29] for the adversarial setting.Section 5 completes the proof that our new LP is a relaxation of the adaptive benchmark(upper bounds the benchmark’s value). Note that unlike previously used LP relaxations for probingproblems, this fact does not seem to have an easy proof, and thus is a technical contribution.Finally, Section 6 shows that 1 − /e is a tight competitive ratio for non-adaptive algorithms inthe random order model, even when the stochastic graph is known (as is the case in Theorem 2.1).We do this by establishing an adaptivity gap between the optimal non-adaptive probing algorithmand the adaptive benchmark we are using.We conclude with a summary of our contributions and some open problems. The online stochastic matching problem generalizes the classical online bipartite setting asfollows. An input to the problem consists of a bipartite stochastic graph , which is a (simple)bipartite graph G = ( U, V, E ) with edge weights ( w e ) e ∈ E and edge probabilities ( p e ) e ∈ E . We shallrefer to U as the offline vertices of G and V as its online nodes. For each e ∈ E of G = ( U, V, E ),the fraction 0 ≤ p e ≤ e . More precisely, each edge e ∈ E is associated with an independent Bernoulli random variable of parameter p e , which wedenote by st( e ), corresponding to the state of the edge. If st( e ) = 1, then we say that e is active ,and otherwise we say that e is inactive .A solution to the online stochastic matching problem is an online probing algorithm . In eachround, an online node v ∈ V arrives, and the online probing algorithm sees all the adjacent edgesof v , denoted ∂ ( v ), as well as their associated probabilities, ( p e ) e ∈ ∂ ( v ) , and weights, ( w e ) e ∈ ∂ ( v ) .However, the edge states (st( e )) e ∈ ∂ ( v ) initially remain hidden to the algorithm. Instead, given e ∈ ∂ ( v ), the algorithm must perform a probing operation on the edge to reveal/expose its5tate, st( e ). As in the classical problem, an online algorithm must decide on a possible matchfor an online node v before seeing the next online node. The algorithm can be non-greedy andnot match a given v ∈ V even though some u ∈ U is still unmatched. The arrival order of V iseither chosen adversarially , in which case we work in the adversarial order model (AOM), or u.a.r. (uniformly at random), in which case we work in the random order model (ROM). Theonline stochastic matching problem simplifies to the classical setting in the appropriate order modelprovided p e = 1 for all e ∈ E .Our probing algorithms function in the probe-commit model , in which there is a commit-ment requirement upon probing an edge. Specifically, if an edge e = ( u, v ) is probed in the onlinestochastic matching problem and turns out to be active, then the probing algorithm must makean irrevocable decision as to whether or not to include e in its matching, prior to probing anysubsequent edges. If M is the matching output by the online probing algorithm, then its goal is tomaximize E [ w ( M )], where w ( M ) := P e ∈M w e . Note that this definition of commitment is the oneconsidered by Gupta et al. [36] , and while seemingly less strict than the commitment requirementof the Chen et al. [20] matching model, these definitions have equivalent power, as an algorithmmay simply pass on probing an edge if it doesn’t intend to add the edge to its matching. Wedescribe our positive results in the Gupta et al. [36] model, as it simplifies both the presentation ofour probing algorithms as well as the definition of non-adaptivity. All our results can be restatedwithout loss while satisfying the Chen et al. commitment requirement, albeit with a very smallchange to the definition of non-adaptive.In past works, each online node v of G is additionally associated with a known patienceparameter (also called timeout parameter) ℓ v which bounds the number of probes that can bemade to ∂ ( v ). In this work, we provide a generalization of the patience framework. Specifically,for each v ∈ V , suppose that ∂ ( v ) ( ∗ ) corresponds to the collection of strings (tuples) formed from distinct edges of ∂ ( v ). Upon the arrival of v , an online probing constraint C v ⊆ ∂ ( v ) ( ∗ ) ispresented to the algorithm, which specifies which sequences of edges of ∂ ( v ) can be probed. Wemake the minimal assumption that C v is substring-closed ; that is, if e ∈ C v , then so is anysubstring of e (thus, the empty string λ is in C v by convention). Observe that C v is a generalenough definition to encode any collection downward-closed constraints one may wish to place onthe feasible edge probes incident to v . In particular, it includes the case when v has a patiencevalue ℓ v , and more generally, when C v corresponds to a matroid or budget constraint on ∂ ( v ). Thatbeing said, allowing for inputs which impose order on the probes of ∂ ( v ) is clearly desirable, as itallows for precedence relations. For instance, perhaps one wishes to ensure that if distinct edges e , e ∈ ∂ ( v ) are each probed, then e is always probed before e .The classical online bipartite matching problems (unweighted, vertex weighted, or edge weighted)for adversarial, ROM, and i.i.d. online vertex arrivals all generalize to the online stochastic match-ing setting. We emphasize that the appropriate online stochastic matching problem generalizesthe corresponding classical online problem, even when restricted to the simplest possible probingconstraints; for instance, when each v ∈ V has unit patience (i.e., ℓ v = 1 for all v ∈ V ).Clearly, in the classical adversarial or ROM settings, if the algorithm knew the input graph G , the online algorithm could compute an optimal solution before seeing the online sequence anduse that optimal solution to determine an optimal matching online. But similar to knowing the Gupta et al. [36] refer to probing algorithms which are required to make irrevocable decisions in this way as“executing online”. We reserve the term “online” to refer to the requirement that the probing algorithm must processthe vertices of V one by one, and has no control on their arrival order. G , namely (st( e )) e ∈ E , so the stochastic matching problem is interesting,whether the stochastic graph G is known or unknown to the algorithm. We are left then with awide selection of problems, depending on whether or not the stochastic graph is known, how inputsequences are determined, and whether or not edges or vertices are weighted. To illustrate ourtechniques, we first focus on the setting of a known stochastic graph with ROM inputs. We thengeneralize to the known i.d. setting and consider the cases of both adversarial as well as ROMarrivals.For stochastic probing problems, it is easy to see we cannot hope to obtain a non-trivial com-petitive bound against the expected value of an optimum matching of the stochastic graph . Thestandard approach in the literature is to instead benchmark against an optimum offline probingalgorithm .A solution to the offline stochastic matching problem on the stochastic graph G is an offline probing algorithm A , which is given access to the stochastic graph G = ( U, V, E ) (andthus ( p e ) e ∈ E , ( w e ) e ∈ E , and ( C v ) v ∈ V ), yet does not initially have access to the edges states (st( e )) e ∈ E .Observe then that ( C v ) v ∈ V induces an offline probing constraint C ⊆ E ( ∗ ) in the natural way.For a tuple e ∈ E ( ∗ ) , let e v be the substring of e formed by restricting the coordinates of e tothose edges which contain v . We define C from ( C v ) v ∈ V , where e is included in C if and only if e v ∈ C v for each v ∈ V . In each step t ≥ A may perform a probing operation to reveal the stateof an edge e t = ( u t , v t ) ∈ E , subject to the constraint that if ( e , . . . , e t − ) were the previouslyprobed edges, then ( e , . . . , e t − , e t ) ∈ C . The probing algorithm A may be adaptive ; that is, inaddition to all the information regarding G , the decision on whether to probe e t may depend onall the previously probed edges e , . . . , e t − , and their revealed states, st( e ) , . . . , st( e t − ). It mustalso respect commitment, in that upon probing st( e t ) and revealing st( e t ) = 1, it must irrevocablydecide whether to include e t in its matching. Note that an online probing algorithm with access to G is a special case of an offline probing algorithm.We say that an offline probing algorithm is non-adaptive , provided the probes of E are a(randomized) function of G . Equivalently, an offline probing algorithm is non-adaptive if its probesare statistically independent from the edge states of G . Note that a non-adaptive probing algorithmnecessarily must still use the previously revealed edge probes to determine which edges to add toits matching. However, it may possibly waste edge probes.We define the adaptive benchmark on the input G as an offline probing algorithm whichreturns a matching whose expected weight is as large as possible, and denote this value by OPT( G ).More precisely, if A ( G ) is the matching returned by an offline probing algorithm A , then OPT( G ) :=sup A E [ w ( A ( G ))], where the supremum is over all offline probing algorithms.Our first result is a non-adaptive online probing algorithm which attains a competitive ratioagainst the adaptive benchmark, provided it is given full access to G = ( U, V, E ), and the verticesof V arrive u.a.r. Theorem 2.1.
Suppose G = ( U, V, E ) is an arbitrary stochastic graph and Algorithm 4 returnsthe matching M when given full access to G . In this case, if the vertices of V arrive u.a.r. , then E [ w ( M )] ≥ (cid:0) − e (cid:1) OPT ( G ) . Moreover, Algorithm 4 is non-adaptive. Consider a single online vertex with patience 1, and n offline (unweighted) vertices where each edge e hasprobability n of being present. The expectation of an online probing algorithm will be at most n while the expectedsize of an optimal matching (over all instantiations of the edge probabilities) will be 1 − (1 − n ) n → − e . Thisexample also clearly shows that no constant ratio is possible if the patience is sub-linear (in n = | U | ). non-adaptive online probingalgorithms in the known stochastic graph setting with ROM arrivals. In fact, in Section 6 we provea hardness result which applies to all non-adaptive probing algorithms (even probing algorithmswhich execute offline, and thus do not respect the arrival order of V ): Theorem 2.2.
No non-adaptive offline probing algorithm can attain an approximation ratio againstthe adaptive benchmark which is greater than − /e . Not only does Algorithm 4 attain a competitive ratio against the adaptive benchmark, it can alsobe implemented efficiently under some mild assumptions. Given the stochastic graph G = ( U, V, E ),observe that the size of C v for v ∈ V may be exponentially large in the size of U . On the other hand,if we denote | G | as the amount of space needed to represent the information associated with G afterexcluding the constraints ( C v ) v ∈ V , then we wish to find online probing algorithms which executein time poly( | G | ). Since the probing algorithm must interact with ( C v ) v ∈ V , we either need to workwith constraint systems which have representations of size poly( | G | ) (such as patience constraints),or we need to work in an oracle model . We take the latter approach, as it allows us to provemore general results.We consider two oracle models. In the first model, which we refer to as the membershipmodel , a probing algorithm may make a membership query to any string e ∈ ∂ ( v ) ( ∗ ) for v ∈ V . More precisely, in a single operation the probing algorithm may determine whether or not e ∈ ∂ ( v ) ( ∗ ) is included in C v .The second model, which we refer to as the demand oracle model , allows the probing algo-rithm far more power. In particular, for any v ∈ V and any selection of real values, ( α u ) u ∈ N ( v ) , thealgorithm may determine in a single operation a solution to the following maximization problem:maximize | e | X i =1 ( w e i − α u i ) · p e i · i − Y j =1 (1 − p e j ) (2.1)subject to e ∈ C v (2.2)Observe that this definition is closely related to the demand oracle model in the context of theiterative auctions, as originally studied by Blumrosen and Nisan [11, 12]. More precisely, ignoringthe edge probabilities ( p e ) e ∈ ∂ ( v ) for now (i.e. p e = 1 for all e ), let us suppose a seller is tryingto allocate the items of U to a number of buyers. We view the vertex v as a buyer who wishesto purchase a subset of items S ⊆ N ( v ), based on their valuation function f ( S ). Assume that v has unit demand , that is f ( S ) := max s ∈ S w s,v . The values ( α s ) s ∈ N ( v ) are viewed as pricesthe buyer must pay , and the demand oracle returns a solution to max S ⊆ N ( v ) ( f ( S ) − P s ∈ S α s ),thereby maximizing the utility of v . Clearly, for the simple case of a unit-demand buyer, anoptimum assignment is the item u ∈ N ( v ) for which w u,v − α u is maximized, and so the notion ofa demand oracle is unnecessary. However, often v is not a unit-demand buyer, and so f is a muchmore general valuation function. In this case, a solution to max S ⊆ N ( v ) ( f ( S ) − P s ∈ S α s,v ) may becomputationally difficult to find, so the demand oracle assumption is convenient to make.Even the case of a unit-demand buyer is a non-trivial optimization problem in the stochasticprobing framework,. Observe that we may view the edge probabilities ( p e ) e ∈ E as modeling thesetting when there is uncertainty in whether or not the purchase proposals will succeed; that is, See Eden et al. [26] for a buyer/seller interpretation of the classical
Ranking algorithm [42] for bipartite matching. u, v ) = 1, provided the seller agrees to sell item u to buyer v . In this interpretation, (2.1) is theexpected utility of the unit-demand buyer v which commits to the first item u ∈ N ( v ) such thatst( u, v ) = 1, at which point v gains utility w u,v − α u . The relevancy of a demand oracle dependson whether or not it is reasonable to assume that each buyer’s optimum greedy probing strategy isreadily available. In Section 3, we introduce a new LP to design probing algorithms whose efficientsolvability is closely tied to the existence of a demand oracle. More precisely, the demand oracleyields a separation oracle for the dual of LP-new, thus allowing the ellipsoid algorithm [53, 33] torun in polynomial time [59, 57, 2, 46].We say that C v is closed under permutations , provided if e ∈ C v , then any permutation of e is also in C v . Observe then that since C v is substring-closed by assumption, this is condition isequivalent to requiring that C v corresponds to a downward-closed family of subsets of ∂ ( v ). Weobserve then the following reduction: Proposition 2.3. If C v is both substring-closed and permutation-closed, then for any selection ofvalues ( α u ) u ∈ N ( v ) , (2.1) can be solved efficiently, assuming access to a membership query oracle for C v . The proof of Proposition 2.3 builds upon the work of Brubach et al. [15] as well as Purohit etal. [52]:
Proof of Proposition 2.3.
Compute e w e := w e − α u for each e = ( u, v ) ∈ ∂ ( v ), and define P := { e ∈ ∂ ( v ) : e w e ≥ } . First observe that if P = ∅ , then (2.1) is maximized by the empty-string λ .Thus, for now on assume that P = ∅ . Since C v is closed under substrings, it suffices to considerthose e ∈ C v whose edges all lie in P . As such, for notational convenience, let us hereby assumethat ∂ ( v ) = P .Now, for any e ∈ C v , let e r be the rearrangement of e , based on the non-increasing order of theweights ( e w e ) e ∈ e . Since C v is closed under permutations, we know that e r is also in C v . Moreover,the evaluation of e r in (2.1) is at least as large as that of e . Hence, let us order the edges of ∂ ( v ) as e , . . . , e k , such that e w e ≥ . . . ≥ e w e k , where k := | ∂ ( v ) | . Observe then that it suffices to maximize(2.1) over those strings within C v which respect the ordering on ∂ ( v ). Stated differently, let us denote I v as the family of subsets of ∂ ( v ) induced by C v , and define the set function f : 2 ∂ ( v ) → [0 , ∞ ).where f ( S ) := P | S | i =1 e w s i · p s i · Q i − j =1 (1 − p s j ) for S = { s , . . . , s | S | } ⊆ ∂ ( v ) (where e w s ≥ . . . ≥ e w s | S | ).Our goal is then to efficiently maximize f over the set-system ( ∂ ( v ) , I v ). Observe that since C v is both substring-closed and permutation-closed, I v is downward-closed. Moreover, clearly we cansimulate oracle access to I v , based on our oracle access to C v .For each i = 0 , . . . , k −
1, denote ∂ ( v ) >i := { e i +1 , . . . , e k } , and ∂ ( v ) >k := ∅ . Moreover, define thefamily of subsets I >iv := { S ⊆ ∂ ( v ) >i : S ∪ { e i } ∈ I v } for each 2 ≤ i ≤ k , and I > v := I v . Observethen that ( ∂ ( v ) >i , I >iv ) is a downward-closed set system, as I v is downward-closed. Moreover, wemay simulate oracle access to I >iv based on our oracle access to I v .Denote OPT( I >iv ) as the maximum value of f over constraints I >iv . Observe then the followingrecursion: OPT( I v ) := max i ∈ [ k ] ( p e i · e w e i + (1 − p e i ) · OPT( I >iv )) (2.3)Hence, given access to the values OPT( I > v ) , . . . , OPT( I >kv ), we can compute OPT( I v ) efficiently.In fact, it is clear that we can use (2.3) to recover an optimum solution to f , and so the prooffollows by an inductive argument on | ∂ ( v ) | . 9s we shall see in Sections 3 and 4, solving (2.1) is the key subroutine needed to execute ouronline probing algorithms efficiently. In particular, we have the following statement in regards tothe efficiency of Algorithm 4: Theorem 2.4.
Suppose that G = ( U, V, E ) is a stochastic graph with (substring-closed) probingconstraints ( C v ) v ∈ V . • In the demand oracle model, Algorithm 4 executes in time poly ( | G | ) . • In the membership oracle model, Algorithm 4 executes in time poly ( | G | ) , provided for each v ∈ V , C v is also permutation-closed. We now introduce the technical definitions necessary to precisely describe the online stochasticmatching problem with known i.d. arrivals . We first describe the randomized procedurefor generating the input the online probing algorithm operates on. Afterwards, we indicate whichinformation the online algorithm has access to, and the precise benchmark it is compared against.Let us suppose that H typ = ( U, B, F ) where F ⊆ U × B is an arbitrary stochastic graph withedge weights ( w f ) f ∈ F , edge probabilities ( p f ) f ∈ F , and online probing constraints ( C b ) b ∈ B . In theknown i.d. setting, we refer to H typ as the stochastic type graph (or type graph when clear),and the vertices of B as the online type nodes of H typ . The online probing algorithm does not execute on H typ . Instead, the online algorithm operates on an (unknown) stochastic graph G = ( U, V, E ). Here, each arriving online vertex v ∈ V , together with ( p e ) e ∈ ∂ ( v ) , ( w e ) e ∈ ∂ ( v ) , and C v ,is chosen by selecting v from B . In this work, we are interested in the case when these selectionsare done randomly according to known distributions on B .More precisely, the adversary fixes parameter n ≥
1, indicating the size of V , and thus thenumber of rounds or arrivals to occur. The adversary also fixes a sequence of distributionsdenoted ( D i ) ni =1 , which are each supported on B . Using H typ , and ( D i ) ni =1 , we construct the instantiated stochastic graph G = ( U, V, E ) by executing the following randomized procedure.Note that technically V and E are each multi-sets, as a type node b ∈ B may appear multipletimes. • For i = 1 , . . . , n : draw v i ∈ B independently using D i , and add a copy of v i to G , with therelevant edge probabilities, edge weights and online probing constraint. Remark 2.5.
We say that v i is has type b ∈ B , provided v i = b . Note that once G is constructed,the edge states (st( e )) e ∈ E are drawn independently of each other, where in particular, (st( e )) e ∈ ∂ ( v i ) and (st( e )) e ∈ ∂ ( v j ) are independent, provided i = j (even if v i and v j have the same type).We denote that the instantiated stochastic graph G = ( U, V, E ) is drawn from ( H typ , ( D i ) ni =1 ) inthis way, by writing G ∼ ( H typ , ( D i ) ni =1 ).An online probing algorithm is given access to H typ and the distributions ( D i ) ni =1 , yet initiallydoes not have access to G . Instead, a permutation π : [ n ] → [ n ] is chosen upon, and for each t = 1 , . . . , n , vertex v π ( t ) , along with its edge weights, probabilities, and probing constraint arepresented to the algorithm. Using all past available information regarding the outcomes of theprobes involving v π (1) , . . . , v π ( t − , together with the edge probabilities and weights adjacent to v π ( t ) , the algorithm may probe the edges of ∂ ( v π ( t ) ), subject to the probing constraint C v π ( t ) . Onceagain, the algorithm must operate in the probe-commit model.10he permutation π is either generated u.a.r. , independently of all other randomization, or by anadversary. In either case, π is unknown to the online probing algorithm. We emphasize that we workwith an oblivious adversary , in that π must be depend solely upon the input ( H typ , ( D i ) ni =1 ).In particular, it cannot depend on the generation of G , nor the decisions of the online probingalgorithm. This is the more standard assumption in prophet problems as recently discussed byEzra et al. [29].Suppose that A ( H typ , ( D i ) ni =1 , π ) is the matching constructed by an online probing algorithm A when presented the online vertices of G ∼ ( H typ , ( D i ) ni =1 ) in order π . The goal of the onlineprobing algorithm is to maximize the expected weight of A ( H typ , ( D i ) ni =1 , π ). Specifically, it aimsto maximize E [ w ( A ( H typ , ( D i ) ni =1 , π ))] , where the expectation is over the construction of G , the edge states (st( e )) e ∈ E , as well as anyrandomized decisions made by the algorithm. We benchmark against the expected performanceof adaptive benchmark on G ∼ ( H typ , ( D i ) ni =1 ), which we denote by OPT( H typ , ( D i ) ni =1 ). Moreprecisely, OPT( H typ , ( D i ) ni =1 ) := E [OPT( G )] , where the expectation is solely over the randomness in generating G .The standard in the literature (see [3, 9, 17]) is to prove competitive ratios against OPT( H typ , ( D i ) ni =1 ).That is, in the adversarial order model, the goal is to find an online probing algorithm A for whichthe (strict) competitive ratio inf ( H typ , ( D i ) ni =1 ,π ) E [ w ( A ( H typ , ( D i ) ni =1 , π ))]OPT( H typ , ( D i ) ni =1 )is as close to 1 as possible. An analogous definition holds in the random order model. We saythat an online probing algorithm is non-adaptive , provided for each t ∈ [ n ], the probes of ∂ ( v π ( t ) )are a (randomized) function of H typ , and the type of arrival v π ( t ) . In particular, when probingedges adjacent to the online node v π ( t ) , the algorithm does not make use of the previously probededge states of the online nodes v π (1) , . . . , v π ( t − (nor the matching decisions made thus far). Thealgorithm will therefore possibly waste some probes to edges ( u, v π ( t ) ) but will not violate thematching constraint. Note that this definition is consistent with the definition considered in theknown stochastic graph case. It also generalizes the classical definition of Manshadi et al. [49]. Wenow present our main results: Theorem 2.6.
Suppose ( H typ , ( D i ) ni =1 ) is a known i.d. input, which Algorithm 6 is given fullaccess to. If M is the matching returned by the algorithm when presented the online vertices of G ∼ ( H typ , ( D i ) ni =1 ) in random order, then E [ w ( M )] ≥ (cid:0) − e (cid:1) OPT ( H typ , ( D i ) ni =1 ) . Moreover,Algorithm 6 is non-adaptive. Theorem 2.7.
Suppose ( H typ , ( D i ) ni =1 ) is a known i.d. input, which Algorithm 8 is given fullaccess to. If M ( π ) is the matching returned by the algorithm when presented the online vertices of G ∼ ( H typ , ( D i ) ni =1 ) in an adversarial order π : [ n ] → [ n ] , then E [ w ( M ( π ))] ≥ OPT ( H typ , ( D i ) ni =1 ) . Other works, for instance Feldman et al. [31], consider an almighty adversary which has access to the in-stantiations of all the random variables when generating π . This definition is far too strong for stochastic probingproblems, as it allows the benchmark to forgo making edge probes. emark 2.8. This is a tight bound since the problem generalizes the classical single item prophetinequality for which is an optimal competitive ratio. Theorem 2.9.
Suppose that ( H typ , ( D i ) ni =1 ) is a known i.d. input, where H typ = ( U, B, F ) has(substring-closed) probing constraints ( C b ) b ∈ B . If | H typ | is the size of H typ (excluding ( C b ) b ∈ B ), and |D i | is the amount of space needed to encode the distribution D i , then the following claims holds: • In the demand oracle model, Algorithms 6 and 8 execute in time poly ( | H typ | , ( |D i | ) ni =1 ) . • In the membership oracle model, Algorithms 6 and 8 execute in time poly ( | H typ | , ( |D i | ) ni =1 ) ,provided for each b ∈ B , C b is also permutation-closed. Remark 2.10. | H typ | may be exponentially large in the size of G ∼ ( H typ , ( D i ) ni =1 ), however foreach ε >
0, our results can be made to run in time poly( | G | , log(1 /ε )) using Monte Carlo simulation(at a loss of (1 − ε ) in performance), assuming we have oracle access to samples drawn from ( D i ) ni =1 . In order to prove our positive results, we require a means to devise online probing algorithms whichdo reasonably well compared to the adaptive benchmark, and yet are efficient. In this section, wedevelop the tools needed to devise such algorithms. We focus on the known stochastic graph settingof Theorem 2.1. In the following section, we generalize our techniques to handle the more generalproblem of known i.d. arrivals.Our goal is to find an online probing algorithm A , such that for each stochastic graph G =( U, V, E ), we have that E [ w ( A ( G ))] ≥ (1 − /e )OPT( G ), provided the vertices of V arrive in randomorder. Towards this goal, we introduce some notation which allows us to derive a new configurationLP. For each e = ( e , . . . , e | e | ) ∈ E ( ∗ ) , define g ( e ) := Q | e | i =1 (1 − p e i ). Notice that g ( e ) correspondsto the probability that all the edges of e are inactive, where g ( λ ) := 1 for the empty string λ . Wealso define e 13e say that A is non-adaptive , provided the probes are a (randomized) function of G . Equiva-lently, A is non-adaptive if the probes of A are statistically independent from (st( e )) e ∈ E . Unlikethe offline stochastic matching problem, there exists a relaxed probing algorithm which is optimum,and yet non-adaptive: Lemma 3.2. For any stochastic graph G = ( U, V, E ) with (substring-closed) probing constraints ( C v ) v ∈ V , there exists an optimum relaxed probing algorithm B which satisfies the following proper-ties:( Q ) If e = ( u, v ) is probed and st ( e ) = 1 , then e is included in B ( G ) , provided v is currentlyunmatched.( Q ) B is non-adaptive on G . We defer the proof of Lemma 3.2 to Section A. Observe that by considering B of Lemma 3.2,and defining x v ( e ) as the probability that B probes the edges of e in order for v ∈ V and e ∈ C v ,properties ( Q ) and ( Q ) ensure that ( x v ( e )) v ∈ V, e ∈C v is a feasible solution to LP-new, such that E [ w ( B ( G ))] = X v ∈ V X e ∈C v val( e ) · x v ( e ) . Thus, the optimality of B implies that OPT rel ( G ) ≤ LPOPT( G ), and so together with (3.4),Theorem 3.1 follows.In fact, we claim the following equivalence between LP-new and the relaxed stochastic matchingproblem, whose proof we defer to Appendix A: Theorem 3.3. For any stochastic graph G with substring-closed probing constraints, OPT rel ( G ) = LPOPT ( G ) . We shall now show how to use LP-new to design online probing algorithms which have accessto the stochastic graph G = ( U, V, E ). Suppose that we are presented a feasible solution, say( x v ( e )) v ∈ V, e ∈C v , to LP-new for G . For each e ∈ E , define e x e := X e ′ ∈C v : e ∈ e ′ g ( e ′ VertexProbe Input: an online vertex s of a stochastic graph, ∂ ( s ), and probabilities ( z ( e )) e ∈C s such that P e ∈C s z ( e ) = 1. Output: an active edge N of ∂ ( s ). Initialize N ← ∅ . Draw e ′ from C s with probability z ( e ′ ). if e ′ = λ then ⊲ the empty string is drawn. return N . else Denote e ′ = ( e ′ , . . . , e ′ k ) for k := | e ′ | ≥ for i = 1 , . . . , k do ⊲ probe the edges of e ′ in order and return the first active edge Probe the edge e ′ i . if st( e ′ i ) = 1, and s is not matched by N then Add e ′ i to N . end if end for end if return N .Observe the following claim, which follows immediately from the definition of the edge variables,( e x e ) e ∈ E : Lemma 3.4. Let G = ( U, V, E ) be a stochastic graph with LP-new solution ( x v ( e )) v ∈ V,∂ ( v ) , andwhose induced edge variables we denote ( e x e ) e ∈ E . If the VertexProbe algorithm is passed a fixednode s ∈ V , then each e ∈ ∂ ( s ) is returned by the algorithm with probability e x e . Remark 3.5. We say that VertexProbe commits to the edge e , provided the algorithm outputsthis edge when executing on the fixed node s ∈ V .To clearly illustrate how we use LP-new in conjunction with VertexProbe to design probingalgorithms, we first consider a simpler algorithm which attains a competitive ratio of 1 / 2, providedthe vertices of G = ( U, V, E ) arrive in random order. Afterwards, we modify this algorithm via anon-greedy contention resolution scheme considered by Lee and Singla [46], allowing us to improvethe competitive ratio to 1 − /e , thus proving Theorem 2.1 Algorithm 2 Known Stochastic Graph Input: a stochastic graph G = ( U, V, E ). Output: a matching M of active edges of G . M ← ∅ . Compute an optimum solution of LP-new for G , say ( x v ( e )) v ∈ V, e ∈C v for s ∈ V in u.a.r. order do Set e ← VertexProbe ( s, ∂ ( s ) , ( x s ( e )) e ∈C s ). if e = ( u, s ) for some u ∈ U , and u is unmatched then ⊲ rule out when e = ∅ Add e to M . end if end for return M . 15 emark 3.6. Technically, line (6) should occur within the VertexProbe subroutine to adhereto the probe-commit model, however we express our algorithms in this way for brevity. Proposition 3.7. If M is the matching returned by Algorithm 2 when executing on the stochasticgraph G = ( U, V, E ) , then E [ w ( M )] ≥ OPT ( G ) , provided the vertices of V arrive u.a.r. . Moreover,the analysis of Algorithm 2 is tight. In order to prove Proposition 3.7, and to later derive the 1 − /e -competitive ratio of Theorem2.1, we review contention resolution schemes, restricted to the simplest case of rank 1 matroids.Given k ≥ 1, consider the ground set [ k ] := { , . . . , k } . Fix z ∈ [0 , k , and let R ( z ) ⊆ [ k ] denotethe random set, where each i ∈ [ k ] is included in R ( z ) independently with probability z i . Let usdenote P := { z ∈ [0 , k : P ki =1 z i ≤ } . Note that P is the convex relaxation of the constraintimposed by the rank 1 matroid on [ k ] (i.e., at most one element of [ k ] may be selected). Definition 1 (Contention Resolution Scheme – Rank 1 Matroid) . A contention resolutionscheme (CRS) for the rank 1 matroid on [ k ] is a (randomized) algorithm ψ , which given z ∈ P and S ⊆ [ k ] as inputs, returns a single element ψ z ( S ) of S . Given c ∈ [0 , ψ is said to be c - selectable , provided for all i ∈ [ k ] and z ∈ P , P [ i ∈ ψ z ( R ( z )) | i ∈ R ( z )] ≥ c, (3.6)where the probability is over the generation of R ( z ), and the potential randomness used by ψ . Remark 3.8. Observe that if f : 2 [ k ] → R is a monotone linear function, then for any z ∈ P ,executing a c -selectable CRS ψ yields an element ψ z ( R ( z )) ∈ R ( z ), such that E [ f ( ψ z ( R ( z )))] ≥ c · E [ f ( R ( z ))]. Thus, c -selectable CRS are useful for designing approximation algorithms, in whichone works with a convex relaxation of the constraint system on [ k ] (in our case, a rank 1 matroid).Much more general results hold, and we refer the reader to the seminal paper by Chekuri, Vondrak,and Zenklusen [57].Feldman et al. [31] considered a more restricted class of contention resolution schemes, called online contention resolution schemes (OCRS). These are schemes in which R ( z ) is not knownto the scheme ahead of time. Instead, the elements of [ k ] are presented to the scheme ψ in adversarialorder, where in each step, an arriving i ∈ [ k ] reveals if it is in R ( z ), at which point ψ must make anirrevocable decision as to whether it wishes to return i as its output. Lee and Singla [46], as wellas Adamczyk and Wlodarczyk [4], considered an extension of this definition to the setting wherethe elements of [ k ] arrive in random order, thus defining random order contention resolutionschemes (RCRS). Both order variants continue to allow ψ to depend on z , and the notion ofselectability extends, where in the case of a RCRS, the random order is incorporated into theprobabilistic computation of (3.6).We can view Algorithm 2 as executing a concurrent contention resolution scheme on each of theoffline vertices. More precisely, given a fixed u ∈ U , observe that Algorithm 2 commits to each edge e = ( u, v ) ∈ ∂ ( u ), independently with probability e x e . Thus, from the perspective of u , Algorithm2 can be viewed as executing a greedy RCRS in which the edges of ∂ ( u ) are considered uniformlyat random, and the first one which commits to u is included in M . This greedy RCRS is 1 / k ] = { , . . . k } , draw Y i ∼ [0 , u.a.r and independently for i = 1 , . . . , k .16 lgorithm 3 RCRS – Lee and Singla [46] Input: z ∈ P , where P ⊆ [0 , k . Output: at most one element of [ k ]. for i ∈ [ k ] in increasing order of Y i do if i ∈ R ( z ) then return i independently with probability exp( − Y i · z i ) end if end for return ∅ . ⊲ pass on returning an element of [ k ]. Theorem 3.9 (Lee and Singla [46]) . Algorithm 3 is a − /e -selectable RCRS for the case of arank matroid. We are now ready to modify Algorithm 2 to attain the desired competitive ratio. For each v ∈ V , draw e Y v ∈ [0 , 1] independently and uniformly at random. We assume the vertices of V arepresented to Algorithm 2 in non-decreasing order, based upon the values ( e Y v ) v ∈ V . Algorithm 4 Known Stochastic Graph – Modified Input: a stochastic graph G = ( U, V, E ). Output: a matching M of G of active edges. M ← ∅ . Compute an optimum solution of LP-new for G , say ( x v ( e )) v ∈ V, e ∈C v . for s ∈ V in increasing order of e Y s do Set e ← VertexProbe ( s, ∂ ( s ) , ( x s ( e )) e ∈C s ). if e = ( u, s ) for some u ∈ U , and u is unmatched then Add e to M independently with probability exp( − e Y s · e x u,s ). end if end for return M . Proof of Theorem 2.1. Given u ∈ U , let M ( u ) denote the edge matched to u by M , where M ( u ) := ∅ if no such edge exists.Observe now that if C ( e ) corresponds to the event in which VertexProbe commits to e ∈ ∂ ( u ),then P [ C ( e )] = e x e by Lemma 3.4. Moreover, the events ( C ( e )) e ∈ ∂ ( u ) are independent, and satisfy X e ∈ ∂ ( u ) P [ C ( e )] = X e ∈ ∂ ( u ) e x e ≤ , (3.7)by constraint (3.1) of LP-new. As such, denote z := ( z e ) e ∈ ∂ ( u ) where z e := e x e , and observe that(3.7) ensures that z ∈ P , where P is the convex relaxation of the rank 1 matroid on ∂ ( u ). Let usdenote R ( z ) as those those e ∈ ∂ ( u ) for which C ( e ) occurs.For each e = ( u, v ) ∈ ∂ ( u ), define Y u,v := e Y v . Observe then that the random variables ( Y e ) e ∈ ∂ ( u ) are independent and drawn u.a.r. from [0 , ψ is the RCRS defined in Algorithm 3, thenwe may pass z to ψ , and process the edges of ∂ ( u ) in non-increasing order based on ( Y e ) e ∈ ∂ ( u ) .Denote the resulting output by ψ z ( R ( z )). By coupling the random draws of lines (3) and (6) of17lgorithms 3 and 4, respectively, we get that w ( M ( u )) = X e ∈ ∂ ( u ) w e · [ e ∈ R ( z )] · [ e ∈ ψ z ( R ( z ))] Thus, after taking expectations, E [ w ( M ( u ))] = X e ∈ ∂ ( u ) w e · P [ e ∈ ψ z ( R ( z )) | e ∈ R ( z )] · P [ e ∈ R ( z )] . Now, Theorem 3.9 ensures that for each e ∈ ∂ ( u ), P [ e ∈ ψ z ( R ( z )) | e ∈ R ( z )] ≥ (cid:0) − e (cid:1) . It followsthat E [ w ( M ( u ))] ≥ (cid:0) − e (cid:1) P e ∈ ∂ ( u ) w e e x e , for each u ∈ U . Thus, E [ w ( M )] = X u ∈ U E [ w ( M ( u ))] ≥ (cid:18) − e (cid:19) X e ∈ ∂ ( u ) w e e x e = (cid:18) − e (cid:19) LPOPT( G ) , where the equality follows since ( x v ( e )) v ∈ V, e ∈C v is an optimum solution to LP-new. On the otherhand, LPOPT( G ) ≥ OPT( G ) by Theorem 3.1, and so the proof is complete.We conclude the section by showing how LP-new be solved efficiently under the assumptions ofTheorem 2.4. Since the remaining steps of Algorithm 4 can clearly be implemented efficiently, thiswill prove Theorem 2.4. Theorem 3.10. Suppose that G = ( U, V, E ) in a stochastic graph with (substring-closed) probingconstraints ( C v ) v ∈ V . • In the demand oracle model, LP-new is efficiently solvable in | G | . • In the membership oracle model, LP-new is efficiently solvable in | G | , provided for each v ∈ V , C v is also permutation-closed. We prove Theorem 3.10 by first considering the dual of LP-new. Note, that in the below LPformulation, if e = ( e , . . . , e k ) ∈ C v , then we set e i = ( u i , v ) for i = 1 , . . . , k for convenience.minimize X u ∈ U α u + X v ∈ V β v (LP-new-dual)subject to β v + | e | X j =1 p e j · g ( e In the demand oracle model, there exists an efficient deterministic algorithm forchecking whether f ( e ′ ) > β v for some e ′ ∈ C v provided C v is substring-closed. Moreover, if such atuple exists, then it can be found efficiently. The same result holds in the membership oracle model,provided C v is also permutation-closed.Proof. We first consider the case of the demand oracle model. In this setting, we can pass thevalues ( α u ) u ∈ N ( v ) to the oracle, and it will return a string e ′ ∈ C v for which f ( e ′ ) = | e ′ | X j =1 ( w e j − α u j ) · p e ′ j · g ( e ′ For any known i.d instance ( H typ , ( D t ) nt =1 ) , it holds that OPT ( H typ , ( D t ) nt =1 ) ≤ LPOPT ( H typ , ( D t ) nt =1 ) . Now, given a feasible solution to LP-new-id, say ( x t ( e || b )) t ∈ [ n ] ,b ∈ B, e ∈C b , for each u ∈ U, t ∈ [ n ]and b ∈ B define e x u,t ( b ) := X e ∈C b :( u,b ) ∈ e g ( e < ( u,b ) ) · p u,b · x t ( e || b ) . (4.4)Suppose now that we fix t ∈ [ n ] and b ∈ B , and consider the variables, ( x t ( e || b )) e ∈C b . Observethat (4.2) ensures that P e ∈C b x t ( e || b ) r t ( b ) = 1 . Hence, if v t is drawn from D t , then VertexProbe may be passed the input( v t , ∂ ( v t ) , ( x t ( e || v t ) /r t ( v t )) e ∈C vt ), no matter which type node v t is instantiated as. As such, if wedefine C ( u, v t ) as the event in which VertexProbe outputs the edge ( u, v t ), then observe thefollowing extension of Lemma 3.4: Lemma 4.2. If VertexProbe is passed ( v t , ∂ ( v t ) , ( x t ( e || v t ) /r t ( v t )) e ∈C vt ) , then P [ C ( u, v t ) | v t = b ] = e x u,t ( b ) r t ( b ) Remark 4.3. As in Lemma 3.4, if C ( u, v t ) occurs, then we say that VertexProbe commits tothe edge ( u, v t ). 20uppose now that each vertex v t has an arrival time, say Y t ∈ [0 , u.a.r. and indepen-dently for t ∈ [ n ]. The values ( Y t ) nt =1 again indicate the increasing order in which the vertices of G ∼ ( H typ , ( D t ) nt =1 ) arrive. We first consider the following online probing algorithm, without anysophisticated CRS: Algorithm 5 Known I.D. – ROM Input: a known i.d. input ( H typ , ( D t ) nt =1 ). Output: a matching M of active edges of G ∼ ( H typ , ( D t ) nt =1 ). M ← ∅ . Compute an optimum solution of LP-new-id for ( H typ , ( D t ) nt =1 ), say ( x t ( e || b )) t ∈ [ n ] ,b ∈ B, e ∈C b . for t ∈ [ n ] in increasing order of Y t do Set e ← VertexProbe ( v t , ∂ ( v t ) , ( x t ( e || v t ) /r t ( v t )) e ∈C vt ). if e = ( u, v t ) for some u ∈ U , and u is unmatched then Add e to M . end if end for return M .Similarly, to Algorithm 2 of Proposition 3.7, one can show that Algorithm 5 attains a competitiveratio of 1 / 2. Interestingly, if the distributions ( D t ) nt =1 are identical – that is, we work in the knowni.i.d. model – then it is easy to show that this algorithm’s competitive ratio improves to 1 − /e . Proposition 4.4. If Algorithm 5 is presented a known i.i.d. input, say the type graph H typ togetherwith the (fixed) distribution D , then E [ w ( M )] ≥ (1 − /e ) OPT ( H typ , D ) . In order to prove Theorems 2.6 and 2.7, we again must apply more sophisticated forms ofcontention resolution. Observe that for each u ∈ U , in the execution of Algorithm 5, the probabilitythat VertexProbe ( v t , ∂ ( v t ) , ( x t ( e || v t ) /r t ( v t )) e ∈C vt ) commits to the edge ( u, v t ) is precisely, z u,t := X b ∈ B e x u,t ( b ) = X b ∈ B X e ∈C b :( u,b ) ∈ e p u,b · g ( e < ( u,b ) ) · x t ( e || b ) (4.5)Moreover, the events ( C ( u, v t )) nt =1 are independent. Thus, it is again natural to again apply con-tention resolution on each vertex u ∈ U , as done in Algorithm 4. Algorithm 6 Known I.D. – ROM – Modified Input: a known i.d. input ( H typ , ( D t ) nt =1 ). Output: a matching M of active edges of G ∼ ( H typ , ( D t ) nt =1 ). M ← ∅ . Compute an optimum solution of LP-new-id for ( H typ , ( D t ) nt =1 ), say ( x t ( e || b )) t ∈ [ n ] ,b ∈ B, e ∈C b . for t ∈ [ n ] in increasing order of Y t do Set e ← VertexProbe ( v t , ∂ ( v t ) , ( x t ( e || v t ) /r t ( v t )) e ∈C vt ). if e = ( u, v t ) for some u ∈ U , and u is unmatched then Add e to M independently with probability exp( − Y t · z u,t ). end if end for return M . 21 roof of Theorem 2.6. Analyzing Algorithm 6 follows in the same way as in the proof of Theo-rem 2.1, with the only change being that the performance of Algorithm 6 is compared againstLPOPT( H typ , ( D i ) ni =1 ), and thus the adaptive benchmark OPT( H typ , ( D i ) ni =1 ) via an applicationof Theorem 4.1.More interesting is how we can adapt Algorithm 5 in the case of adversarial arrivals. For this,we again make use of existing contention resolution schemes. In particular, we adapt the OCRSused by Ezra et al. [29].Given the ground set [ k ] = { , . . . k } , suppose the elements of [ k ] are presented according tosome permutation π : [ k ] → [ k ] (i.e., π (1) , . . . , π ( k )), and z ∈ [0 , k satisfies P ki =1 z i ≤ 1. Uponthe arrival of element π ( t ) ∈ [ k ], compute q t := 12 − P t − i =1 z π ( i ) . Observe that 0 ≤ q t ≤ 1, as 0 ≤ P ki =1 z i ≤ k ] to the algorithm upfront. Algorithm 7 OCRS – Ezra et al. [29] Input: z ∈ P , where P ⊆ [0 , k . ⊲ P is the convex relaxation of the rank 1 matroid Output: at most one element of [ k ]. for t = 1 , . . . , k do if π ( t ) ∈ R ( z ) then ⊲ π ( t ) is in R ( z ) with probability z π ( t ) Compute q t , based on the arrivals π (1) , . . . , π ( t − return π ( t ) independently with probability q t . end if end for return ∅ . ⊲ pass on returning an element of [ k ] Theorem 4.5 (Ezra et al. [46]) . Algorithm 7 is an OCRS for a rank matroid which is / -selectable. We return to the problem of designing a modification of Algorithm 5 that works for adversarialvertex arrivals. Assume vertices v , . . . , v n are presented to the online probing algorithm using thepermutation π : [ n ] → [ n ] (i.e., v π (1) , . . . v π ( n ) ). For each t ∈ [ n ] and u ∈ U , define q u,t := 12 − P t − i =1 z u,π ( i ) , (4.6)where we recall z u,j := P b ∈ B e x u,j ( b ). Clearly, P j ∈ [ n ] z u,j ≤ 1, by constraint (4.1) of LP-new-id,and so we get the following online probing algorithm:22 lgorithm 8 Known I.D. – AOM – Modified Input: a known i.d. input ( H typ , ( D i ) ni =1 ). Output: a matching M of active edges of G ∼ ( H typ , ( D t ) nt =1 ). M ← ∅ . Compute an optimum solution of LP-new-id for ( H typ , ( D i ) ni =1 ), say ( x i ( e || b )) i ∈ [ n ] ,b ∈ B, e ∈C b . for t = 1 , . . . , n do Based on the previous arrivals v π (1) , . . . , v π ( t − before v π ( t ) , compute values ( q u,t ) u ∈ U . Set e ← VertexProbe ( v t , ∂ ( v t ) , ( x t ( e || v t ) /r t ( v t )) e ∈C vt ). if e = ( u, v t ) for some u ∈ U , and u is unmatched then Add e to M independently with probability q u,t . end if end for return M . Proof of Theorem 2.7. Analysing Algorithm 8 follows the proof of Theorem 2.1.We conclude the section by noting that LP-new-id can be solved in time poly( | H typ | , ( r t ( b )) t ∈ [ n ] ,b ∈ B )in both the membership and demand oracle models, under the assumptions of Theorem 2.9. Theargument follows as in the proof of Theorem 3.10, with a slight adjustment to handle the values( r t ( b )) t ∈ [ n ] ,b ∈ B and so we defer the details. The efficiency of Algorithms 6 and 8 thereby follows, asclaimed in Theorem 2.9. Let us suppose that G = ( U, V, E ) is a stochastic graph with substring-closed probing constraints( C v ) v ∈ V . In order to prove Lemma 3.2, we must show that there exists an optimum relaxed probingalgorithm which is non-adaptive and satisfies ( Q ). Our high level approach is to consider an opti-mum relaxed probing algorithm A which satisfies ( Q ), and then to construct a new non-adaptivealgorithm B by stealing the strategy of A , without any loss in performance. More specifically, weconstruct B by writing down for each v ∈ V and e ∈ C v the probability that A probes the edgesof e in order. These probabilities necessarily satisfy certain inequalities which we make use of indesigning B . In order to do so, we need a technical randomized rounding procedure whose preciserelevance will become clear in the proof of Lemma 3.2.Suppose that e ∈ E ( ∗ ) , and for each j ≥ 0, denote e j as the j th edge of e j , where e j := λ if j = 0 or j > | e | (recall that λ is the empty-string). Let us now assume that ( y v ( e )) e ∈C v is acollection of values which satisfies y v ( λ ) = 1, and X e ∈ ∂ ( v ):( e ′ ,e ) ∈C v y v ( e ′ , e ) ≤ y v ( e ′ ) , (5.1)for each e ′ ∈ C v . Proposition 5.1. Given a collection of values ( y v ( e )) e ∈C v which satisfy (5.1) , there exists a dis-tribution D v supported on C v , such that if Y ∼ D v , then for each e ∈ C v with k := | e | ≥ , it holds hat P [( Y , . . . , Y k ) = ( e , . . . , e k )] = y v ( e ) , (5.2) where Y , . . . , Y k are the first k edges of Y .Proof. Observe first that for each e ′ ∈ E ( ∗ ) , we have that X e ∈ ∂ ( v ):( e ′ ,e ) ∈C v y v ( e ′ , e ) y v ( e ′ ) ≤ y v ( λ ) := 1). We thus define for each e ′ ∈ C v , z v ( e ′ ) := 1 − X e ∈ ∂ ( v ):( e ′ ,e ) ∈C v y v ( e ′ , e ) y v ( e ′ ) , (5.4)which we observe has the property that 0 ≤ z v ( e ′ ) ≤ 1. This leads to the following random-ized rounding algorithm, which we claim outputs a random string Y which satisfies the desiredproperties: Algorithm 9 VertexRound Input: a collection of values ( y v ( e )) e ∈C v satisfying (5.1). Output: a random string Y = ( Y , Y , . . . , Y | U | ) supported on C v . Set e ′ ← λ . Initialize Y i = λ for each i = 0 , . . . , | U | − for i = 0 , . . . , | U | do Exit the “for loop” with probability z v ( e ′ ). ⊲ pass with a certain probability – see (5.4) Draw e ∈ ∂ ( v ) satisfying ( e ′ , e ) ∈ C v with probability y v ( e ′ , e ) / ( y v ( e ′ ) (1 − z v ( e ′ ))). Set Y i = e . e ′ ← ( e ′ , e ). end for return Y = ( Y , Y , . . . , Y | U | ). ⊲ concatenate the edges in order and return the resulting stringClearly, the random string Y is supported on C v , thanks to line 5 of Algorithm 9. We now showthat (5.2) holds. As such, let us first assume k = 1, and e ∈ ∂ ( v ) satisfies ( e ) ∈ C v . Observe that P [ Y = e ] = (1 − z v ( λ )) y v ( e )1 − z v ( λ ) = y v ( e ) , as the algorithm exits the “for loop” with probability z v ( λ ) = 1 − y v ( λ ) = 0, and then draws e withprobability y v ( e ).In general, take k ≥ 2, and assume that for each e ′ ∈ C v with | e ′ | < k , it holds that P [( Y , . . . , Y k ) = e ′ ] = y v ( e ′ ) . If we now fix e = ( e , . . . , e k ) ∈ C v with | e | = k , observe that e 1. We knowhowever that P [ Y k = e k | ( Y , . . . , Y k − ) = e Suppose that A is an optimum relaxed probing algorithm which returns theone-sided matching M after executing on the stochastic graph G = ( U, V, E ). In a slight abuse ofterminology, we say that e is matched by A , provided e is included in M . We shall also make thesimplifying assumption that p e < e ∈ E , as the proof can be clearly adapted to handlethe case when certain edges have p e = 1.Observe that since A is optimum, it is clear that we may assume the following properties holdwithout loss of generality: For each e ∈ E ,1. e is probed only if e can be added to the currently constructed one-sided matching.2. If e is probed and st( e ) = 1, then e is included in M .Thus, in order to prove the lemma, we must find an alternative algorithm B which is non-adaptive,yet continues to remain optimum. To this end, we shall first express E [ w ( M ( v ))] in a convenientform for each v ∈ V , where w ( M ( v )) is the weight of the edge matched to v (which is 0 if no matchoccurs).Given v ∈ V and 1 ≤ i ≤ | U | , we define X vi to be the i th edge which includes v that is probedby A . This is set equal to λ by convention, provided no such edge exists. We may then define X v := ( X v , . . . , X v | U | ). Moreover, given e = ( e , . . . , e k ) ∈ E ( ∗ ) , define S ( e ) to be the event inwhich e k is the only active edge amongst e , . . . , e k . Observe then that (1) and (2) ensure that E [ w ( M ( v ))] = X e =( e ,...,e k ) ∈C v w e k P [ S ( e ) ∩ { X v ≤ k = e } ] , where X v ≤ k := ( X v , . . . , X vk ). Moreover, (1) and (2) imply that if e = ( e , . . . , e k ) ∈ C v , then P [ { st( e k ) = 1 } ∩ { X v ≤ k = e } ] = P [ S ( e ) ∩ { X v ≤ k = e } ] . (5.5)Thus, E [ w ( M ( v ))] = X e =( e ,...,e k ) ∈C v w e k P [ S ( e ) ∩ { X v ≤ k = e } ]= X e =( e ,...,e k ) ∈C v w e k P [ { st( e k ) = 1 } ∩ { X v ≤ k = e } ]= X e =( e ,...,e k ) ∈C v w e k p e k P [ X v ≤ k = e ] , A must decide on whether to probe e k prior to revealing st( e k ).As a result, after summing over v ∈ V , E [ w ( M )] = X v ∈ V X e =( e ,...,e k ) ∈C v w e k p e k P [ X v ≤ k = e ] . (5.6)Our goal is to find a non-adaptive relaxed probing algorithm which matches the value of (5.6).Thus, for each v ∈ V and e = ( e , . . . , e k ) ∈ C v , define x v ( e ) := P [ X v ≤ k = e ] , where x v ( λ ) := 1. This gives us a collection of values, namely ( x v ( e )) e ∈C v , for which the followingconditions hold: For each e ′ ∈ C v , X e ∈ ∂ ( v ):( e ′ ,e ) ∈C v x v ( e ′ , e ) ≤ (1 − p e ) x v ( e ′ ) . (5.7)Now, given e = ( e , . . . , e k ) ∈ C v , define y v ( e ) := x v ( e ) Q | e |− j =1 (1 − p e j ) . (5.8)Observe that (5.7) ensures that for each e ′ ∈ C v X e ∈ ∂ ( v ):( e ′ ,e ) ∈C v y v ( e ′ , e ) ≤ y v ( e ′ ) . (5.9)As a result, Proposition 5.1 implies that for each v ∈ V , there exists a distribution D v such that if Y v ∼ D v , then for each e ∈ C v with | e | = k ≥ P [ Y v ≤ k = e ] = y v ( e ) . (5.10)Moreover, Y v is drawn independently from the edge states, (st( e )) e ∈ E . Consider now the followingalgorithm B , which clearly satisfies the desired properties ( Q ) and ( Q ) of Lemma 3.2: Algorithm 10 Algorithm B Input: a stochastic graph G = ( U, V, E ). Output: a one-sided matching N of G of active edges. Set N ← ∅ . Draw ( Y v ) v ∈ V according to the product distribution Q v ∈ V D v . for v ∈ V do for i = 1 , . . . , | Y v | do Set e ← Y vi . Probe the edge e , revealing st( e ). if st( e ) = 1 and v is unmatched by N then Add e to N . end if end for end for return N . 26sing (5.10) and the non-adaptivity of B , it is clear that for each v ∈ V , E [ w ( N ( v ))] = X e =( e ,...,e k ) ∈C v w e k P [ S ( e )] · P [ Y v ≤ k = e ]= X e =( e ,...,e k ) ∈C v w e k p e k | e |− Y j =1 (1 − p e j ) y v ( e )= X e =( e ,...,e k ) ∈C v w e k p e k x v ( e )= E [ w ( M ( v ))] . Thus, after summing over v ∈ V , it holds that E [ w ( N )] = E [ w ( M )] = OPT rel ( G ), and so inaddition to satisfying ( Q ) and ( Q ), B is optimum. Finally, it is easy to show that each u ∈ U is matched by N at most once in expectation, and so B is a relaxed probing algorithm which isoptimum and satisfies the required properties of Lemma 3.2. Similarly, to the definition of the adaptive benchmark, we define the non-adaptive benchmark ,as the optimum performance of a non-adaptive probing algorithm on G . That is, OPT n-adap ( G ) :=sup B E [ w ( B ( G ))], where the supremum is over all offline non-adaptive probing algorithms. Theupper bound (negative result) of Theorem 2.2 can thus be viewed a statement regarding the powerof adaptivity. More precisely, we define the adaptivity gap of the bipartite stochastic matchingproblem, as the ratio inf G OPT n-adap ( G )OPT( G ) , (6.1)where the infimum is over all bipartite stochastic graphs G .We can therefore restate Theorem 2.2 in the following terminology: Theorem 6.1. The adaptivity gap of the bipartite stochastic matching problem is no smaller than − /e . Theorem 6.1 follows by considering a sequence of stochastic graphs. In particular, given n ≥ p = p ( n ) and s = s ( n ) which satisfy the following:1. p ≪ / √ n and s → ∞ as n → ∞ .2. s ≤ pn and s = (1 − o (1)) pn .Consider now a an unweighted stochastic graph G n = ( U, V, E ) with unit patience values, andwhich satisfies | U | = s and | V | = n . Moreover, assume that p u,v = p for all u ∈ U and v ∈ V .Observe that G n corresponds to the bipartite Erd˝os–R´enyi random graph G ( s, n, p ). Lemma 6.2. The adaptive benchmark returns a matching of size asymptotically equal to s whenexecuting on G n ; that is, OPT ( G n ) = (1 + o (1)) s . 27e defer the proof of Lemma 6.2, as it is routine analysis of the Erd˝os–R´enyi random graph G ( s, n, p ). Instead, we focus on proving the following lemma, which together with Lemma 6.2implies the upper bound of Theorem 6.1: Lemma 6.3. The non-adaptive benchmark returns in expectation a matching of size at most (1 + o (1)) (cid:0) − e (cid:1) s when executing on G n . That is,OPT n-adp ( G ) ≤ (1 + o (1)) (cid:18) − e (cid:19) s. Proof. Let A be a non-adaptive probing algorithm, which we may assume is deterministic withoutloss of generality. As the probes of A are determined independently of the random variables(st( e )) e ∈ E , we can define x e ∈ { , } for each e ∈ E to indicate whether or not A probes the edge e . Now, if A ( G ) is the matching returned by A , then using the independence of the edge states(st( e )) e ∈ E , we get that P [ u matched by A ( G )] = P (cid:20) ∪ v ∈ V : x u,v =1 st( u, v ) = 1 (cid:21) (6.2) ≥ − Y v ∈ V (1 − px u,v ) (6.3)and so, E [ |A ( G )) | ] ≤ s − X u ∈ U Y v ∈ V (1 − px u,v ) . As such, if we can show that X u ∈ U Y v ∈ V (1 − px u,v ) ≥ (1 − o (1)) se , then this will imply that E [ |A ( G ) | ] ≤ (1 + o (1)) (cid:18) − e (cid:19) s. To see this, first observe that since p ( n ) → n → ∞ , we know that1 − px u,v = (1 + o (1)) exp( − px u,v )for each v ∈ V . In fact, since px u,v ≤ p for all v ∈ V , the asymptotics are uniform across V . Moreprecisely, there exists C > 0, such that for n sufficiently large,1 − px u,v ≥ (1 − Cp ) exp( − px u,v )for all v ∈ V . As a result, Y v ∈ V (1 − px u,v ) ≥ (1 − Cp ) n exp − X v ∈ V px u,v ! = (1 + o (1)) exp − X v ∈ V px u,v ! , p ≪ / √ n by assumption. On the other hand, Jensen’s inequalityensures that X u ∈ U exp (cid:0) − P v ∈ V px u,v (cid:1) s ≥ exp (cid:18) − P u ∈ U,v ∈ V px u,v n (cid:19) . However, P u ∈ U x u,v ≤ v ∈ V . Thus, P u ∈ U,v ∈ V px u,v ≤ pn , and soexp (cid:18) − P u ∈ U,v ∈ V px u,v s (cid:19) ≥ exp (cid:16) − pns (cid:17) ≥ e , where the last line follows since pn ≤ s . It follows that X u ∈ U Y v ∈ V (1 − px u,v ) ≥ (1 + o (1)) se , and so E [ |A ( G ) | ] ≤ (1 + o (1)) (cid:18) − e (cid:19) s. As the asymptotics hold uniformly across each deterministic non-adaptive algorithm A , this com-pletes the proof.Theorems 2.1 and 6.1 exactly characterize the adaptivity gap of the bipartite stochastic match-ing problem: Corollary 6.4. The adaptivity gap of the bipartite stochastic matching problem is − /e . We have considered the stochastic bipartite matching problem (with probing constraints) in a fewsettings. As discussed, our results generalize both the classical bipartite matching problem thatdoes not have probing constraints and the prophet inequality and prophet secretary problems. Ouralgorithms are polynomial time assuming a mild assumption on the probing constraints which, inparticular, generalizes the standard patience constraints.Our main results concern stochastic graphs generated from i.d. distributions for which weobtain an optimal competitive ratio for adversarial input sequences and a 1 − /e competitiveratio for random order input sequences. While the i.d. setting subsumes the known stochasticgraph setting, this latter problem is of independent interest as this is the probing model studied byChen et al. [20]. Unlike the classical setting, it is not clear if the Karande et at. [41] result that“ROM implies known (and unknown) i.i.d” holds in the known stochastic graph model.There are some basic questions that are unresolved. Perhaps the most basic question which isalso unresolved in the classical setting is to bridge the gap between the positive 1 − /e competitiveratio and inapproximations in the context of random order input sequences. In terms of the singleitem prophet secretary problem (without probing), Correa et al. [22] obtain a .669 competitiveratio following Azar et al. [8] who were the first to surpass the 1 − /e “barrier”. Note thatin an i.i.d. setting there is no difference between adversarial input sequences and random ordersequences. Correa et al. [22] also establish a .732 inapproximation. Our adaptivity gap proves the29ptimality of the 1 − /e competitive ratio for non-adaptive algorithms. 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The Design of Approximation Algorithms . Cam-bridge University Press, USA, 1st edition, 2011. A Section 3 Additions Suppose that we are given an arbitrary stochastic graph G = ( U, V, E ). Let us restate LP-new forconvenience:maximize X v ∈ V X e ∈C v val( e ) · x v ( e ) (LP-new)subject to X v ∈ V X e ∈C v :( u,v ) ∈ e p u,v · g ( e < ( u,v ) ) · x v ( e ) ≤ ∀ u ∈ U (A.1) X e ∈C v x v ( e ) ≤ ∀ v ∈ V, (A.2) x v ( e ) ≥ ∀ v ∈ V, e ∈ C v (A.3) Proof of Theorem 3.3. Suppose we are presented a feasible solution ( x v ( e )) v ∈ V, e ∈C v to LP-new.Consider then the following relaxed probing algorithm:1. M ← ∅ .2. For each v ∈ V , set e ← VertexProbe ( v, ∂ ( v ) , ( x v ( e )) e ∈C v ). If e = ∅ , then let e = ( u, v )and set M ( v ) = u .3. Return M .Using Lemma 3.4, it is clear that E [ w ( M )] = X v ∈ V X e ∈C v val( e ) · x v ( e ) . Moreover, each vertex u ∈ U is matched by M at most once in expectation, as a consequence ofconstraint (A.1) of LP-new.In order to complete the proof, it remains to show that if A is an optimum relaxed probingalgorithm, then there exists a solution to LP-new whose value is equal to E [ w ( A ( G ))]. Observe35hat w.l.o.g. we may assume that A satisfies properties ( Q ) and ( Q ) of Lemma 3.2. Now, foreach v ∈ V and e = ( e , . . . , e k ) ∈ C v with k := | e | we can define x v ( e ) := P [ A probes the edges ( e i ) ki =1 in order] . Setting M = A ( G ) for convenience, observe that if w ( M ( v )) corresponds to the weight of the edgeassigned to v (which is 0 if no assignment is made), then E [ w ( M ( v ))] = X e ∈C v val( e ) · x v ( e ) , by properties ( Q ) and ( Q ). Similarly, for each u ∈ U , X v ∈ V X e ∈C v :( u,v ) ∈ e p u,v · g ( e < ( u,v ) ) · x v ( e ) ≤ Q ) and ( Q ). The proof is therefore complete. B Section 4 Additions Proof of Theorem 4.1. Suppose that ( H typ , ( D t ) nt =1 ) is a known i.d. instance, where H typ = ( U, B, F ).Recall that C b corresponds to the online probing constraint of each type node b ∈ B . For conve-nience, we denote I := ⊔ b ∈ B C b . We can then define the following collection of random variables,denoted ( X t ( e )) t ∈ [ n ] , e ∈I , based on the following randomized procedure: • Draw the instantiated graph G ∼ ( H typ , ( D t ) nt =1 ), whose vertex arrivals we denote by v , . . . , v n . • Compute an optimum solution of LP-new for G , which we denote by ( x v t ( e )) t ∈ [ n ] , e ∈C vt . • For each t = 1 , . . . , n and e ∈ I , set X t ( e ) = x v t ( e ) if e ∈ C v t , otherwise set X t ( e ) = 0.Observe then that since by definition ( X v t ( e )) t ∈ [ n ] , e ∈C vt is a feasible solution to LP-new for G ,it holds that for each t = 1 , . . . , n X e ∈I X t ( e ) = 1 , (B.1)and X t ∈ [ n ] ,b ∈ B X e ∈I :( u,b ) ∈ e p u,b · g ( e < ( u,b ) ) · X t ( e ) ≤ , (B.2)for each u ∈ U . Moreover, ( X t ( e )) t ∈ [ n ] , e ∈C vt is a optimum solution to LP-new for G , so Theorem3.1 implies that OPT( G ) ≤ LPOPT new ( G ) = n X t =1 X e ∈I val( e ) · X t ( e ) . (B.3)In order to make use of these inequalities in the context of the type graph H typ , let us first fixa type node b ∈ B and a string e ∈ C b . For each t ∈ [ n ], we can then define x t ( e || b ) := E [ X t ( e ) · [ v t = b ] ] , (B.4)36here the randomness is over the generation of G . Observe that by definition of the ( X t ( e )) t ∈ [ n ] , e ∈I values, x t ( e || b ) = 0 , provided e / ∈ C b .We claim that ( x t ( e || b )) b ∈ B,t ∈ [ n ] , e ∈C b is a feasible solution to LP-new-id. To see this, firstobserve that if we multiply (B.1) by the indicator random variable [ b t = v ] , then we get that X e ∈I X t ( e ) · [ v t = b ] = [ v t = b ] . As a result, if we take expectation over this inequality, X e ∈I x t ( e || b ) = X e ∈I E (cid:2) X t ( e ) · [ v t = b ] (cid:3) = P [ v t = b ]=: r t ( b ) , for each b ∈ B and t ∈ [ n ]. Let us now fix u ∈ U . Observe since X t ( e ) · [ v t = b ] = X t ( e ) for each e ∈ C b , (B.2) ensures that X t ∈ [ n ] ,b ∈ B X e ∈C b :( u,b ) ∈ e p u,b · g ( e < ( u,b ) ) · X t ( e ) · [ v t = b ] = X t ∈ [ n ] ,b ∈ B X e ∈C b :( u,b ) ∈ e p u,b · g ( e < ( u,b ) ) · X t ( e ) ≤ X t ∈ [ n ] ,b ∈ B X e ∈C b :( u,b ) ∈ e p u,b · g ( e < ( u,b ) ) · x t ( e || b ) ≤ , for each u ∈ U .Since ( x t ( e || b )) t ∈ [ n ] ,b ∈ B, e ∈C b satisfies these inequalities, and the variables are clearly all non-negative, it follows that ( x t ( e || b )) t ∈ [ n ] ,b ∈ B, e ∈C b is a feasible solution to LP-new-id.In order to complete the proof, let us express the right-hand side of (B.3) as in (B.5) and takeexpectations. We then get that E [OPT( G )] ≤ X b ∈ B,t ∈ [ n ] X e ∈I val( e ) · x t ( e || b ) . Now, OPT( H typ , ( D i ) ni =1 = E [OPT( G )] by definition, so since ( x t ( e || b )) b ∈ B,t ∈ [ n ] , e ∈C b is feasible, itholds that OPT( H typ , ( D i ) ni =1 ) ≤ LPOPT new − id ( H typ , ( D i ) ni =1 ) , thus completing the proof. 37 Extended Related Works Our results pertain to the online stochastic matching problem which (loosely speaking) is onlinebipartite matching where edges are associated with their probabilities of existence. There is asubstantial body of research pertaining to the “classical” (i.e. non stochastic) online bipartite modelin the fully adversarial online model, the random order model, and the i.i.d. input model. The evergrowing interest in various online bipartite matching problems is a reflection of the importanceof online advertising but there are many other natural applications. The literature concerningcompetitive analysis of online bipartite matching is too extensive to do justice to many importantpapers. We refer the reader to the excellent 2013 survey by Mehta [50] with emphasis on onlinevariants relating to ad-allocation. Given the continuing interest in ad-allocation, the survey is notcurrent but does describe the basic results.The seminal result for unweighted online bipartite matching is due to Karp, Vazirani, andVazirani [42]. They gave the randomized Ranking algorithm that achieves competitive ratio 1 − /e in the adversarial online setting which they show is the best possible ratio for any randomizedalgorithm. There have been many proofs of this seminal result, such as the primal-dual approachdue to Devanur et al. [25]. Any greedy algorithm (i.e., one that always makes a match whenpossible) has a 0 . . − /e competitive ratio by their PerturbedRanking algorithm. Huang et al. [39] show that the Perturbed Ranking algorithm obtains a . − /e inapproximation for bipartite matching that applies to the fullyadversarial online model. The i.i.d. online bipartite model has been studied for the unweightedand edge weighted models. The most recent competitive ratios for integral arrival rates are dueto Brubach et al. [16] in which they derive a 0 . . 705 ratio for edge weighted graphs. Karande et al. [41] show that any competitive ratiofor the ROM model applies to the unknown (and therefore known) i.i.d. models. It follows thatany inapproximation for the known i.i.d. model applies to the ROM model. Kesselheim et al.[43] extend the classical secretary result and established the optimal 1 /e ROM ratio for bipartitematching with edge weights.An early example of stochastic probing without commitment is the Pandora’s box problemattributed to Weitzman [58]. In Weitzman’s Pandora’s box problem, a set of boxes is given, whereeach box contains a stochastic value from a known distribution and a cost for opening (i.e., probing)the box. The algorithm has the option at any time of accepting the value of any opened box andpays the total cost of all opened boxes. This is an offline probing problem in that boxes can beopened in any order. An online version of the Pandora’s box problem has recently been studied in Initially, competitive analysis refered to the relative performance (i.e., the competitive ratio) of an online algorithmas compared to an optimal solution (in the worst case over all input sequences determined adversarially). We extendthe meaning of the competitive ratio to also refer to input sequences generated in the ROM model as well as sequencesgenerated i.i.d. from a known or unknown distribution; that is, whenever the algorithm has no control over the orderof input arrivals. greedy algorithm in the unweighted case for arbitrary patience values.They conjectured that their greedy algorithm was a 2-approximation. Subsequently, Adamczyk [1]confirmed that the greedy algorithm is a 2-approximation for the unweighted problem and that thisapproximation is tight. Bansal et al. [9] established a 4-approximation for the edge weighted casewith arbitrary patience and a 3-approximation for the special case of bipartite graphs. Adamczyket al. [3] improved the Bansal et al. bounds providing an approximation algorithm with a ratioof 2 . 845 for bipartite graphs and an algorithm with a ratio of 3 . 709 for general graphs. Bavejaet al. [10] recently improved the analysis of the original algorithm of Bansal et al., yielding anapproximation ratio of 3 . 224 for general graphs.Of particular importance to our paper is the known stochastic matching framework with ROMarrivals, as defined precisely in Section 2. Gamlath et al. [32] presented a probing algorithm whichis a 1 − e -approximation for the bipartite case in the full patience setting ; that is, when there are nopatience restrictions for nodes on either side of the bipartition. The full patience setting is closelyrelated to the bipartite matching algorithm studied by Ehsani et al. [27], which they prove is a1 − e -approximation as a corollary of their work in the more general combinatorial auctions prophetsecretary problem . While not explicitly stated in [27], their bipartite matching algorithm can beinterpreted as an adaptive probing algorithm in the known stochastic matching framework withROM arrivals, attaining the same 1 − e non-adaptive approximation ratio as Gamlath et al.. Veryrecently, Tang et al. [55] provided an alternative algorithm also attaining the same approximationratio of 1 − e in the more general oblivious bipartite matching setting, however their algorithm doesnot execute in an online fashion, and so is incomparable. See also Tang et al. [56] for an onlinegreedy algorithm achieving a 0 . 501 ratio for a known stochastic graph with edge weights.Mehta and Panigrahi [51] adapted the stochastic matching problem to the online setting prob-lem with unit patience where the stochastic graph is not known to the algorithm. They specificallyconsidered the unweighted case for unit patience (for the online nodes) and uniform edge probabil-ities (i.e,, for every edge e , p e = p for some fixed probability p ). They showed that every greedyalgorithm has competitive ratio . In the same online setting, they provided a greedy algorithmthat achieves competitive ratio (1 + (1 − p ) /p ) which limits to (1 + e − ) ≈ . 567 as p → 0. Theyalso show that against a “standard linear programming (LP)” benchmark, that the best possibleratio is 0 . < − e . However, this does not preclude a 1 − e competitive ratio for a stricter LPbound on an optimal stochastic probing algorithm. Preceding the Mehta and Panigrahi work is aresult in Bansal et al. [9] where they consider a known stochastic (type) graph with a distributionon the online nodes. This can be called the stochastic matching problem with known i.i.d. inputs. Unfortunately, approximation and competitive bounds for maximization problems are sometimes representedboth as ratios > < 1. We shall report these ratios as stated in the relevant papers. Our resultswill be stated as fractions. . 92 competitive ratio (or approximately, 0 . 13 as a fraction) in this stochas-tic i.i.d. model. This was improved to 0 . 24 by Adamczyk [3] and most recently, by Brubach et al.[17] where they obtain a 0 . 46 competitive ratio and a 1 − e inapproximation against a standard LP.Returning to the unknown stochastic graph setting, there are recent independent papers byGoyal and Udwani [34] and Brubach et al. [15]. Goyal and Udwani consider the vertex weightedunit patience problem and establish a (best possible) 1 − e competitive ratio against an LP that actsas an upper bound on an optimum offline probing algorithm (the adaptive benchmark) under theassumption that the edge probabilities are decomposable (i.e., p u,v = p u · p v ) and a . 596 competitiveratio for vanishingly small edge probabilities. In a recent paper, Huang and Zhang [40] provide arandomized algorithm for unit patience and offline vertex weights in the online stochastic matchingframework. In the limit as edge probabilities decrease, their algorithm achieves a . 572 competitiveratio. Brubach et al. use and motivate the “ideal stochastic benchmark” (for arbitrary patience)and an LP relaxation for that ideal benchmark. They establish a best possible deterministic competitive ratio against their LP for the vertex weighted online stochastic matching problem. In[13], the authors generalized the patience setting of Brubach et al. to the case of a downward-closed set system on an online node’s allowable sequences of edge probes, thereby attaining thesame 1 / − /e for a number of settings. Finally, we showed thatan (optimal) asymptotic competitive ratio of 1 /e/e