Attenuate Locally, Win Globally: Attenuation-Based Frameworks for Online Stochastic Matching with Timeouts

Abstract

Online matching problems have garnered significant attention in recent years due to numerous applications in e-commerce, online advertisements, ride-sharing, etc. Many of them capture the uncertainty in the real world by including stochasticity in both the arrival and matching processes. The online stochastic matching with timeouts problem introduced by Bansal et al. (Algorithmica, 2012) models matching markets (e.g., E-Bay, Amazon). Buyers arrive from an independent and identically distributed (i.i.d.) known distribution on buyer profiles and can be shown a list of items one at a time. Each buyer has some probability of purchasing each item and a limit (timeout) on the number of items they can be shown. Bansal et al. (Algorithmica, 2012) gave a 0.12-competitive algorithm which was improved by Adamczyk et al. (ESA, 2015) to 0.24. We present several online attenuation frameworks that use an algorithm for offline stochastic matching as a black box. On the upper bound side, we show that one framework, combined with a black-box adapted from Bansal et al. (Algorithmica, 2012), yields an online algorithm which nearly doubles the ratio to 0.46. Additionally, our attenuation frameworks extend to the more general setting of fractional arrival rates for online vertices. On the lower bound side, we show that no algorithm can achieve a ratio better than 0.632 using the standard LP for this problem. This framework has a high potential for further improvements since new algorithms for offline stochastic matching can directly improve the ratio for the online problem. Our online frameworks also have the potential for a variety of extensions. For example, we introduce a natural generalization: online stochastic matching with two-sided timeouts in which both online and offline vertices have timeouts. Our frameworks provide the first algorithm for this problem achieving a ratio of 0.30. We once again use the algorithm of Bansal et al. (Algorithmica, 2012) as a black-box and plug it into one of our frameworks.

Appendices

Appendix

1.1 Error Accumulation in the First Attenuation Framework

Our simulation-based edge-attenuation approach is very similar to that shown in [1]. For more details, please refer to the Appendix B in [1]. Here we assume that with probability $(1-\epsilon )$, we can similarly obtain that all safe edges with $f_e \ge \epsilon /n$ should be probed with probability $[\alpha f_e/(1+\epsilon ), \alpha f_e/(1-\epsilon )]$ in all rounds. For those edges with $f_e <\epsilon /n$, we add no attenuation.

Now we show that the error from simulation accumulates at most O(1) times in the final ratio. Consider an edge $e=(u,v)$. From the analysis in Theorem 2, we have that in each round, u will be matched with probability at most $F'_u=\frac{1}{n}\sum _{e \sim \partial (u)} \frac{\alpha f_e p_e}{1-\epsilon }+\frac{\epsilon }{n} \le \frac{\alpha }{n}+\frac{2\epsilon }{n}$, if u is safe. Therefore e will be probed with probability at least

$$\begin{aligned} \Pr [e \hbox { is probed}]\ge & {} (1-\epsilon ) \sum _{t=1}^{n} \frac{1}{n} \frac{\alpha f_{e}}{1+\epsilon } \left( 1-\frac{\alpha +2\epsilon }{n}\right) ^{t-1}\\\ge & {} f_e \Big (1-(1-\alpha /n)^n-O(\epsilon )\Big ) \end{aligned}$$

1.2 Simulation-Based Vertex-Attenuation in the Second Attenuation Framework

The vertex-attenuation approach is slightly more involved compared to edge-attenuation. Consider a node u and let $\beta _t$ and $\beta '_t$ be the probability that u is safe at time t before and after attenuation, respectively. Define the event $E_t$ as $\beta '_t \in \left[ (1-1/n)^{t-1}\frac{1}{1+\epsilon }, (1-1/n)^{t-1}\frac{1}{1-\epsilon } \right]$. Here we show how to achieve the goal that $\left( \bigwedge _{t\in [n]} E_t \right)$ occurs with probability at least $1-\epsilon$.

For $t=1$, we do not need any attenuation. From Lemma 3.1, we have that $\beta _2 \ge (1-1/n)$. Let ${\hat{\beta }}_2$ be the estimation obtained from N experiments. Thus we see

$$\begin{aligned} \Pr \left[ |{\hat{\beta }}_2-\beta _2| \ge \epsilon \beta _2 \right] \le 2 \exp \left( -\frac{\epsilon ^2}{3 e} N \right) \end{aligned}$$

Thus by setting $N=\ln (2/\epsilon ) (3e/\epsilon ^2 )$, we claim that with probability at least $1-\epsilon$, ${\hat{\beta }}_2 \in [\beta _2 (1-\epsilon ), \beta _2 (1+\epsilon )]$. The resulting attenuation factor is defined as follows: $\sigma _2=1$ if ${\hat{\beta }}_2 <(1-1/n)$ and $\sigma _2=\frac{1-1/n}{{\hat{\beta }}_2}$ if ${\hat{\beta }}_2 \ge (1-1/n)$. At the beginning of $t=2$, we keep u with probability $\sigma _2$ and throw away u otherwise, and do this independently for other LHS nodes.

Now assume ${\hat{\beta }}_2 \in \left[ \beta _2 (1-\epsilon ), \beta _2 (1+\epsilon ) \right]$ occurs. Note that $\beta '_2=\beta _2*\sigma _2$. We have two cases.

• ${\hat{\beta }}_2 <(1-1/n)$. In this case, we have $\beta _2<(1-1/n)\frac{1}{1-\epsilon }$. And hence, $\beta '_2=\beta _2 \in \left[ (1-1/n), (1-1/n)\frac{1}{1-\epsilon } \right]$.

• ${\hat{\beta }}_2 \ge (1-1/n)$. In this case we have $\beta '_2=\beta _2 \frac{1-1/n}{{\hat{\beta }}_2} \in \left[ (1-1/n)\frac{1}{1+\epsilon }, \right. \left. (1-1/n)\frac{1}{1-\epsilon } \right]$.

Therefore at $t=2$ with probability $1-\epsilon$, the event $E_2$ occurs. Now assume $E_2$ occurs. From Lemma 3.1, we see $\beta _3 \ge (1-1/n) \beta _2' \ge (1-1/n)^2\frac{1}{1+\epsilon }$. Using the same analysis as above and the same value of N, we have that with probability $1-\epsilon$, the event $E_3$ occurs. Similarly the analysis carries through from $E_3$, $E_4$, $\ldots$, $E_n$. Therefore we claim that $\Pr [ \bigwedge _{t\in [n]} E_t] \ge (1-\epsilon )^n \ge 1- n\epsilon$. Finally, in each round we scale down the error probability by a factor of 1 / n.

1.3 Error Accumulation in the Second Attenuation Framework

From the analysis above, we can safely assume that in the second attenuation framework, with probability at least $(1-\epsilon )$, $\Pr [\mathcal {S}_{u,t}] \in [(1-1/n)^{t-1}(1-\epsilon ), (1-1/n)^{t-1}(1+\epsilon )]$ for all u and t. We can achieve this by setting the simulation number in each round as $N=\ln (2nm/\epsilon ) (3e /\epsilon ^2 )$. Condition on this, we show how the error accumulates in the final ratio. Consider a given edge $e=(u,v)$. Notice that $\mathbb {E}[R_{e,\mathbf{f }_{t,v}}\bigm |{\mathcal {S}_{u,t}}] \le (1-1/n)^{t-1}(1+\epsilon )$, which implies that

$$\begin{aligned} \mathbb {E}[{\mathsf {R}}_{{\mathsf {BB}}}[R_{e,\mathbf{f }_{t,v}}]] \ge {\mathsf {R}}_{{\mathsf {BB}}}[(1-1/n)^{t-1}(1+\epsilon )] \ge {\mathsf {R}}_{{\mathsf {BB}}}\left[ (1-1/n)^{t-1}\right] - \epsilon M \end{aligned}$$

where M is a constant upper bound for absolute value of the first derivative of ${\mathsf {R}}_{{\mathsf {BB}}}$ over [0, 1]. Recall that $\mathcal {A}_{e,t}$ is the event that e is effectively probed during round t. Applying the same analysis in Theorem 3, we have

$$\begin{aligned} \Pr [\hbox {e is probed}]\ge & {} (1-\epsilon )\sum _{t=1}^n \Pr [\mathcal {A}_{e,t}] \\\ge & {} (1-\epsilon )^2 \sum _{t=1}^n \frac{f_e}{n}\left( 1-\frac{1}{n}\right) ^{t-1} \left( {\mathsf {R}}_{{\mathsf {BB}}} \left[ \left( 1-\frac{1}{n}\right) ^{t-1}\right] -\epsilon M \right) \\= & {} \int _0^1 e^{-x} {\mathsf {R}}_{{\mathsf {BB}}} [e^{-x}] dx -O(\epsilon ) \end{aligned}$$

Therefore by setting $\epsilon$ small enough, we can get a ratio of $\int _0^1 e^{-x} {\mathsf {R}}_{{\mathsf {BB}}} [e^{-x}] dx -\epsilon$ for any given $\epsilon >0$.

1.4 Simulation-Based Attenuation in the Third Attenuation Framework

For the third attenuation framework, we need the following key ingredient. Suppose we have a random variable X with $\mathbb {E}[X]=\mu \in [\beta -\epsilon ,1]$ where $0<\beta <1$. The random variable models the event that an edge e is probed in some round or a LHS node is safe at some round t. Through analytical analysis, we know a good lower bound $\beta$ for the unknown mean value $\mu$ with error $\epsilon$. Now we need to compute a proper attenuation factor $\sigma \in [0,1]$ such that $\sigma \mu$ is very close to $\beta$ with high probability. Consider the following simulation-based approach: we sample the random variable X for N times and let ${\hat{\mu }}$ be the sample mean; define $\sigma =\beta /{\hat{\mu }}$ if ${\hat{\mu }} \ge \beta$ and $\sigma =1$ otherwise. Assume $\mu \ge \beta -\epsilon \ge \beta /2$.

Lemma A.1

When $N= \frac{6}{\epsilon ^2 \beta } \ln \frac{2}{\delta }$, we have that with probability at least $1-\delta$, $\sigma \mu \in [\beta -\epsilon , \frac{\beta }{1-\epsilon }]$

Proof

By applying Chernoff Bound, we see

$$\begin{aligned} \Pr [|{\hat{\mu }}-\mu | \ge \epsilon \mu ] \le 2 \exp \left( -\frac{\epsilon ^2}{3} N \mu \right) \le 2 \exp \left( -\frac{\epsilon ^2}{3} \frac{\beta N}{2} \right) = \delta \end{aligned}$$

Thus with probability $1-\delta$, ${\hat{\mu }} \in [(1-\epsilon )\mu , (1+\epsilon )\mu ]$. Assume this occurs. Consider the first case ${\hat{\mu }} \ge \beta$. Then $\sigma \mu =\beta \frac{\mu }{{\hat{\mu }}} \in [\beta /(1+\epsilon ), \beta /(1-\epsilon )]$. We are done since $\beta /(1+\epsilon ) \ge \beta -\epsilon$. For the second case ${\hat{\mu }} < \beta$, we see that $\mu \le \beta /(1-\epsilon )$. Thus we have $\sigma \mu =\mu \in [\beta -\epsilon , \beta /(1-\epsilon )]$. $\square$

WLOG assume all $f_e \ge \epsilon /n$ and ${\mathsf {R}}_{{\mathsf {BB}}}$ have finite first derivative and upper bounded by 1 / 2. Recall that in Eq. (3.2), $\alpha _t \ge \alpha _1>0$ and $\gamma _t \in [1/e, 1]$ for all $t\in [n]$.

Consider the first round $t=1$. Consider an edge e and let $\beta '_{e,1}$ be the probability that e is probed during the round $t=1$ before attenuation. Through the analysis in Sect. 3.1, we see $\beta '_{e,1} \ge \alpha _1 f_e$. Let $\beta ''_{e,1}$ be the probability that e is probed during the round $t=1$ after attenuation. From Lemma A.1 and by setting $N=O(\ln (1/\delta )n/\epsilon ^3)$, we have that with probability $1-\delta$, $\beta ''_{e,1} \in [\alpha _1 f_e-\epsilon ,\alpha _1 f_e/(1-\epsilon ) ]$. Let $A_1$ be the event that during round $t=1$, $\beta ''_{e,1} \in [\alpha _1 f_e-\epsilon ,\alpha _1 f_e/(1-\epsilon ) ]$ for all e. By union bound and by setting $N=O(\ln (nm/\delta )n/\epsilon ^3)$ (where $m=|U|$ and $|E| \le mn$), we can ensure $A_1$ occurs with probability $1-\delta$.

Now condition on $A_1$. Consider a node u at the beginning of $t=2$. Let $\gamma '_{u,2}$ and $\gamma ''_{u,2}$ be the probability that u is safe at the beginning of $t=2$ before and after attenuation. We have $\gamma '_{u,2} \ge 1-\frac{\alpha _1 /(1-\epsilon )}{n} \ge \gamma _2-\epsilon _2$ where $\epsilon _2\doteq 2\epsilon /n$. Here WLOG assume $\epsilon <1/2$. Let $B_2$ be the event that $\gamma ''_{u,2}\in [\gamma _2-\epsilon _2, \gamma _2/(1-\epsilon _2)]$ for all $u \in U$. Similarly we can ensure $B_2$ occurs with probability $1-\delta$ by setting $N=O(\ln (m/\delta )n^2/\epsilon ^2)$.

Now condition on both $A_1$ and $B_2$. For a given safe edge e, let $\beta '_{e,2}$ be the probability that e is probed during the round $t=2$ before attenuation. From previous analysis, we have

$$\begin{aligned} \beta '_{e,2} \ge f_e {\mathsf {R}}_{{\mathsf {BB}}}[\gamma _2''] \ge f_e ({\mathsf {R}}_{{\mathsf {BB}}}[\gamma _2+2\epsilon _2] ) \ge f_e \alpha _2-\epsilon _2 \end{aligned}$$

Let $\beta ''_{e,2}$ be the probability that e is probed during the round $t=2$ after attenuation and $A_2$ is the event that $\beta ''_{e,2}\in [f_e \alpha _2-\epsilon _2, f_e \alpha _2/(1-\epsilon _2)]$ for all safe e at $t=2$. Applying Lemma A.1 and union bound, we can make sure that $A_2$ occurs with probability $1-\delta$, by setting $N=O(\ln (mn/\delta ) n^3/\epsilon ^3)$.

Now condition on $A_1, B_2, A_2$. Similarly, let $\gamma '_{u,3}$ and $\gamma ''_{u,3}$ be the probability that u is safe at time $t=3$ before and after attenuation. Note that

$$\begin{aligned} \gamma '_{3,u}\ge & {} \gamma ''_{u,2}\left( 1-\frac{\alpha _2/(1-\epsilon _2)}{n} \right) \ge \left( \gamma _2-\epsilon _2\right) \left( 1-\frac{\alpha _2}{n}-\frac{2\epsilon _2}{n}\right) \\\ge & {} \gamma _3-\epsilon _2 \left( 1+\frac{2}{n}\right) \doteq \gamma _3-\epsilon _3 \end{aligned}$$

Define $B_3$ as the event that $\gamma ''_{u,3}\in [\gamma _3-\epsilon _3, \gamma _3/(1-\epsilon _3)]$ for all $u \in U$. Similarly we can ensure $B_3$ occurs with probability $1-\delta$ by setting $N=O(\ln (m/\delta )n^2/\epsilon ^2)$.

Let $\gamma ''_{u,t}$ be the probability that u is safe at time t and $\beta ''_{e,t}$ the probability that e is probed during round t when it is safe, after attenuation. Define $\epsilon _t=\frac{2\epsilon }{n}\left( 1+\frac{2}{n}\right) ^{t-2}$ for each $t \ge 2$ and $\epsilon _1=\epsilon$ for $t=1$. Similarly, let $B_t$ with $t>1$ be the event that $\gamma ''_{u,t}\in [\gamma _t-\epsilon _t,\gamma _t/(1-\epsilon _t)]$ for all u and $A_t$ with $t \ge 1$ be the event that $\beta ''_{e,t}\in [f_e \alpha _t-\epsilon _t,f_e \alpha _t/(1-\epsilon _t)]$ for all safe e. Doing a similar analysis as above we have the following two observations.

• Condition on $\{A_{t'}, B_{t'} | t'<t\}$. We can ensure that $B_t$occurs with probability $1-\delta$by setting $N=O(\ln (m/\delta )n^2/\epsilon ^2)$.

• Condition on $\{A_{t'}, B_{t'} | t'<t\}$and $B_t$. We can ensure that $A_t$occurs with probability $1-\delta$by setting $N=O(\ln (mn/\delta )n^3/\epsilon ^3)$.

Therefore by setting $\delta =\epsilon /(2n)$, we achieve that with probability $1-\epsilon$, all events in $\{A_t, B_t| t\in [n]\}$ occur. Note that during each round and for each edge or LHS node, our sampling size is $N=O(\ln (mn^2/\epsilon ) n^3/\epsilon ^2)$.

1.5 Error Accumulation in the Third Attenuation Framework

Now assume all events in $\{A_t, B_t| t\in [n]\}$ occur. Consider an edge $e=(u,v)$. The performance should be at least

$$\begin{aligned} (1-\epsilon )\sum _{t=1}^n \frac{\gamma ''_{u,t} \beta ''_{e,t}}{n}\ge (1-\epsilon ) \sum _{t=1}^n \frac{(\gamma _t-\epsilon )(\alpha _t f_e-\epsilon )}{n}=\sum _{t=1}^n \frac{\gamma _t \alpha _t f_e}{n}-O(\epsilon ) \end{aligned}$$

This completes the description of the error analysis.

Summary of Notation

In this section, we summarize all of the notation used throughout the paper.

Notation	Usage
n	Total number of online rounds
$r_v$	Expected number of arrivals of vertex v in the n online rounds
$\partial (u)$, $\partial (e)$	The set of edges incident to vertex u and edge e respectively
${\mathbf {f}}$	Optimal solution to LP (2.1)
$F_u$	$\sum _{e \in \partial (u)} f_e p_e$
${\mathbf {g}}$	Scaled fractional solution feasible to LP (2.1)
$G_t(v)$	The (random) star graph incident to v with all of the safe neighbors at time t in a run of the algorithm
$\hat{{\mathbf {G}}}$	(Random) integral solution obtained by running GKPS on ${\mathbf {g}}$
$S_{u, t}$	Event (random) that u is safe at time t in any run of the algorithm
$\mathcal {A}_{e, t}$	Event (random) that for an edge $e=(u, v)$, v comes at time t, u is safe and e is probed
$\lambda (e, {\mathbf {g}})$	$\sum _{e' \ne e} g_{e'} p_{e'}$