Online Matching with Stochastic Rewards: Optimal Competitive Ratio via Path Based Formulation

TL;DR

This paper advances online stochastic matching by introducing a path-based analysis, achieving optimal competitive ratios for vertex-weighted cases with product-form probabilities and providing a new deterministic algorithm for small heterogeneous probabilities.

Contribution

It presents a novel path-based analysis method and improves competitive ratios for vertex-weighted online matching with stochastic rewards, including a new deterministic algorithm.

Findings

Achieves a (1-1/e) competitive ratio for product-form probabilities.

Provides a deterministic 0.596 competitive algorithm for small heterogeneous probabilities.

Introduces a path-based analysis technique for comparing online algorithms to offline benchmarks.

Abstract

The problem of online matching with stochastic rewards is a generalization of the online bipartite matching problem where each edge has a probability of success. When a match is made it succeeds with the probability of the corresponding edge. Introducing this model, Mehta and Panigrahi (FOCS 2012) focused on the special case of identical edge probabilities. Comparing against a deterministic offline LP, they showed that the Ranking algorithm of Karp et al. (STOC 1990) is 0.534 competitive and proposed a new online algorithm with an improved guarantee of $0.567$ for vanishingly small probabilities. For the case of vanishingly small but heterogeneous probabilities Mehta et al. (SODA 2015), gave a 0.534 competitive algorithm against the same LP benchmark. For the more general vertex-weighted version of the problem, to the best of our knowledge, no results being $1/2$ were previously known even for identical probabilities. We focus on the vertex-weighted version and give two improvements. First, we show that a natural generalization of the Perturbed-Greedy algorithm of Aggarwal et al. (SODA 2011), is $(1-1/e)$ competitive when probabilities decompose as a product of two factors, one corresponding to each vertex of the edge. This is the best achievable guarantee as it includes the case of identical probabilities and in particular, the classical online bipartite matching problem. Second, we give a deterministic $0.596$ competitive algorithm for the previously well studied case of fully heterogeneous but vanishingly small edge probabilities. A key contribution of our approach is the use of novel path-based analysis. This allows us to compare against the natural benchmarks of adaptive offline algorithms that know the sequence of arrivals and the edge probabilities in advance, but not the outcomes of potential matches.

Tables

Table 1. Table 1: Summary of best known competitive ratio guarantees in relevant settings. Here c i subscript 𝑐 𝑖 c_{i} represents the number of copies of each vertex i ∈ I 𝑖 𝐼 i\in I available, with c i = 1 subscript 𝑐 𝑖 1 c_{i}=1 being the most general case. Entries in bold represent our contributions.

	Identical rewards	Vertex-weighted rewards
		$c_i=1$	$c_i\to \infty$
$p_{it}=p\to 0$	$0.567\to \left(\mathrm{𝟏}-\mathrm{𝟏}/𝒆\right)$	$0.5\to \left(\mathrm{𝟏}-\mathrm{𝟏}/𝒆\right)$	$\left(1-1/e\right)$
$p_{it}=p_i{p}_t$	$0.500\to \left(\mathrm{𝟏}-\mathrm{𝟏}/𝒆\right)$
$p_{it}\to 0$	$0.534\to 0.596$	$0.5\to 0.596$
General $p_{it}$	1/2	1/2

Equations

$$\underset{G}{\min }\frac{A(G)}{OPT(G)}.$$

$$OPT(LP)=$$

$$\min$$

Full text

\OneAndAHalfSpacedXI\TheoremsNumberedThrough\ECRepeatTheorems\EquationsNumberedThrough

\RUNAUTHOR

Goyal and Udwani

\RUNTITLE

Online Matching with Stochastic Rewards \TITLEOnline Matching with Stochastic Rewards: Optimal Competitive Ratio via Path Based Formulation

\ARTICLEAUTHORS\AUTHOR

Vineet Goyal \AFFColumbia University, \EMAILvgoyal@ieor.columbia.edu \AUTHOR Rajan Udwani \AFFUC Berkeley, \EMAILrudwani@berkeley.edu

\ABSTRACT

The problem of online matching with stochastic rewards is a generalization of the online bipartite matching problem where each edge has a probability of success. When a match is made it succeeds with the probability of the corresponding edge. Introducing this model, deb focused on the special case of identical edge probabilities. Comparing against a deterministic offline LP, they showed that the Ranking algorithm of Karp et al. (STOC 1990) is 0.534 competitive and proposed a new online algorithm with an improved guarantee of $0.567$ for vanishingly small probabilities. For the case of vanishingly small but heterogeneous probabilities, mehta gave a 0.534 competitive algorithm against the same LP benchmark.

We consider the more general vertex-weighted version of the problem and give two new results. First, we show that a natural generalization of the Perturbed-Greedy algorithm of goel, is $(1-1/e)$ competitive when probabilities decompose as a product of two factors, one corresponding to each vertex of the edge. This is the best achievable guarantee as it includes the case of identical probabilities and in particular, the classical online bipartite matching problem. Second, we give a deterministic $0.596$ competitive algorithm for the previously well studied case of fully heterogeneous but vanishingly small edge probabilities. A key contribution of our approach is the use of novel path-based formulations and a generalization of the primal-dual scheme of devanur. These allow us to compare against the natural benchmarks of adaptive offline algorithms that know the sequence of arrivals and the edge probabilities in advance, but not the outcomes of potential matches. These ideas may be of independent interest in other online settings with post-allocation stochasticity.

\KEYWORDS

Online Matching, Stochastic Rewards, Competitive Ratio, Path Based Analysis, Primal-Dual

1 Introduction

Online bipartite matching and its variants and generalizations have been intensely studied for past few decades (survey). Internet advertising is a major application domain for these problems, along with Crowdsourcing (crowd1, crowd2), personalized recommendations (negin, will), ridesharing (ashlagi, ali, rad), online order order fulfillment (fultut) and many others. Despite a rich body of work on these problems, there are several basic questions that remain open. In this work, we study one such problem. To introduce our setting, we start with the classic online bipartite matching problem.

Online bipartite matching (kvv): Consider a graph $G(I,T,E)$ where the vertices in $I$, called * resources* (may correspond to advertisers in Internet advertising and tasks in Crowdsourcing), are known in advance. Vertices in $T$, called arrivals (corresponding to customers, ad slots, workers), are sequentially revealed one at a time. When a vertex $t\in T$ arrives, the set of edges $(i,t)\in E$ is revealed. Without loss of generality (w.l.o.g.), the vertices in $T$ arrive in the order of their index. After each arrival, the (online) algorithm must make an irrevocable decision to offer at most one (available/unmatched) vertex $i$ with an edge to $t$. Unmatched arrivals depart immediately and cannot be matched later. The goal is to maximize the total number of matches and the performance of online algorithm is compared against an optimal offline algorithm that knows the entire graph $G$ in advance. Formally, let $OPT(G)$ denote the optimal achievable by an offline algorithm and ${A}(G)$ the expected value achieved by a (possibly randomized) online algorithm $A$. The goal is to design an algorithm that maximizes the competitive ratio,

[TABLE]

kvv proposed and analyzed the Ranking algorithm that orders all offline vertices in a random permutation and matches every online vertex to the highest ranked neighbor available. They showed that Ranking attains the best possible competitive ratio of $(1-1/e)$ for online bipartite matching (see baum, goel2, devanur for simplified analysis).

In many applications, matches may fail with non-zero probability and the success or failure of a match is only revealed after the match has been made. This post-allocation stochasticity arises, for instance, in Internet advertising where with some probability a user may not click on the ads shown. In the popular pay-per-click model, an advertiser pays only if the user actually clicks i.e., the match succeeds. Similarly, in personalized recommendations new revenue is generated only if the customer buys an offered product.

Online matching with stochastic rewards: Motivated by the observations above, deb introduced a generalization of online bipartite matching where the objective is to maximize the expected number of successful matches. Each edge $(i,t)$ has a success probability $p_{it}$ that is revealed online along with the edge. When an algorithm makes a match, the arrival accepts (successful match or reward) with probability given by the corresponding edge. The success of a given match is independent of past ones and the outcome of the stochastic reward is revealed only after the match is made. An unsuccessfully matched arrival departs immediately, leaving the resource available for future re-match. A natural generalization of this problem allows vertex-weighted rewards where each resource $i\in I$ has a (possibly different) reward $r_{i}$ 111If rewards are allowed to depend on both the arrival and resource the problem becomes much harder. It was shown in goel that in this case no algorithm with competitive ratio better than O\big{(}\frac{1}{|T|}\big{)} is possible.. This is the setting we study.

The introduction of stochastic rewards raises an interesting question regarding the nature of the offline benchmark to compare against online algorithms. In addition to knowing all edges and their probabilities in advance, should the offline algorithm know the realizations of the rewards too? Unsurprisingly, such a benchmark turns out to be too strong for any meaningful bound222Consider a single arrival with an edge to $n\geq 1$ identical resources. Each edge has a probability of success $1/n$. A benchmark that knows realizations of rewards attains expected reward at least $(1-1/e)$. However, no online algorithm can achieve a reward better than $1/n$.. A natural alternative is to compare against adaptive offline algorithms that know the arrivals, edges and edge probabilities in advance but must sequentially adapt to the outcomes of matches. In particular, rewards are realized only after matching decisions are made and offline can match any arrival at most once. Though offline knows all arrivals in advance, the sequence in which offline makes matching decisions gives rise to different benchmarks.

• •

Clairvoyant: An often used benchmark where the offline algorithm makes matching decisions in order of the arrival sequence i.e., vertex $t\in T$ is matched only after matching decisions for all vertices $t^{\prime}<t$ have been made (negin, will, chan, reuse, feng).

• •

Fully Offline: Algorithm can adaptively choose an order for matching vertices in $T$, given the realized outcomes of previous matches (brubach2, toronto).

Note that in the deterministic case (all probabilities are one) these benchmarks are identical since the order in which vertices are matched does not matter given knowledge of the entire arrival sequence. In general, there are merits to using either benchmark. On one hand, fully offline is a stronger benchmark than clairvoyant. On the other hand, clairvoyant benchmark generalizes quite naturally to other settings with post-allocation stochasticity. For instance, in case of reusable resources where successfully matched resources can be re-matched after some stochastic duration (reuse), allowing offline to match in an order different from the order of arrivals does not lead to a meaningful benchmark. Due to this generalizability, our main focus in this paper is to compare against clairvoyant benchmark. Interestingly, we find that our results hold more strongly against fully offline as well. For simplicity, we refer to clairvoyant and fully offline collectively as offline.

Unfortunately, the performance of offline can be hard to characterize exactly. However, observe that an upper bound on offline is easily given by the optimum of the following LP (see will, negin for a formal proof).

[TABLE]

here $x_{it}$ is a decision variable corresponding to the fraction of resource $i$ matched to arrival $t$. Constraints (1) enforce that the capacity constraint is satisfied in expectation, for every resource. We let the capacity of each resource be 1 w.l.o.g..333Results for the case of unit capacity of each resource generalize to arbitrary capacities. Given a setting with inventory $c_{i}$ for resource $i$, consider a unit inventory setting where in place of resource $i$ we now have $c_{i}$ resources each with inventory of 1 and arrivals that have edges to all $c_{i}$ vertices for every edge $(i,t)$ in the original instance. Offline algorithm knows all arrivals in advance and thus knows these copies represent the same resource. Therefore, the optimal offline value remains unchanged and an algorithm for the unit inventory case can be used for arbitrary inventory levels without loss in guarantee. Similarly, constraints (1) enforce that the matched fractions of every arrival sum to at most 1. We call this the expectation LP as the capacity constraints need only hold in expectation. Due to the simplicity of this LP, when evaluating competitive ratios against offline one often chooses to use the upper bound $OPT(LP)$.

Previous work: negin and mehta showed that it is possible to obtain $1/2$ competitive guarantee against the expectation LP benchmark using deterministic greedy policies. Finding an algorithm that is provably better than naïve greedy is typically the main objective in this class of online allocation problems (msvv, deb, survey). This has proven to be a major challenge for the stochastic rewards setting. Progress has been made in special cases with remarkable new insights. In applications such as online-advertisement, the edge probabilities or click-through-rates are sometimes quite small (survey). Consequently, this is a previously well studied case for the stochastic rewards problem. In particular, deb found that for identical rewards and identical probabilities i.e., $r_{i}=1,\forall i\in I$ and $p_{it}=p\,\forall(i,t)\in E$: Assigning arrivals to available neighbors with the least number of failed matches in the past (called StochasticBalance) is 0.567 competitive as $p\to 0$. They also showed that Ranking is 0.534 approximate for arbitrary value of $p$. Complementing these lower bounds, they showed that no algorithm has a competitive ratio better than $0.621<(1-1/e)$ against the LP. This indicates that perhaps the problem is strictly harder than the classical setting of kvv, where a $(1-1/e)$ guarantee is possible. mehta studied the case of heterogeneous and vanishingly small edge probabilities and gave a 0.534 competitive algorithm (called SemiAdaptive). No results beating $1/2$ are known to us for other cases, or when the rewards $r_{i}$ are not identical, even for identical probabilities.

1.1 Our Results

Our first result establishes the best possible guarantee of $(1-1/e)$ for a special case of online matching with stochastic rewards where every edge probability is decomposable i.e., is a product of two terms corresponding to each of the two vertices444Note that similar special cases have been considered in other online matching settings. For instance, charikar consider a deterministic online matching problem where each edge has a weight $w_{it}$ and the goal is find a matching with maximum total weight. They consider the special case where $w_{it}=w_{i}\times w_{t},\,\forall(i,t)\in E$, and give a $(1-1/e)$ competitive algorithm for this setting..

Theorem 1.1

For decomposable probabilities $p_{it}=p_{i}p_{t}\,\forall(i,t)\in E$, Algorithm 1 is $(1-1/e)$ competitive against offline (both clairvoyant and fully offline). This is the best possible competitive ratio guarantee.

Algorithm 1 computes a randomly perturbed per-unit reward for every resource and makes each match with the goal of maximizing the expected perturbed reward from the match. The algorithm is formally presented in Section 3. Observe that the case of decomposable probabilities (with otherwise arbitrary magnitudes) includes the case of identical probabilities $p_{it}=p\,\forall(i,t)\in E$, for which deb showed that there is no $0.621<(1-1/e)$ competitive algorithm when comparing with $OPT(LP)$. Theorem 1.1 establishes that this upper bound is largely due to the limitations of the expectation LP benchmark (see Section 3.1). To obtain a tighter bound on offline we introduce a novel and intuitive path based program. In contrast to the expectation LP, this program has a constraint for every possible sample path instance of the stochastic reward process, along with constraints that enforce the independence of offline’s decisions from the outcome of stochastic reward. This allows us to achieve the best possible guarantee of $(1-1/e)$ with a natural generalization of an algorithm from deterministic online matching.

Let $p=\max_{(i,t)\in E}p_{it}$, denote the maximum edge probability in a given instance. For general (not necessarily) decomposable probabilities, we establish the following guarantee.

Theorem 1.2

*There exists a constant $\eta>0$ such that Algorithm LABEL:fullada is $(1-\eta\,\sqrt{p})\,0.596$ competitive against offline for $p\leq 0.38$. *

Similar to Algorithm 1, Algorithm LABEL:fullada matches each arrival greedily based on perturbed rewards. However, the perturbations are deterministic and change over time. The algorithm is formally presented in Section LABEL:sec:vanish. Observe that, for instances with vanishingly small edge probabilities i.e., $p\to 0$, the theorem implies a guarantee of 0.596. Recall that the importance of the special case of small probabilities stems from the observation that in applications such as online-advertisement, the probabilities or click-through-rates are often quite small (survey). The path based program that we introduce to prove Theorem 1.1 appears to be insufficient for analyzing non-decomposable probabilities even when they are vanishingly small. To show Theorem 1.2, we develop a novel LP free analysis technique that bypasses the need for an LP based upper bound and uses path-based arguments to directly compare online algorithms with offline.

A well studied regime for online resource allocation problems, including the stochastic rewards setting, is when we have a large number of copies $c_{i}$ for every vertex $i\in I$. In this large capacity regime, previous results show that one can achieve the best possible $(1-1/e)$ guarantee against expectation LP (negin, msvv). Why does the expectation LP benchmark admit this guarantee only for large inventory? To answer this question, in Section LABEL:asymptote we show that when $c_{i}\to+\infty\,\,\forall i\in I$, the expectation LP and offline benchmark converge to the same optimal value.

1.2 Related Work

We briefly review the vast body of literature on online resource allocation, including work we discussed previously to put things into perspective. kvv introduced the online bipartite matching problem and proposed the optimal $(1-1/e)$ competitive Ranking algorithm. baum, goel2 considerably simplified (and completed) the original analysis. Subsequently, goel, proposed the Perturbed-Greedy algorithm and showed that it is $(1-1/e)$ competitive more generally for the case of online vertex-weighted bipartite matching. devanur gave an elegant and intuitive randomized primal-dual framework for proving these results. Their framework applies to and simplifies several other settings, such the related AdWords problem of msvv (and the earlier result for $b$-matchings by pruhs). There have also been a series of results on different types of stochastic arrival models, where there is distributional information in the arrivals that can be exploited for better results (for example, devhay), as well as simultaneous guarantees in adversarial and stochastic arrival settings (for instance, mirrokni). For a detailed survey of these models and results we refer the reader to a monograph by survey.

Moving on to settings with stochastic rewards, deb introduced the model we consider here and focused on the case where all rewards and edge probabilities are identical. They showed the first results beating $1/2$ (and achieving 0.534) for this setting and motivated the case of small edge probabilities. negin, considered a broad generalization of stochastic rewards where one offers a set or an assortment of resources to each arrival. The arrival then chooses at most one resource from this set based on a choice model that is revealed online at the time of arrival. With the objective of maximizing total expected reward, they showed that when the number of copies (inventory) of each resource approaches $+\infty$, an inventory balancing algorithm555A generalization of the algorithm for AdWords in msvv is asymptotically $(1-1/e)$ competitive. For the most general case of unit inventory, their algorithm reduces to the Greedy algorithm and is $1/2$ competitive. Of course, this also establishes a $1/2$ competitive result for stochastic rewards with arbitrary vertex-weights and edge probabilities. The $1/2$ competitiveness of a non-adaptive Greedy algorithm for identical rewards was also shown independently by mehta. They also gave a 0.534 competitive algorithm for identical rewards and heterogeneous but vanishingly small edge probabilities. Note that in contrast to deb and mehta, negin evaluate competitiveness against the clairvoyant benchmark. However, for the sake of analysis they resort to an upper bound on the performance of clairvoyant and use the expectation LP. Clairvoyant algorithms are a commonly used benchmark in various other settings that include the element of stochastic rewards and more generally, other forms of post-allocation stochasticity (for instance, see will, chan, reuse, reuseopt).

Since the online/arXiv version of this paper first appeared, there have been further developments of interest. In concurrent work, brubach2 consider at a setting where for each arrival a finite number of rematches are allowed in case of unsuccessful matches. They observe that in this case the problem for a single arrival is optimally solved by a dynamic program (DP). Using this insight show that the DP based greedy algorithm is 0.5 competitive. Further, they highlight several limitations of the LP benchmark and advocate the use of fully offline as a more natural benchmark for future work. Subsequently, huang gave a 0.572 competitive for heterogeneous and strongly vanishing probabilities ($p=1/f(n)$, where $f(n)$ is a strictly increasing function of the number of resources $n$). Their result is shown against the expectation LP benchmark. Thus, while their benchmark is stronger than offline the stronger assumption on smallness of probabilities and different numerical guarantee (0.572 in huang vs. $1-1/e$ and $0.596$ here), renders their results incomparable with ours. Very recently, unkads analyzed a variant of Algorithm 1 for an online allocation setting with unknown resource capacities. A byproduct of his result is that Algorithm 1 is at least 0.522 competitive for stochastic rewards with vanishing edge probabilities.

There is also a stream of work that looks at the aspect of stochastic rewards with stochastic arrivals and other related (including offline) versions of the problem. In particular, bansal were the first to study stochastic rewards in the context of bipartite matching. They considered an offline probe-commit version of the problem where all edges and probabilities are known in advance and one must probe edges with the restriction that if a probed edge exists, it must be included in the matching. We refer the reader to brubach2 and toronto for a more detailed review of these settings.

Outline for rest of the paper:

We start by introducing notation in Section 2. In Section 3, we consider the case of decomposable probabilities and present Algorithm 1. Section 3.1 discusses the challenges with using the randomized primal-dual framework of devanur for the stochastic rewards problem. This motivates the consideration of our path based analysis in Section LABEL:sec:pbp and Section LABEL:sec:pbdf. In Section LABEL:sec:vanish, we consider the case of small (not necessarily decomposable) probabilities. We start by presenting Algorithm LABEL:fullada, followed by the proof of Theorem 1.2 (Section LABEL:sec:vanish1 to Section LABEL:sec:fsb). Section LABEL:asymptote completes the picture with a convergence result in the large inventory regime. Finally, in Section LABEL:open we conclude with a review of some relevant open problems.

2 Preliminaries

Recall the problem definition. We have a bipartite graph $G$ with a set $I$ of $n$ offline vertices, also called resources. An arbitrary number of vertices $t\in T$ arrive online. We use the index of arrivals $t$ to also denote their order in time. So assume vertex $t\in T$ arrives at time $t$. All edges $(i,t)$ incident on vertex $t$ are revealed when $t$ arrives, along with a corresponding probability of success $p_{it}$. On each new arrival, we must make an irrevocable decision to match the arrival to any one of the available offline neighbors. Once a match, say $(i,t)$, is made, it succeeds with probability $p_{it}$ independent of all other events. If the match succeeds, resource $i$ unavailable to future arrivals and leads to a reward $r_{i}$. The objective is to maximize the expected reward summed over all arrivals. Throughout the paper, we use

Algorithm 1

to denote the online algorithm under consideration as well its expected total reward. Similarly, OPT denotes the (fully) offline benchmark and its expected total reward. W.l.o.g., let OPT be a deterministic algorithm.

Lemma 2.1

The optimal fully offline and clairvoyant algorithms are deterministic and given by a (possibly exponential time) dynamic program.

We provide a proof in Appendix LABEL:appx:trunc. While the ideas in this paper can be used for comparison with randomized benchmark algorithms, the fact that OPT is deterministic simplifies presentation of the key ideas.

For edge $(i,t)\in E$, let $\mathbbm{1}(i,t)$ be an indicator random variable that denotes the success/failure of the edge. So $\mathbbm{1}(i,t)$ equals 1 w.p. $p_{it}$. Recall that, the success/failure of an edge is independent of the outcome of other edges. Let $\omega$ denote a sample path over the success/failure of all edges. Formally, $\omega$ defines for every edge $(i,t)\in E$, the value $\mathbbm{1}(i,t)=\mathbbm{1}^{\omega}(i,t)$ for the success/failure of $(i,t)$. Conditioned on $\omega$, any given (online or offline) algorithm observes the outcome $\mathbbm{1}^{\omega}(i,t)$ after matching $i$ to $t$. Outcomes for edges not included in the matching are not revealed to the algorithm. Let $\Omega$ denote the universe of all sample paths $\omega$. For simplicity, we use $E_{\omega}[\cdot]$ to denote expectation over the randomness in outcomes of all edges. We use $\omega_{\text{-}it}$ to represent part of sample path $\omega$ that defines the success/failure of all edges except edge $(i,t)\in E$. In other words, $\omega_{\text{-}it}$ determines values of random variables $\mathbbm{1}(j,t^{\prime})$, for all edges $(j,t^{\prime})\neq(i,t)$. Overloading notation, we let $E_{\omega}[\cdot\mid\omega_{\text{-}it}]$ denote expectation over randomness in outcomes of edge $(i,t)$, with the outcomes of all other edges fixed according to $\omega_{\text{-}it}$. Let $E_{\omega_{\text{-}it}}[\cdot]$ denote expectation over the randomness in outcomes of all edges except $(i,t)$.

3 Decomposable Probabilities ($\bm{p_{it}=p_{i}\cdot p_{t}}$)

Our first algorithm for online matching with stochastic rewards computes a randomly perturbed per-unit reward for every resource and makes each match with the goal of maximizing the expected perturbed reward from the match.

Algorithm 1 matches greedily w.r.t. expected reduced rewards $p_{it}r_{i}(1-g(y_{i}))$. We refer to random variable $y_{i}$ as the seed of resource $i$. The choice of scaling function $g(\cdot)$ naturally influences the performance of the algorithm. We let the analysis dictate this choice. As it turns out, $g(x)=e^{x-1}$ achieves the best possible performance guarantee of $(1-1/e)$.

When $p_{it}=1\,\;\forall(i,t)\in E$, Algorithm 1 is exactly the Perturbed-Greedy algorithm of goel. devanur show that Perturbed-Greedy is $(1-1/e)$ competitive for deterministic online matching using the primal-dual approach. To analyze Algorithm 1 for stochastic rewards, we first explore a similar primal-dual analysis with the expectation LP benchmark. Since Algorithm 1 offers a natural extension of Perturbed-Greedy, at the outset it might even seem that the analysis in devanur may generalize directly. In the next section, we demonstrate the challenge with this approach through a simple example. Subsequently, we introduce a stronger benchmark, called the path based program (PBP), that we use to show a tight guarantee for Algorithm 1 against offline. Finally, in Section LABEL:sec:obs we discuss (through an example) the challenges with extending the analysis to non-decomposable probabilities.

3.1 Challenges with using Expectation LP Benchmark

Let OPT denote the expected reward of offline. In absence of the stochastic component (all probabilities one), the offline and LP converge and we have $\textsc{OPT}=OPT(LP)$. In general, it is known that there are simple instances with a gap of up to $(1-1/e)$ between OPT and $OPT(LP)$ i.e., $OPT(LP)$ can be as large as $(1-1/e)^{-1}\textsc{OPT}$.666Consider one unit of a single resource with unit reward and $m$ identical arrivals that have an edge to the resource. Suppose that all edge probabilities are identically equal to $1/m$. The best that any offline algorithm can do is to match each arrival as long as the resource is still available. While this has expected reward $(1-1/e)$ for $m\to+\infty$, the optimal value of LP equals 1 (brubach2, deb). However, the gap between $OPT(LP)$ and OPT does not preclude the possibility of showing a strong competitive ratio guarantee against $OPT(LP)$. After all, a $1/2$ competitive result is known against the expectation LP benchmark in even more general settings (negin).

The LP based primal-dual framework of buchbind and devanur is, arguably, the most versatile and general technique for proving guarantees for online matching and related problems. For the stochastic rewards problem, there are fundamental obstacles to using this technique with the expectation LP as a benchmark, and previous work explores novel approaches instead (deb, mehta). Let us understand one such hurdle from the context of the framework in devanur. We start with the dual of the expectation LP,

[TABLE]

Primal-dual certificate (devanur):* To prove $\alpha$ competitiveness for an online algorithm with expected reward *