Near Optimal Online Algorithms and Fast Approximation Algorithms for Resource Allocation Problems

TL;DR

This paper develops near-optimal online algorithms for resource allocation that adapt to unknown distributions and adversarial inputs, providing fast approximation solutions for large-scale LPs with broad applications.

Contribution

It introduces a unified framework for robust online resource allocation algorithms that work under both stochastic and adversarial models, with near-optimal tradeoffs and fast LP-solving methods.

Findings

Achieved a $1- ext{epsilon}$ competitive ratio for online algorithms under unknown distributions.

Designed algorithms that perform well even with adversarially controlled request distributions.

Provided fast approximation algorithms for large LPs with packing and covering constraints.

Abstract

We present prior robust algorithms for a large class of resource allocation problems where requests arrive one-by-one (online), drawn independently from an unknown distribution at every step. We design a single algorithm that, for every possible underlying distribution, obtains a $1-\epsilon$ fraction of the profit obtained by an algorithm that knows the entire request sequence ahead of time. The factor $\epsilon$ approaches $0$ when no single request consumes/contributes a significant fraction of the global consumption/contribution by all requests together. We show that the tradeoff we obtain here that determines how fast $\epsilon$ approaches $0$, is near optimal: we give a nearly matching lower bound showing that the tradeoff cannot be improved much beyond what we obtain. Going beyond the model of a static underlying distribution, we introduce the adversarial stochastic input model, where an adversary, possibly in an adaptive manner, controls the distributions from which the requests are drawn at each step. Placing no restriction on the adversary, we design an algorithm that obtains a $1-\epsilon$ fraction of the optimal profit obtainable w.r.t. the worst distribution in the adversarial sequence. In the offline setting we give a fast algorithm to solve very large LPs with both packing and covering constraints. We give algorithms to approximately solve (within a factor of $1+\epsilon$) the mixed packing-covering problem with $O(\frac{\gamma m \log (n/\delta)}{\epsilon^2})$ oracle calls where the constraint matrix of this LP has dimension $n\times m$, the success probability of the algorithm is $1-\delta$, and $\gamma$ quantifies how significant a single request is when compared to the sum total of all requests. We discuss implications of our results to several special cases including online combinatorial auctions, network routing and the adwords problem.

Tables

LP relaxations for random and expected instances

(1)

LP relaxations for Adwords and Display Ads

(2)

LPs for random and expected instances

(3)

Primal and dual LPs defining $e_r$

(9)

Primal and dual LPs defining the expected instance

(10)

Primal and dual LPs defining the expected instance

(15)

Primal and dual LPs corresponding to integer program in (6)

(23)

Sampled primal and dual LPs

(24)

Equations

$$\text{Maximize }\underset{i\in \mathcal{A}}{\min }\sum _{j\in \mathcal{J}}{\mathbf{w}}_{i,j}\cdot {\mathbf{x}}_j\text{ s.t.}$$

$$\forall i\in \mathcal{A},\sum _{j\in \mathcal{J}}{\mathbf{a}}_{i,j}\cdot {\mathbf{x}}_j\le c_i$$

$$\forall j\in \mathcal{J},\sum _{k\in \mathcal{K}}{x}_{j,k}\le 1$$

$$\forall j\in \mathcal{J},k\in \mathcal{K},x_{j,k}\ge 0$$

$$\mathrm{Pr}[X<\mu (1-ϵ)]<\exp \left(\frac{-{ϵ}^2\mu }{2B}\right).$$

$$X-\mu \ge -\sqrt{2\mu B\ln (1/\delta )}.$$

$$\mathrm{Pr}[X>\mu (1+ϵ)]<\exp \left(\frac{-{ϵ}^2\mu }{4B}\right).$$

$$X-\mu \le \sqrt{4\mu B\ln (1/\delta )}.$$

$$\mathrm{Pr}[X>\mu (1+ϵ)]<2^{-(1+ϵ)\mu /B}.$$

$$\displaystyle\mathop{\operatorname{\bf Pr}}\bigg{[}\sum_{t=1}^{m}\widetilde{X}_{i,t}\geq\frac{c_{i}}{1+\epsilon}(1+\epsilon)\bigg{]}$$

$$\displaystyle=\mathop{\operatorname{\bf Pr}}\bigg{[}(1+\epsilon)^{\frac{\sum_{t=1}^{m}\widetilde{X}_{i,t}}{\gamma c_{i}}}\geq(1+\epsilon)^{\frac{1}{\gamma}}\bigg{]}$$

$$\displaystyle\leq\mathop{\operatorname{\bf E}}\bigg{[}(1+\epsilon)^{\frac{\sum_{t=1}^{m}\widetilde{X}_{i,t}}{\gamma c_{i}}}\bigg{]}/(1+\epsilon)^{\frac{1}{\gamma}}$$

$$\displaystyle=\mathop{\operatorname{\bf E}}\bigg{[}\prod_{t=1}^{m}(1+\epsilon)^{\frac{\widetilde{X}_{i,t}}{\gamma c_{i}}}\bigg{]}/(1+\epsilon)^{\frac{1}{\gamma}}$$

$$\displaystyle\leq\mathop{\operatorname{\bf E}}\bigg{[}\prod_{t=1}^{m}\left(1+\epsilon{\frac{\widetilde{X}_{i,t}}{\gamma c_{i}}}\right)\bigg{]}/(1+\epsilon)^{\frac{1}{\gamma}}$$

$$\displaystyle\leq\bigg{[}\prod_{t=1}^{m}\left(1+\frac{\epsilon}{(1+\epsilon)\gamma m}\right)\bigg{]}/(1+\epsilon)^{\frac{1}{\gamma}}$$

$$\le {\left(\frac{e^{ϵ}}{(1+ϵ{)}^{1+ϵ}}\right)}^{\frac{1}{\gamma (1+ϵ)}}$$

$$\le e^{\frac{-{ϵ}^2}{4\gamma }\frac{1}{1+ϵ}}$$

$$\le \frac{ϵ}{2n}$$

$$\displaystyle\mathop{\operatorname{\bf Pr}}\bigg{[}\sum_{t=1}^{m}\widetilde{Y}_{i,t}\leq\frac{W_{E}}{1+\epsilon}(1-\epsilon)\bigg{]}$$

$$\displaystyle=\mathop{\operatorname{\bf Pr}}\bigg{[}(1-\epsilon)^{\frac{\sum_{t=1}^{m}\widetilde{Y}_{i,t}}{\gamma W_{E}}}\geq(1-\epsilon)^{\frac{1-\epsilon}{\gamma(1+\epsilon)}}\bigg{]}$$

$$\displaystyle\leq\mathop{\operatorname{\bf E}}\bigg{[}(1-\epsilon)^{\frac{\sum_{t=1}^{m}\widetilde{Y}_{i,t}}{\gamma W_{E}}}\bigg{]}/(1-\epsilon)^{\frac{1-\epsilon}{\gamma(1+\epsilon)}}$$

$$\displaystyle=\mathop{\operatorname{\bf E}}\bigg{[}\prod_{t=1}^{m}(1-\epsilon)^{\frac{\widetilde{Y}_{i,t}}{\gamma W_{E}}}\bigg{]}/(1-\epsilon)^{\frac{1-\epsilon}{\gamma(1+\epsilon)}}$$

$$\displaystyle\leq\mathop{\operatorname{\bf E}}\bigg{[}\prod_{t=1}^{m}\left(1-\epsilon{\frac{\widetilde{Y}_{i,t}}{\gamma W_{E}}}\right)\bigg{]}/(1-\epsilon)^{\frac{1-\epsilon}{\gamma(1+\epsilon)}}$$

$$\displaystyle\leq\bigg{[}\prod_{t=1}^{m}\left(1-\frac{\epsilon}{(1+\epsilon)\gamma m}\right)\bigg{]}/(1-\epsilon)^{\frac{1-\epsilon}{\gamma(1+\epsilon)}}$$

$$\le {\left(\frac{e^{-ϵ}}{(1-ϵ{)}^{1-ϵ}}\right)}^{\frac{1}{\gamma (1+ϵ)}}$$

$$\le e^{\frac{-{ϵ}^2}{2\gamma }\frac{1}{1+ϵ}}$$

$$\le \frac{ϵ}{2n}$$

$$\displaystyle\mathop{\operatorname{\bf Pr}}\bigg{[}\sum_{t=1}^{m}X_{i,t}^{A}\geq\frac{c_{i}}{1+\epsilon}(1+\epsilon)\bigg{]}$$

$$\displaystyle=\mathop{\operatorname{\bf Pr}}\bigg{[}(1+\epsilon)^{\frac{\sum_{t=1}^{m}X_{i,t}^{A}}{\gamma c_{i}}}\geq(1+\epsilon)^{\frac{1}{\gamma}}\bigg{]}$$

$$\displaystyle\leq\mathop{\operatorname{\bf E}}\bigg{[}(1+\epsilon)^{\frac{\sum_{t=1}^{m}X_{i,t}^{A}}{\gamma c_{i}}}\bigg{]}/(1+\epsilon)^{\frac{1}{\gamma}}$$

$$\displaystyle=\mathop{\operatorname{\bf E}}\bigg{[}\prod_{t=1}^{m}(1+\epsilon)^{\frac{X_{i,t}^{A}}{\gamma c_{i}}}\bigg{]}/(1+\epsilon)^{\frac{1}{\gamma}}$$

$$\displaystyle\mathop{\operatorname{\bf Pr}}\bigg{[}\sum_{t=1}^{m}Y_{i,t}^{A}\leq\frac{W_{E}}{1+\epsilon}(1-\epsilon)\bigg{]}$$

$$\displaystyle=\mathop{\operatorname{\bf Pr}}\bigg{[}(1-\epsilon)^{\frac{\sum_{t=1}^{m}Y_{i,t}^{A}}{\gamma W_{E}}}\geq(1-\epsilon)^{\frac{1-\epsilon}{\gamma(1+\epsilon)}}\bigg{]}$$

$$\displaystyle\leq\mathop{\operatorname{\bf E}}\bigg{[}(1-\epsilon)^{\frac{\sum_{t=1}^{m}Y_{i,t}^{A}}{\gamma W_{E}}}\bigg{]}/(1-\epsilon)^{\frac{1-\epsilon}{\gamma(1+\epsilon)}}$$

$$\displaystyle=\mathop{\operatorname{\bf E}}\bigg{[}\prod_{t=1}^{m}(1-\epsilon)^{\frac{Y_{i,t}^{A}}{\gamma W_{E}}}\bigg{]}/(1-\epsilon)^{\frac{1-\epsilon}{\gamma(1+\epsilon)}}$$

$$\sum_{i}\frac{\mathop{\operatorname{\bf E}}\bigg{[}\prod_{t=1}^{m}\left(1+\epsilon\right)^{\frac{X_{i,t}^{A}}{\gamma c_{i}}}\bigg{]}}{(1+\epsilon)^{\frac{1}{\gamma}}}+\sum_{i}\frac{\mathop{\operatorname{\bf E}}\bigg{[}\prod_{t=1}^{m}\left(1-\epsilon\right)^{\frac{Y_{i,t}^{A}}{\gamma W_{E}}}\bigg{]}}{(1-\epsilon)^{\frac{1-\epsilon}{\gamma(1+\epsilon)}}}\leq\epsilon$$

$$\sum_{i}\frac{\mathop{\operatorname{\bf E}}\bigg{[}\left(1+\epsilon\right)^{\frac{S_{s}\left(X_{i}^{A}\right)}{\gamma c_{i}}}\prod_{t=s+1}^{m}\left(1+\epsilon\right)^{\frac{\widetilde{X}_{i,t}}{\gamma c_{i}}}\bigg{]}}{(1+\epsilon)^{\frac{1}{\gamma}}}+\sum_{i}\frac{\mathop{\operatorname{\bf E}}\bigg{[}\left(1-\epsilon\right)^{\frac{S_{s}\left(Y_{i}^{A}\right)}{\gamma W_{E}}}\prod_{t=s+1}^{m}\left(1-\epsilon\right)^{\frac{\widetilde{Y}_{i,t}}{\gamma W_{E}}}\bigg{]}}{(1-\epsilon)^{\frac{1-\epsilon}{\gamma(1+\epsilon)}}},$$

$$\sum_{i}\frac{\mathop{\operatorname{\bf E}}\bigg{[}\left(1+\epsilon\right)^{\frac{S_{s}\left(X_{i}^{A}\right)}{\gamma c_{i}}}\prod_{t=s+1}^{m}\left(1+\epsilon{\frac{\widetilde{X}_{i,t}}{\gamma c_{i}}}\right)\bigg{]}}{(1+\epsilon)^{\frac{1}{\gamma}}}+\sum_{i}\frac{\mathop{\operatorname{\bf E}}\bigg{[}\left(1-\epsilon\right)^{\frac{S_{s}\left(Y_{i}^{A}\right)}{\gamma W_{E}}}\prod_{t=s+1}^{m}\left(1-\epsilon{\frac{\widetilde{Y}_{i,t}}{\gamma W_{E}}}\right)\bigg{]}}{(1-\epsilon)^{\frac{1-\epsilon}{\gamma(1+\epsilon)}}}.$$

$$\sum_{i}\frac{\mathop{\operatorname{\bf E}}\bigg{[}\left(1+\epsilon\right)^{\frac{S_{s}\left(X_{i}^{A}\right)}{\gamma c_{i}}}\left(1+\frac{\epsilon}{(1+\epsilon)\gamma m}\right)^{m-s}\bigg{]}}{(1+\epsilon)^{\frac{1}{\gamma}}}+\sum_{i}\frac{\mathop{\operatorname{\bf E}}\bigg{[}\left(1-\epsilon\right)^{\frac{S_{s}\left(Y_{i}^{A}\right)}{\gamma W_{E}}}\left(1-\frac{\epsilon}{(1+\epsilon)\gamma m}\right)^{m-s}\bigg{]}}{(1-\epsilon)^{\frac{1-\epsilon}{\gamma(1+\epsilon)}}}.$$

$$\mathcal{F}[A^{p+1}{\tilde{P}}^{m-p-1}]$$

$$\displaystyle\qquad\qquad\qquad\sum_{i}\frac{\mathop{\operatorname{\bf E}}\bigg{[}(1-\epsilon)^{\frac{S_{p+1}\left(Y_{i}^{A}\right)}{\gamma W_{E}}}\left(1-\frac{\epsilon}{(1+\epsilon)\gamma m}\right)^{m-p-1}\bigg{]}}{(1-\epsilon)^{\frac{1-\epsilon}{\gamma(1+\epsilon)}}}$$

$$\displaystyle\leq\sum_{i}\frac{\mathop{\operatorname{\bf E}}\bigg{[}(1+\epsilon)^{\frac{S_{p}\left(X_{i}^{A}\right)}{\gamma c_{i}}}\left(1+\epsilon{\frac{X_{i,p+1}^{A}}{\gamma c_{i}}}\right)\left(1+\frac{\epsilon}{(1+\epsilon)\gamma m}\right)^{m-p-1}\bigg{]}}{(1+\epsilon)^{\frac{1}{\gamma}}}\qquad+$$

$$\displaystyle\qquad\qquad\qquad\sum_{i}\frac{\mathop{\operatorname{\bf E}}\bigg{[}(1-\epsilon)^{\frac{S_{p}\left(Y_{i}^{A}\right)}{\gamma W_{E}}}\left(1-\epsilon{\frac{Y_{i,p+1}^{A}}{\gamma W_{E}}}\right)\left(1-\frac{\epsilon}{(1+\epsilon)\gamma m}\right)^{m-p-1}\bigg{]}}{(1-\epsilon)^{\frac{1-\epsilon}{\gamma(1+\epsilon)}}}$$

$${\varphi }_{i,s}$$

Full text

Going beyond the model of a static underlying distribution, we introduce the adversarial stochastic input model, where an adversary, possibly in an adaptive manner, controls the distributions from which the requests are drawn at each step. Placing no restriction on the adversary, we design an algorithm that obtains a $1-\epsilon$ fraction of the optimal profit obtainable w.r.t. the worst distribution in the adversarial sequence. Further, if the algorithm is given one number per distribution, namely, the optimal profit possible for each of the adversary’s distribution, we design an algorithm that achieves a $1-\epsilon$ fraction of the weighted average of the optimal profit of each distribution the adversary picks.

In the offline setting we give a fast algorithm to solve very large LPs with both packing and covering constraints. We give algorithms to approximately solve (within a factor of $1+\epsilon$) the mixed packing-covering problem with $O(\frac{\gamma m\log(n/\delta)}{\epsilon^{2}})$ oracle calls where the constraint matrix of this LP has dimension $n\times m$, the success probability of the algorithm is $1-\delta$, and $\gamma$ quantifies how significant a single request is when compared to the sum total of all requests.

We discuss implications of our results to several special cases including online combinatorial auctions, network routing and the adwords problem.

1 Introduction & Summary of Results

There has been an increasing interest in online algorithms for resource allocation problems motivated by their wide variety of applications in Internet advertising, allocating multi-leg flight seats for customers online, allocating goods to customers arriving online in a combinatorial auction etc. Designing efficient resource allocation algorithms has significant scientific and commercial value. The traditional computer science approach to deal with uncertain future inputs has been the worst-case competitive analysis. Here nothing is assumed about the sequence of requests that arrive online, and the benchmark is the optimal algorithm that knows the entire sequence of requests ahead of time. Several problems in this space have been analyzed in the traditional framework, exemplified, for instance, in the well-studied Adwords problem introduced by Mehta et al. (2005). While worst-case analysis is a robust framework, for many problems it leads to pessimistic bounds that rule out obtaining more than a constant fraction of the optimal profit. Consequently, there has been a drive in the last few years to go beyond worst-case analysis. A frequently used alternative is to perform stochastic analysis: assume that the input is drawn from a known distribution and optimize the objective w.r.t. this distribution. While stochastic analysis circumvents the impossibility results in worst-case analysis, any error in the knowledge of distribution could render the algorithm suboptimal, and sometimes even infeasible.

In this paper, we study a middle ground between worst-case and stochastic settings. We assume that the input is drawn from an underlying distribution that is unknown to the algorithm designer. We present a single algorithm, that for every distribution performs nearly as well as the optimal algorithm that knows the entire sequence of requests ahead of time. In this sense, the algorithm is prior robust.

We now give an informal description of the resource allocation framework and our main contributions. See Section 2 for a formal description and theorem statements. We consider a resource allocation setting where requests arrive online; every request can be served by some subset of several available options; each (request, option) pair consumes some amount of every resource, and generates some profit. There is a given budget for each resource. Requests are drawn i.i.d. from an unknown distribution. The goal is to maximize the total profit generated while making sure that the total consumption of each resource is no more than the corresponding budget. We compare the profit of the algorithms against the offline optimum and prove competitive ratio bounds. Even for very restricted special cases of this problem, the worst-case setting cannot yield anything beyond a $1-\frac{1}{e}$ competitive ratio Kalyanasundaram and Pruhs (1996); Mehta et al. (2005). While the stochastic setting with a fully known distribution can give near optimal performance guarantees, it often leads to very distribution dependent algorithms (e.g. see Alaei et al. (2012) for the special case of the adwords problem, which requires knowledge of the entire distribution to perform the optimization). Hence both these approaches are not satisfactory, and this problem lends itself well to the middle ground of prior robust analysis.

Going beyond i.i.d., our work introduces the adversarial stochastic input (ASI) model as a more realistic model for analyzing online algorithmic problems. Here the distribution from which the requests arrive is allowed to change over time (unlike i.i.d., where it stays identical for every request). The adversary decides how to pick the distributions, and is even allowed to pick them adaptively. For many practical applications such as in display advertising, the distribution of requests shows trends that change over the course of time: mornings are different from evenings and weekdays are different from weekends. Thus a time varying distributional model is more realistic than the i.i.d. model. A keen reader might notice that the above description includes the worst-case setting as well, therefore we have to make some extra assumptions, either by restricting how these distributions can be picked, or by allowing the algorithm some extra information about the distributions. We will describe these in greater detail later on.

Apart from the theoretical contribution, the algorithms we design for the ASI models were used to completely overhaul the display advertising management system at Microsoft, leading to a significant improvement in revenue ($\approx 10\%$), better system manageability and enabling new capabilities.111This system had been globally operational from 2011 to 2015, when Microsoft made a deal with AOL to allow AOL to sell the display advertisement on behalf of Microsoft. We believe that our results make a significant contribution to the search for “allows-positive-results-yet-realistic” models for online algorithms.

First Result: Near-Optimal Prior Robust Online Algorithms for Resource Allocation

Problems

A key parameter on which algorithms for several resource allocation problems depend on is the relative significance of any single request when compared to the entire sequence of requests. For instance, for the special case of the Adwords problem, this is the ratio of a single bid to an advertiser’s budget. For the Adwords problem, Mehta et al. (2005) and Buchbinder et al. (2007) design an algorithm that achieves a worst case competitive ratio that tends to $1-1/e$ as the bid to budget ratio (which we denote by $\gamma$) tends to 0.222Note that $\gamma$ approaching zero is the easiest case. Even with $\gamma$ approaching zero, $1-1/e$ is the best competitive ratio that any randomized algorithm can achieve in the worst case, illustrating how worst-case analysis leads to pessimistic bounds. Devanur and Hayes (2009) studied the same problem in the random permutation model, and showed that the competitive ratio tends to 1 as $\gamma$ tends to 0. This result showed that competitive ratio of algorithms in stochastic models could be much better than that of algorithms in the worst case. The important question since then has been to determine the optimal trade-off between $\gamma$ and the competitive ratio. Devanur and Hayes (2009) showed how to get a 1- $O(\epsilon)$ competitive ratio when $\gamma$ is at most $O(\frac{\epsilon^{3}}{n\log(mn/\epsilon)})$ where $n$ is the number of advertisers and $m$ is the number of keywords. Subsequently Agrawal et al. (2014) improved the bound on $\gamma$ to $O(\frac{\epsilon^{2}}{n\log(m/\epsilon)})$. The papers of Feldman et al. (2010) and  Agrawal et al. (2014) have also shown that the technique of Devanur and Hayes (2009) can be extended to other online problems.

The first main result in this paper is the following $3$-fold improvement of previous results (Theorems 2.2 and 2.3), for the i.i.d. with unknown distributions model. All our results apply to the general class of problems that we call the resource allocation framework. A formal definition of the framework is presented in Section 2.2 and a discussion of many interesting special cases including online network routing and online combinatorial auctions is presented in Section 7.

We give an algorithm which guarantees a $1-\epsilon$ approximation factor when $\gamma=O(\frac{\epsilon^{2}}{\log(n/\epsilon)})$. 2. 2.

We show that our bound on $\gamma$ is almost optimal; we show that no algorithm, even if it knew the distribution, can guarantee a $1-\epsilon$ approximation when $\gamma=\omega(\frac{\epsilon^{2}}{\log(n)})$. 3. 3.

Our algorithms lend themselves to natural generalizations that provide identical guarantees in the more general adversarial stochastic input (ASI) model that was described earlier. We provide three different versions of the ASI model in Section 3.5.

Significance

Regarding the bound on $\gamma$, we remove a factor of $n$ from $\gamma$, making the algorithm more practical. Consider for instance the Adwords problem and suppose that the bids are all in [0,1]. The earlier bound implies that the advertiser budgets need to be of the order of $n\log n/\epsilon^{2}$ in order to get a $1-\epsilon$ competitive algorithm, where $n$ is the number of advertisers. With realistic values for these parameters, it seems unlikely that this condition would be met. While with the improved bounds presented in this paper, we only need the advertiser budget to be of the order of $\log n/\epsilon^{2}$ and this condition is met for reasonable values of the parameters. Furthermore, in the more general resource allocation framework, the previous best upper bound on $\gamma$ is from Agrawal, Wang and Ye Agrawal et al. (2014) and equals $O(\frac{\epsilon^{2}}{n\log(mK/\epsilon)})$. Here $K$ is the number of available “options” (see Section 2.2) and in typical applications like network routing, $K$ could be exponential in $n$, and thus, the factor saved by our algorithm becomes quadratic in $n$. 2. 2.

Our ASI models are realistic models of time varying distributions for which we present algorithms with asymptotically optimal performance guarantees. We consider three different benchmarks, each progressively stronger than the previous, and require different levels of information about the distributions to achieve near optimal performance guarantees. For the weakest benchmark, we need just one parameter from the distribution, while for the strongest benchmark, we still need only $2mn$ parameters. Note that the distributions themselves can have an arbitrarily large support size333 We even allow continuous distributions which have an infinite support. and hence the amount of information we need is much smaller than the description of all the distributions. Our results for the ASI model can be thought of as generalizations of the “Prophet Inequality”.444The Prophet Inequality is essentially a $1/2$-competitive algorithm for the following problem: a sequence of values is drawn independently from different distributions and presented one at a time. The algorithm may pick at most one of these values in an online manner, given some knowledge of the distributions, such as the expectation of the maximum of these values. The goal is to maximize the value picked. See Samuel-Cahn (1984) Finally, as mentioned earlier, our algorithms for this model have made a significant impact on the practice of display advertising management at Microsoft.

Second Result: Prior Robust $1-1/e$ approximation Greedy Algorithm

for Adwords

A natural algorithm for the Adwords problem that is widely used for its simplicity is the greedy algorithm: always match an incoming query to the advertiser that has the maximum effective bid (the minimum of bid and remaining budget) for that query. Because of its wide use, previously the performance of the greedy algorithm has been analyzed by Goel and Mehta Goel and Mehta (2008) who showed that in the random permutation and the i.i.d. models, it has a competitive ratio of $1-1/e$ with an assumption which is essentially that $\gamma$ tends to 0.

It has been an important open problem to analyze the performance of greedy algorithm in a stochastic setting for unbounded $\gamma$, i.e., for all $0\leq\gamma\leq 1$. The best factor known so far is $1/2$, and this works for the worst case also. Nothing better was known, even in the stochastic models. The second result in this paper is that for the Adwords problem in the i.i.d. unknown distributions model, with no assumption on $\gamma$ (i.e., $\gamma$ could be as big as $1$), the greedy algorithm gets an approximation factor of $1-1/e$ against the optimal fractional solution to the expected instance (Theorem 2.4).

Our proof technique for this result has been subsequently used to prove a similar result for the greedy algorithm in online submodular welfare maximization Kapralov et al. (2013). We note here that there are other algorithms that achieve a $1-1/e$ approximation for the Adwords problem with unbounded $\gamma$, but the greedy algorithm is the only prior robust (i.e., distribution independent) algorithm known, and it is quite simple too. For example Alaei, Hajiaghayi and Liaghat  Alaei et al. (2012) design a randomized algorithm that obtains a $1-1/e$ approximation, but requires the knowledge of the entire distribution. Devanur, Sivan and Azar Devanur et al. (2012) design a deterministic algorithm that obtains a $1-1/e$ approximation, but requires a few parameters from the distribution.

Third Result: Fast Approximation Algorithms for Mixed Packing and

Covering Integer Programs

Charles et al. (2010) considered the following (offline) problem: given a lopsided bipartite graph $G=(L,R,E)$, that is a bipartite graph where $m=|L|\gg|R|=n$, does there exist an assignment $M:L\rightarrow R$ with $(j,M(j))\in E$ for all $j\in L$, and such that for every vertex $i\in R$, $|M^{-1}(i)|\geq B_{i}$ for some given values $B_{i}$. Even though this is a classic problem in combinatorial optimization with well known polynomial time algorithms, the instances of interest are too large to use traditional approaches to solve this problem. (The value of $m$ in particular is very large.) The approach used by Charles et al. (2010) was to essentially design an online algorithm in the i.i.d. model: choose vertices from $L$ uniformly at random and assign them to vertices in $R$ in an online fashion. The online algorithm is guaranteed to be close to optimal, as long as sufficiently many samples are drawn. Therefore it can be used to solve the original problem (approximately): the online algorithm gets an almost satisfying assignment if and only if the original graph has a satisfying assignment (with high probability).

The third result in this paper is a generalization of this result to get fast approximation algorithms for a wide class of mixed packing and covering integer programs (IPs) inspired by problems in the resource allocation framework (Theorem 2.5). Problems in the resource allocation framework where the instances are too large to use traditional algorithms occur fairly often, in particular in the management of display advertising systems, where these algorithms are being used. Formal statements and a more detailed discussion are presented in Section 2.4.

High Level Description of Techniques

The underlying idea used for all these results can be summarized at a high level as thus: consider a hypothetical algorithm called Hypothetical-Oblivious that knows the distribution from which the input is drawn and uses an optimal solution w.r.t. this distribution. Now suppose that we can analyze the performance of Hypothetical-Oblivious by considering a potential function and showing that it decreases by a certain amount in each step. Now we can design an algorithm that does not know the distribution as follows: consider the same potential function, and in every step choose the option that minimizes the potential function. Since the algorithm minimizes the potential in each step, the decrease in the potential for this algorithm is better than that for Hypothetical-Oblivious and hence we obtain the same guarantee as that for Hypothetical-Oblivious. The choice of potential function varies across the results; also, whether we minimize or maximize the the potential function varies across the results.

For instance, our first result (Theorem 2.2), the performance of Hypothetical-Oblivious is analyzed using Chernoff bounds. The Chernoff bounds are proven by showing bounds on the expectation of the moment generating function of a random variable. Thus the potential function is the sum of the moment generating functions for all the random variables that we apply the Chernoff bounds to. The proof shows that in each step this potential function decreases by some multiplicative factor. The algorithm is then designed to achieve the same decrease in the potential function. A particularly pleasing aspect about this technique is that we obtain very simple proofs. E.g., the proof of the second result mentioned above (that greedy is $1-1/e$ competitive, Theorem 2.4 ) is extremely simple: the potential function in this case is simply the total amount of unused budgets and we show that this amount (in expectation) decreases by a factor of $1-1/m$ in each step where there are $m$ steps in all.

Multiplicative-Weight Updates

Our techniques and the resulting algorithms for our first and third results (Theorem 2.2 and Theorem 2.5) are similar to the algorithms of Young (1995, 2001) for derandomizing randomized rounding and the fast approximation algorithms for solving covering/packing LPs of Plotkin, Shmoys, and Tardos (1991), Garg and Koenemann (1998), Fleischer (2000). In fact Arora et al. (2005) showed that all these algorithms are related to the multiplicative weights update method for solving the experts problem and especially highlighted the similarity between the potential function used in the analysis of the multiplicative update method and the moment generating function used in the proof of Chernoff bounds and Young’s algorithms. Hence it is no surprise that our algorithm which uses Chernoff bounds is also a multiplicative update algorithm. Our algorithm is closer in spirit to Young’s algorithms than others. The main difference is that our algorithm solves an online problem, rather than an offline one, and hence will run short of essential distribution dependent parameters to run the multiplicative weights based algorithm directly: we show that these parameters can be estimated near optimally. And further, we introduce more adversarial models of online input, namely, the varying ASI models, and come up with varying levels of knowledge of the distribution that are sufficient to be able to design good algorithms for these models. And for the offline case, a basic difference of our algorithm from this previous set of results is that our algorithm uses the special structure of the polytope $\sum_{k}x_{j,k}\leq 1$ (as against the more general polytopes in these works) in giving a more efficient solution. For instance, for our offline problem the number of oracle calls required will have a quadratic dependence on $\gamma m$ if we used the Plotkin et al. (1991) algorithm, where as using the special structure of the polytope, we obtain a linear dependence on $\gamma m$.

It is possible that our algorithm can also be interpreted as an algorithm for the experts problem. In fact Mehta et al. (2005) asked if there is a $1-o(1)$ competitive algorithm for Adwords in the i.i.d model with small bid to budget ratio, and in particular if the algorithms for experts could be used. They also conjectured that such an algorithm would iteratively adjust a budget discount factor based on the rate at which the budget is spent. Our algorithms for resource allocation problem when specialized for Adwords look exactly like that, but we do not provide formal connections to the experts framework. This was done in follow-up works Agrawal and Devanur (2015); Gupta and Molinaro (2014) which showed that essentially the same algorithm as ours can be thought of as using a subroutine of Multiplicative-weight updates on a suitably defined learning with experts problem.

Follow-up work

There has been a number of follow-up papers since the conference version of this paper has been published. Alaei et al. (2012) show that for the Adwords problem with a known distribution, it is enough for $\gamma$ to be $O(\epsilon^{2})$ to get a $1-\epsilon$ approximation. Simultaneously, Devanur et al. (2012) showed the same dependence of $\gamma=O(\epsilon^{2})$ for the Adwords problem, but requiring only a few parameters from the distribution. Kapralov et al. (2013) study a generalized version of the adwords problem where, an advertiser’s profit, instead of being budget-additive, could be an arbitrary submodular function of the queries assigned to him. For this problem in the worst case setting, they show that no algorithm can obtain better than a $\frac{1}{2}$ approximation, which the greedy algorithm already achieves. For the same problem in the i.i.d. setting, they show, using techniques we develop in this work, that the greedy algorithm obtains a $1-\frac{1}{e}$ approximation.  Kesselheim et al. (2014) gave similar guarantees as us, for the random permutation model (i.i.d. without replacement), and also get the improved bound of $\gamma=O(\epsilon^{2})$ for the special case of the Adwords problem. On the other hand, the algorithms of Kesselheim et al. (2014) are computationally expensive, requiring to solve a linear program for serving every single request, where as our algorithm performs a much simpler optimization in every step: for the adwords problem for instance, it takes only a linear time to perform each step’s optimization. Both Agrawal and Devanur (2015) and Gupta and Molinaro (2014) showed that essentially the same algorithm as ours also works for the random permutation model, with the same guarantees, while also relating it formally to the learning from experts framework. Agrawal and Devanur (2015) also greatly generalize the resource allocation framework, to handle arbitrary concave objectives and convex constraints. Eghbali et al. (2014) interpret our algorithm as an exponentiated sub-gradient algorithm, show that it works for the random permutation model, and give a slight generalization to handle additively separable concave reward functions.

2 Preliminaries & Main Results

2.1 Resource Allocation Framework

We consider the following framework of optimization problems. There are $n$ resources, with resource $i\in\mathcal{A}$ having a capacity of $c_{i}$. There are $m$ requests; each request $j\in\mathcal{J}$ can be satisfied by a vector $\mathbf{x}_{j}\in\{0,1\}^{K}$, with coordinates $x_{j,k}$, such that $\sum_{k}x_{j,k}\leq 1$. Think of vector $\mathbf{x}_{j}$ as picking a single option to satisfy a request from a total of $K$ options. We use $\mathcal{K}$ to denote the set of options. The vector $\mathbf{x}_{j}$ consumes $\mathbf{a}_{i,j}\cdot\mathbf{x}_{j}$ amount of resource $i$, and gives $\mathbf{w}_{i,j}\cdot\mathbf{x}_{j}$ amount of type $i$ profit555While this notation seems to imply that the number of resource-types is equal to the number/set of profit types, namely $n$, this choice was made purely to reduce clutter in notation. In general the number/set of resource-types could be different from that of the number of profit-types, and it’s straight-forward to verify that our proofs go through for the general case.. The $\mathbf{a}_{i,j}$’s and $\mathbf{w}_{i,j}$’s are non-negative vectors of length $K$ (and so are the $\mathbf{x}_{j}$’s). The co-ordinates of the vectors $\mathbf{a}_{i,j}$ and $\mathbf{w}_{i,j}$ will be denoted by $a_{ijk}$ and $w_{ijk}$ respectively, i.e., the $k^{\rm th}$ option consumes $a_{ijk}$ amount of resource $i$ and gives a type $i$ profit of $w_{ijk}$. The objective is to maximize the minimum among all types of profit subject to the capacity constraints on the resources. The following is the linear program relaxation of the resource allocation problem:

[TABLE]

Note that dropping a request by not picking any option at all is feasible too. For expositional convenience, we will denote not picking any option at all as having picked the $\bot$ option ($\bot$ may not be in the set $\mathcal{K}$) for which $a_{ij\bot}=0$ and $w_{ij\bot}=0$ for all $i,j$.

We consider two versions of the above problem. The first is an online version with stochastic input: requests are drawn from an unknown distribution. The second is an offline problem when the number of requests is much larger than the number of resources, and our goal is to design a fast PTAS for the problem.

2.2 Near-Optimal Online Algorithm for Resource Allocation

We now consider an online version of the resource allocation framework. Here requests arrive online. We consider the i.i.d. model, where each request is drawn independently from a given distribution. The distribution is unknown to the algorithm. The algorithm knows $m$, the total number of requests. To define our benchmark, we now define the expected instance.

Expected Instance

Consider the following expected instance of the problem, where everything happens as per expectation. It is a single offline instance which is a function of the given distribution over requests and the total number of requests $m$. Every request in the support of the distribution is also a request in this instance. The capacities of the resources in this instance are the same as in the original instance. Suppose request $j$ has a probability $p_{j}$ of arriving in the given distribution. The resource consumption of $j$ in the expected instance is given by $mp_{j}\mathbf{a}_{i,j}$ for all $i$ and the type $i$ profit is $mp_{j}\mathbf{w}_{i,j}$. The intuition is that if the requests were drawn from this distribution then the expected number of times request $j$ is seen is $mp_{j}$. To summarize, the LP relaxations of a random instance with set of requests $R$, and the expected instance $E$ are as follows (slightly rewritten for convenience).

We now prove that the fractional optimal solution to the expected instance $W_{E}$ is an upper bound on the expectation of $W_{R}$, where $W_{R}$ is the offline fractional optimum of the actual sequence of requests in a random instance $R$.

Lemma 2.1

$W_{E}\geq\mathop{\operatorname{\bf E}}[W_{R}]$**

Proof: The average of optimal solutions for all possible sequences of requests is a feasible solution to the expected instance with a profit equal to $\mathop{\operatorname{\bf E}}[W_{R}]$. Thus the optimal profit for the expected instance could only be larger.

The approximation factor of an algorithm in the i.i.d. model is defined as the ratio of the expected profit of the algorithm to the fractional optimal profit $W_{E}$ for the expected instance. Let $\gamma=\max\left(\left\{\frac{a_{ijk}}{c_{i}}\right\}_{i,j,k}\cup\left\{\frac{w_{ijk}}{W_{E}}\right\}_{i,j,k}\right)$ be the parameter capturing the significance of any one request when compared to the total set of requests that arrive online. The main result is that as $\gamma$ tends to zero, the approximation factor ratio tends to 1. In fact, we give the almost optimal trade-off.

Theorem 2.2

For any $\epsilon\geq 1/m$, Algorithm 2 achieves an objective value of $W_{E}(1-O(\epsilon))$ for the online resource allocation problem with probability at least $1-\epsilon$, assuming $\gamma=O(\frac{\epsilon^{2}}{\log(n/\epsilon)})$. Algorithm 2 does not require any knowledge of the distribution at all.

Theorem 2.3

There exist instances with $\gamma=\frac{\epsilon^{2}}{\log(n)}$ such that no algorithm, even with complete knowledge of the distribution, can get a $1-o(\epsilon)$ approximation factor.

Oracle Assumption

We assume that we have the following oracle available to us: given a request $j$ and a vector $\mathbf{v}$, the oracle returns the vector $\mathbf{x}_{j}$ that maximizes $\mathbf{v}.\mathbf{x}_{j}$ among all $\mathbf{x}_{j}$s in $\{0,1\}^{K}$ satisfying $\sum_{k\in\mathcal{K}}x_{j,k}\leq 1$. This assumption boils down to being able to find the maximum among $K$ numbers, but $K$ may be exponential in some cases. For the Adwords and display ads problem (described below), $K$ is actually equal to $n$, and this is trivial. For network routing (described in Section 7), $K$ could be exponential in the size of the network, and this assumption corresponds to being able to find the shortest path in a graph in polynomial time. For combinatorial auctions (described in Section 7), this corresponds to the demand query assumption: given prices on various items, the buyer should be able to decide in polynomial time which bundle gives her the maximum utility. (While this is not always achievable in polynomial time, there cannot be any hope of a posted pricing solution for combinatorial auction without this minimum assumption. )

Extensions and Special Cases

The extensions of Theorem 2.2 to the various generalizations of the i.i.d. model, including the adversarial stochastic input model are presented in Section 3.5. We refer the reader to Section 7 for a discussion on several problems that are special cases of the resource allocation framework and have been previously considered. Here, we discuss two special cases — the Adwords problem and display ads problem.

Adwords. In the adwords problem, there are $n$ advertisers with a daily budget of $B_{i}$ for advertiser $i$. There are $m$ keywords/queries that arrive online, and advertiser $i$ has a bid of $b_{ij}$ for query $j$. This is a special case of the resource allocation framework where the set of options $\mathcal{K}$ matches the set of resources/advertisers $\mathcal{A}$, i.e., each query can be given to at most one advertiser, and will consume budget just from that advertiser. Let $x_{ij}$ denote the indicator variable for whether or not query $j$ was allocated to agent $i$. After all allocation is over, agent $i$ pays $\min(\sum_{j\in\mathcal{J}}b_{ij}x_{ij},B_{i})$, i.e., the minimum of the sum of the bids for queries allocated to $i$ and his budget $B_{i}$. The objective is to maximize the sum of the payments from all advertisers — this is again a special case of the resource allocation framework where this only a single profit type, and we just want to maximize it. One could raise a technical objection that this is not a special case of the resource allocation framework because the budget constraint is not binding: the value of the allocated bids to an advertiser can exceed his budget, although the total payment from the advertiser will be at most the budget. But it is not difficult to see that the LP relaxation of the offline problem can be written as in LP (2), which is clearly a special case of resource allocation framework LP. Note that the benchmark is anyway an upper bound even on the expected optimal fractional solution. Therefore, any algorithm that gets an $\alpha$ approximation factor for resource allocation is also guaranteed to get the same approximation factor for Adwords. The only notable thing being that an algorithm for resource allocation when used for adwords will treat the budget constraints as binding, and obtain the guarantee promised in Theorem 2.2 (In our $1-1/e$ approximation algorithm for adwords in Section 5 that holds for all values of $\gamma$ ($\leq 1$ of course), we use this facility to exceed budget). 2. 2.

Display Ads. In the display ads problem, there are $n$ advertisers and $m$ impressions arrive online. Advertiser $i$ has wants $c_{i}$ impressions in total and pays $v_{ij}$ for impression $j$, and will get paid a penalty of $\rho_{i}$ for every undelivered impression. If over-delivered, he will pay his bid for the first $c_{i}$ impressions delivered. Letting $b_{ij}=v_{ij}+\rho_{i}$, we can write the LP relaxation of the offline display ads problem as in LP (2), which is clearly a special case of the resource allocation LP, where just like the Adwords special case the set of options $\mathcal{K}$ is equal to the set of resources/advertisers $\mathcal{A}$, and there is only a single profit type.

2.3 Greedy Algorithm for Adwords

As noted in the introduction, the greedy algorithm is widely implemented due to its simplicity, but its performance was known to be only a $1/2$ approximation even in stochastic models. We show that the greedy algorithm obtains a $1-1/e$ approximation for all $\gamma$, i.e., $0\leq\gamma\leq 1$.

Theorem 2.4

The greedy algorithm achieves an approximation factor of $1-1/e$ for the Adwords problem in the i.i.d. unknown distributions model for all $\gamma$, i.e., $0\leq\gamma\leq 1$.

We note here that the competitive ratio of $1-1/e$ is tight for the greedy algorithm Goel and Mehta (2008). It is however not known to be tight for an arbitrary algorithm.

2.4 Fast Approximation Algorithms for Large Mixed Packing and

Covering Integer Programs

Charles et al. (2010) consider the following problem: given a bipartite graph $G=(L,R,E)$ where $m=|L|\gg|R|=n$, does there exist an assignment $M:L\rightarrow R$ with $(j,M(j))\in E$ for all $j\in L$, and such that for every vertex $i\in R$, $|M^{-1}(i)|\geq B_{i}$ for some given values $B_{i}$. Since $m$ is very large classic matching algorithms are not useful. Charles et al. (2010) gave an algorithm that runs in time linear666In fact, the algorithm makes a single pass through this graph. in the number of edges of an induced subgraph obtained by taking a random sample from $L$ of size $O\left(\frac{m\log n}{\min_{i}\{B_{i}\}\epsilon^{2}}\right)$, for a gap-version of the problem with gap $\epsilon$. Such an algorithm is very useful in a variety of applications involving ad assignment for online advertising, particularly when ${\min_{i}\{B_{i}\}}$ is large.

We consider a generalization of the above problem inspired by the resource allocation framework. In fact, we consider the following mixed covering-packing integer program. Suppose that there are $n$ packing constraints, one for each $i\in[n]$ of the form $\sum_{j=1}^{m}\mathbf{a}_{i,j}\cdot\mathbf{x}_{j}\leq c_{i}$ and $n$ covering constraints, one for each $i\in[n]$ of the form $\sum_{j=1}^{m}\mathbf{w}_{i,j}\cdot\mathbf{x}_{j}\geq d_{i}$. Each $\mathbf{x}_{j}$ (with coordinates $x_{j,k}$) is constrained to be in $\{0,1\}^{K}$ and satisfy $\sum_{k}x_{j,k}\leq 1$. The $\mathbf{a}_{i,j}$’s and $\mathbf{w}_{i,j}$’s (and hence $\mathbf{x}_{j}$’s) are non-negative vectors of length $K$ with coordinates $a_{ijk}$ and $w_{ijk}$. Does there exist a feasible solution to this system of constraints? The gap-version of this problem is as follows. Distinguish between the two cases, with a high probability, say $1-\delta$:

• •

YES: There is a feasible solution.

• •

NO: There is no feasible solution even if all the $c_{i}$’s are multiplied by $1+\epsilon$ and all the $d_{i}$’s are multiplied by $1-\epsilon$.

We note that solving (offline) an optimization problem in the resource allocation framework can be reduced to the above problem through a binary search on the objective function value.

Let $\gamma=\max\left(\left\{\frac{a_{ijk}}{c_{i}}\right\}_{i,j,k}\cup\left\{\frac{w_{ijk}}{d_{i}}\right\}_{i,j,k}\right)$.

Theorem 2.5

For any $\epsilon>0$, assuming $\gamma=O(\frac{\epsilon^{2}}{\log(n/\epsilon)})$, Algorithm 5 solves the gap version of the mixed covering-packing integer program with $\Theta(\frac{\gamma m\log(n/\delta)}{\epsilon^{2}})$ oracle calls.

2.5 Chernoff Bounds

We present here the form of Chernoff bounds that we use throughout the rest of this paper. Let $X=\sum_{i}X_{i}$, where $X_{i}\in[0,B]$ are i.i.d random variables. Let $\mathop{\operatorname{\bf E}}[X]=\mu$. Then, for all $\epsilon>0$,

[TABLE]

Consequently, for all $\delta>0$, with probability at least $1-\delta$,

[TABLE]

Similarly, for all $\epsilon\in[0,2e-1]$,

[TABLE]

Consequently, for all $\delta>\exp(\frac{-(2e-1)^{2}\mu}{4B})$, with probability at least $1-\delta$,

[TABLE]

For $\epsilon>2e-1$,

[TABLE]

3 Near-Optimal Prior Robust Online Algorithms

for Resource Allocation

For convenience, we begin by rewriting the LP relaxation of a random instance $R$ of the online resource allocation problem and the expected instance (already defined in Section 2.2 as LP (1)).

We showed in Lemma 2.1 that $W_{E}\geq\mathop{\operatorname{\bf E}}[W_{R}]$. All our approximation guarantees are w.r.t. the stronger benchmark of $W_{E}$ which is the optimal fractional solution of the expected instance. We would like to remind the reader that while the benchmark is allowed to be fractional, the online algorithm of course is allowed to find only integral solutions.

We divide the rest of this section into four subsections. The subsections progressively weaken the assumptions on knowledge of the distribution of the input.

In section 3.1 we develop a hypothetical algorithm called Hypothetical-Oblivious-Conservative, denoted by $\widetilde{P}$, that achieves an objective value of $W_{E}(1-2\epsilon)$ w.p. at least $1-\epsilon$ assuming $\gamma=O\left(\frac{\epsilon^{2}}{\log(n/\epsilon)}\right)$. Theorem 3.1 is the main result of this section. The algorithm is hypothetical because it assumes knowledge of the entire distribution, where as the goal of this paper is to develop algorithms that work without distributional knowledge. 2. 2.

In section 3.2 we design an algorithm for the online resource allocation problem that achieves the same guarantee as the Hypothetical-Oblivious-Conservative algorithm $\widetilde{P}$, without any knowledge of the distribution except for a single parameter of the distribution — the value of $W_{E}$. Theorem 3.2 is the main result of this section. 3. 3.

In section 3.3 we design an algorithm for the online resource allocation problem that achieves an objective value of at least $W_{E}(1-O(\epsilon))$ w.p. at least $1-\epsilon$ assuming $\gamma=O(\frac{\epsilon^{2}}{\log(n/\epsilon)})$ without any knowledge at all about the distribution. The algorithm in Section 3.2 serves as a good warm-up for the algorithm in this section. Theorem 2.2 is the main result of this section. 4. 4.

In section 3.5 we relax the assumption that the distribution from which the requests are drawn is i.i.d.; we give three different generalizations of the i.i.d. model with strong revenue guarantees as in the i.i.d. model.

3.1 Completely Known Distributions

When the distributions are completely known, we first compute the expected instance and solve its LP relaxation (LP (3)) optimally. Let $x_{jk}^{\ast}$ denote the optimal solution to the expected LP (3). The Hypothetical-Oblivious algorithm $P$ works as follows: when request $j$ arrives, it serves it using option $k$ with probability $x_{jk}^{\ast}$. Let $X_{i,t}^{\ast}$ denote the amount of resource $i$ consumed in step $t$ for the algorithm $P$. Thus the total amount of resource $i$ consumed over the entire $m$ steps of algorithm $P$ is $\sum_{t=1}^{m}X_{i,t}^{\ast}$. Note that $\mathop{\operatorname{\bf E}}[X_{i,t}^{\ast}]=\sum_{j,k}p_{j}a_{ijk}x_{jk}^{\ast}\leq\frac{c_{i}}{m}$. Thus, we can bound the probability that $\mathop{\operatorname{\bf Pr}}[\sum_{t=1}^{m}X_{i,t}^{\ast}\geq c_{i}(1+\epsilon)]$ using Chernoff bounds. We explicitly derive this bound here since we use this derivation in designing the algorithm in Section 3.2.

Since we cannot exceed $c_{i}$ amount of resource consumption by any non-zero amount, we need to be more conservative than $P$. So we analyze the following algorithm $\widetilde{P}$, called Hypothetical-Oblivious-Conservative, instead of $P$: when request $j$ arrives, it serves it using option $k$ with probability $\frac{x_{jk}^{\ast}}{1+\epsilon}$, where $\epsilon$ is an error parameter of algorithm designer’s choice. Let $\widetilde{X}_{i,t}$ denote the amount of resource $i$ consumed in step $t$ for the algorithm $\widetilde{P}$. Note that $\mathop{\operatorname{\bf E}}[\widetilde{X}_{i,t}]\leq\frac{c_{i}}{(1+\epsilon)m}$. Thus, even with a $(1+\epsilon)$ deviation using Chernoff bounds, the resource consumption is at most $c_{i}$.

We begin by noting that $\widetilde{X}_{i,t}\leq\gamma c_{i}$ by the definition of $\gamma$. For all $\epsilon\in[0,1]$ we have,

[TABLE]

where the first inequality follows from Markov’s inequality, the second from convexity of exponential function together with with the fact that $\widetilde{X}_{i,t}\leq\gamma c_{i}$, the third from $\mathop{\operatorname{\bf E}}[\widetilde{X}_{i,t}]\leq\frac{c_{i}}{(1+\epsilon)m}$, and the fourth from $1+x\leq e^{x}$, the fifth is standard for all $\epsilon\in[0,1]$, and the sixth follows from $\gamma=O(\epsilon^{2}/\log(n/\epsilon))$ for an appropriate choice of constant inside the big-oh coupled with $n\geq 2$.

Remark 3.1

At first sight this bound might seem anomalous — the bound $\frac{\epsilon}{2n}$ is increasing in $\epsilon$, i.e., the probability of a smaller deviation is smaller than the probability of a larger deviation! The reason for this anomaly is that $\gamma$ is related to $\epsilon$ as $\gamma=O(\frac{\epsilon^{2}}{\log(n/\epsilon)})$, and smaller the $\gamma$, the better revenue we can get (i.e., more granular requests leads to lesser wastage from errors, and hence more revenue). Thus a small deviation for small $\gamma$ has a smaller probability than a larger deviation for a larger $\gamma$.

Similarly let $\widetilde{Y}_{i,t}$ denote the revenue obtained from type $i$ profit in step $t$ for the algorithm $\widetilde{P}$. Note that $\mathop{\operatorname{\bf E}}[\widetilde{Y}_{i,t}]=\sum_{j,k}p_{j}w_{ijk}\frac{x_{jk}^{\ast}}{1+\epsilon}\geq\frac{W_{E}}{(1+\epsilon)m}$. By the definition of $\gamma$, we have $\widetilde{Y}_{i,t}\leq\gamma W_{E}$. For all $\epsilon\in[0,1]$ we have,

[TABLE]

Thus, we have all the capacity constraints satisfied, (i.e., $\sum_{i}\widetilde{X}_{i,t}\leq c_{i}$), and all resource profits are at least $\frac{W_{E}}{1+\epsilon}(1-\epsilon)$ (i.e., $\sum_{i}\widetilde{Y}_{i,t}\geq\frac{W_{E}}{1+\epsilon}(1-\epsilon)\geq W_{E}(1-2\epsilon)$), with probability at least $1-2n\cdot\epsilon/2n=1-\epsilon$. This proves the following theorem:

Theorem 3.1

For any $\epsilon>0$, the Hypothetical-Oblivious-Conservative algorithm $\widetilde{P}$ achieves an objective value of $W_{E}(1-2\epsilon)$ for the online resource allocation problem with probability at least $1-\epsilon$, assuming $\gamma=O(\frac{\epsilon^{2}}{\log(n/\epsilon)})$.

3.2 Unknown Distribution, Known $W_{E}$

We now design an algorithm777Note that the notation $\mathcal{A}$ that we use for the set of advertisers/resources is different from the non-calligraphic $A$ that we use for an algorithm. Also, it is immediate from the context which one we are referring to. $A$ without knowledge of the distribution, but just knowing a single parameter $W_{E}$. Let $A^{s}\widetilde{P}^{m-s}$ be a hybrid algorithm that runs $A$ for the first $s$ steps and $\widetilde{P}$ for the remaining $m-s$ steps. Let $\epsilon\in[0,1]$ be the error parameter, which is the algorithm designer’s choice. Call the algorithm a failure if at least one of the following fails:

For all $i$, $\sum_{t=1}^{m}X_{i,t}^{A}\leq c_{i}$. 2. 2.

For all $i$, $\sum_{t=1}^{m}Y_{i,t}^{A}\geq W_{E}(1-2\epsilon)$.

For any algorithm $A$, let the amount of resource $i$ consumed in the $t$-th step be denoted by $X_{i,t}^{A}$ and the amount of resource $i$ profit be denoted by $Y_{i,t}^{A}$. Let $S_{s}\left(X_{i}^{A}\right)=\sum_{t=1}^{s}X_{i,t}^{A}$ denote the amount of resource $i$ consumed in the first $s$ steps, and let $S_{s}\left(Y_{i}^{A}\right)=\sum_{t=1}^{s}Y_{i,t}^{A}$ denote the resource $i$ profit in the first $s$ steps. Similar to the derivation in Section 3.1 which bounded the failure probability of $\widetilde{P}$, we can bound the failure probability of any algorithm $A$, i.e.,

[TABLE]

[TABLE]

In Section 3.1 our algorithm $A$ was $\widetilde{P}$ \bigg{(}and therefore we can use $\mathop{\operatorname{\bf E}}[\widetilde{X}_{i,t}]\leq\frac{c_{i}}{(1+\epsilon)m}$ and \mathop{\operatorname{\bf E}}[\widetilde{Y}_{i,t}]\geq\frac{W_{E}}{(1+\epsilon)m}\bigg{)}, the total failure probability which is the sum of (3.2) and (3.2) for all the $i$’s would have been $n\cdot\left[\frac{\epsilon}{2n}+\frac{\epsilon}{2n}\right]=\epsilon$. The goal is to design an algorithm $A$ for stage $r$ that, unlike $P$, does not know the distribution and knows just $W_{E}$, but obtains the same $\epsilon$ failure probability. That is we want to show that the sum of (3.2) and (3.2) over all $i$’s is at most $\epsilon$:

[TABLE]

For the algorithm $A^{s}\widetilde{P}^{m-s}$, the above quantity can be rewritten as

[TABLE]

which, by using $(1+\epsilon)^{x}\leq 1+\epsilon x$ for $0\leq x\leq 1$, is in turn upper bounded by

[TABLE]

Since for all $t$, the random variables $\widetilde{X}_{i,t}$, $X_{i,t}^{A}$, $\widetilde{Y}_{i,t}$ and $Y_{i,t}^{A}$ are all independent, and $\mathop{\operatorname{\bf E}}[\widetilde{X}_{i,t}]\leq\frac{c_{i}}{(1+\epsilon)m}$ and $\mathop{\operatorname{\bf E}}[\widetilde{Y}_{i,t}]\geq\frac{W_{E}}{(1+\epsilon)m}$, the above is in turn upper bounded by

[TABLE]

Let $\mathcal{F}[A^{s}\widetilde{P}^{m-s}]$ denote the quantity in  (6), which is an upper bound on failure probability of the hybrid algorithm $A^{s}\widetilde{P}^{m-s}$. By Theorem 3.1, we know that $\mathcal{F}[\widetilde{P}^{m}]\leq\epsilon$. We now prove that for all $s\in\{0,1,\dots,m-1\}$, $\mathcal{F}[A^{s+1}\widetilde{P}^{m-s-1}]\leq\mathcal{F}[A^{s}\widetilde{P}^{m-s}]$, thus proving that $\mathcal{F}[A^{m}]\leq\epsilon$, i.e., running the algorithm $A$ for all the $m$ steps results in a failure with probability at most $\epsilon$. To design such an $A$ we closely follow the derivation of Chernoff bounds, which is what established that $\mathcal{F}[\widetilde{P}^{m}]\leq\epsilon$ in Theorem 3.1. However the design process will reveal that unlike algorithm $\widetilde{P}$ which needs the entire distribution, just the knowledge of $W_{E}$ will do for bounding the failure probability by $\epsilon$.

Assuming that for all $s<p$, the algorithm $A$ has been defined for the first $s+1$ steps in such a way that $\mathcal{F}[A^{s+1}\widetilde{P}^{m-s-1}]\leq\mathcal{F}[A^{s}\widetilde{P}^{m-s}]$, we now define $A$ for the $p+1$-th step in a way that will ensure that $\mathcal{F}[A^{p+1}\widetilde{P}^{m-p-1}]\leq\mathcal{F}[A^{p}\widetilde{P}^{m-p}]$. We have

[TABLE]

Define

[TABLE]

Define the step $p+1$ of algorithm $A$ as picking the following option $k^{*}$ for request $j$, where:

[TABLE]

For the sake of clarity, the entire algorithm is presented in Algorithm 1.

By the definition of step $p+1$ of algorithm $A$ given in equation (8), it follows that for any two algorithms with the first $p$ steps being identical, and the last $m-p-1$ steps following the Hypothetical-Oblivious-Conservative algorithm $\widetilde{P}$, algorithm $A$’s $p+1$-th step is the one that minimizes expression (7). In particular it follows that expression (7) is upper bounded by the same expression where the $p+1$-the step is according to $\widetilde{X}_{i,p+1}$ and $\widetilde{Y}_{i,p+1}$, i.e., we replace $X_{i,p+1}^{A}$ by $\widetilde{X}_{i,p+1}$ and $Y_{i,p+1}^{A}$ by $\widetilde{Y}_{i,p+1}$. Therefore we have

[TABLE]

This completes the proof of the following theorem.

Theorem 3.2

For any $\epsilon>0$, Algorithm 1 achieves an objective value of $W_{E}(1-2\epsilon)$ for the online resource allocation problem with probability at least $1-\epsilon$, assuming $\gamma=O(\frac{\epsilon^{2}}{\log(n/\epsilon)})$. The algorithm $A$ does not require any knowledge of the distribution except for the single parameter $W_{E}$.

3.3 Completely Unknown Distribution

We first give a high-level overview of this section before going into the details. In this section, we design an algorithm $A$ without any knowledge of the distribution at all. The algorithm is similar in spirit to the one in Section 3.2 except that since we do not have knowledge of $W_{E}$, we divide the algorithm into many stages. In each stage, we run an algorithm similar to the one in Section 3.2 except that instead of $W_{E}$, we use an estimate of $W_{E}$ that gets increasingly accurate with each successive stage.

More formally, the algorithm runs in $l$ stages $\{0,1,\dots,l-1\}$, where $l$ is such that $\epsilon 2^{l}=1$, and $\epsilon\in[1/m,1/2]$ (we need $\epsilon\leq 1/2$ so that $l$ is at least $1$) is the error parameter of algorithm designer’s choice. Further we need $m\geq\frac{1}{\epsilon}$ so that $\epsilon m\geq 1$. We assume that $\epsilon m$ is an integer for clarity of exposition. Stage $r$ handles $t_{r}=\epsilon m2^{r}$ requests for $r\in\{0,\dots l-1\}$. The first $\epsilon m$ requests are used just for future estimation, and none of them are served. For convenience we sometimes call this pre-zero stage as stage $-1$, and let $t_{-1}=\epsilon m$. Stage $r\geq 0$ serves $t\in[t_{r}+1,t_{r+1}]$. Note that in the optimal solution to the expected instance of stage $r$, no resource $i$ gets consumed by more than $\frac{t_{r}c_{i}}{m}$, and every resource $i$ gets a profit of $\frac{t_{r}W_{E}}{m}$, i.e., consumption and profit have been scaled down by a factor of $\frac{t_{r}}{m}$. As in the previous sections, with a high probability, we can only reach close to $\frac{t_{r}W_{E}}{m}$. Further, since stage $r$ consists of only $t_{r}$ requests, which is much smaller than $m$ for small $r$, it follows that for small $r$, our error in how close to we get to $\frac{t_{r}W_{E}}{m}$ will be higher. Indeed, instead of having the same error parameter of $\epsilon$ in every stage, we set stage-specific error parameters which get progressively smaller and become close to $\epsilon$ in the final stages. These parameters are chosen such that the overall error is still $O(\epsilon)$ because the later stages having more requests matter more than the former. There are two sources of error/failure which we detail below.

The first source of failure stems from not knowing $W_{E}$. Instead we estimate a quantity $Z_{r}$ which is an approximation we use for $W_{E}$ in stage $r$, and the approximation gets better as $r$ increases. We use $Z_{r}$ to set a profit target of $\frac{t_{r}Z_{r}}{m}$ for stage $r$. Since $Z_{r}$ could be much smaller than $W_{E}$ our algorithm could become very suboptimal. We prove that with a probability of at least $1-2\delta$ we have $W_{E}(1-3\epsilon_{x,r-1})\leq Z_{r}\leq W_{E}$ (see next for what $\epsilon_{x,r}$ is), where $\delta=\frac{\epsilon}{3l}$. Thus for all the $l$ stages, these bounds are violated with probability at most $2l\delta=2\epsilon/3$. 2. 2.

The second source of failure stems from achieving this profit target in every stage. We set error parameters $\epsilon_{x,r}$ and $\epsilon_{y,r}$ such that for every $i$ stage $r$ consumes at most $\frac{t_{r}c_{i}}{m}(1+\epsilon_{x,r})$ amount of resource $i$, and for every $i$ we get a profit of at least $\frac{t_{r}Z_{r}}{m}(1-\epsilon_{y,r})$, with probability at least $1-\delta$. Thus the overall failure probability, as regards falling short of the target $\frac{t_{r}Z_{r}}{m}$ by more than $\epsilon_{y,r}$ and exceeding $\frac{t_{r}c_{i}}{m}$ by more than $\epsilon_{x,r}$, for all the $l$ stages together is at most $\delta\cdot l=\epsilon/3$.

Thus summing over the failure probabilities we get $\epsilon/3+2\epsilon/3=\epsilon$. We have that with probability at least $1-\epsilon$, for every $i$, the total consumption of resource $i$ is at most $\sum_{r=0}^{l-1}\frac{t_{r}c_{i}}{m}(1+\epsilon_{x,r})$, and total profit from resource $i$ is at least $\sum_{r=0}^{l-1}\frac{t_{r}W_{E}}{m}(1-3\epsilon_{x,r-1})(1-\epsilon_{y,r})$. We set $\epsilon_{x,r}=\sqrt{\frac{4\gamma m\ln(2n/\delta)}{t_{r}}}$ for $r\in\{-1,0,1,\dots,l-1\}$, and $\epsilon_{y,r}=\sqrt{\frac{2w_{\max}m\ln(2n/\delta)}{t_{r}Z_{r}}}$ for $r\in\{0,1,\dots,l-1\}$ (we define $\epsilon_{x,r}$ starting from $r=-1$, with $t_{-1}=\epsilon m$, just for technical convenience). From this it follows that $\sum_{r=0}^{l-1}\frac{t_{r}c_{i}}{m}(1+\epsilon_{x,r})\leq c_{i}$ and $\sum_{r=0}^{l-1}\frac{t_{r}W_{E}}{m}(1-3\epsilon_{x,r-1})(1-\epsilon_{y,r})\geq W_{E}(1-O(\epsilon))$, assuming $\gamma=O(\frac{\epsilon^{2}}{\log(n/\epsilon)})$. The algorithm is described in Algorithm 2. This completes the high-level overview of the proof. All that is left to prove is the points 1 and 2 above, upon which we would have proved our main theorem, namely, Theorem 2.2, which we recall below. Theorem 2.2. For any $\epsilon\geq 1/m$, Algorithm 2 achieves an objective value of $W_{E}(1-O(\epsilon))$ for the online resource allocation problem with probability at least $1-\epsilon$, assuming $\gamma=O(\frac{\epsilon^{2}}{\log(n/\epsilon)})$. Algorithm 2 does not require any knowledge of the distribution at all.

Detailed Description and Proof

We begin with the first point in our high-level overview above, namely by describing how $Z_{r}$ is estimated and proving its concentration around $W_{E}$. After stage $r$ (including stage $-1$), the algorithm computes the optimal fractional objective value $e_{r}$ to the following instance $\mathcal{I}_{r}$: the instance has the $t_{r}$ requests of stage $r$, and the capacity of resource $i$ is capped at $\frac{t_{r}c_{i}}{m}$. Using the optimal fractional objective value $e_{r}$ of this instance, we set $Z_{r+1}=\frac{e_{r}}{1+\epsilon_{x,r}}\cdot\frac{m}{t_{r}}.$ The first task now is to prove that $Z_{r+1}$ as estimated above is concentrated enough around $W_{E}$. This basically requires proving concentration of $e_{r}$.

Lemma 3.3

With a probability at least $1-2\delta$, we have

[TABLE]

Proof: We prove that the lower and upper bound hold with probability $1-\delta$ each, thus proving the lemma.

We begin with the lower bound on $e_{r}$. Note that the expected instance of the instance $\mathcal{I}_{r}$ has the same optimal solution $x_{jk}^{\ast}$ as the optimal solution to the full expected instance (i.e., the one without scaling down by $\frac{t_{r}}{m}$). Now consider the algorithm $\widetilde{P}(r)$, which is the same as the $\widetilde{P}$ defined in Section 3.1 except that $\epsilon$ is replaced by $\epsilon_{x,r}$, i.e., it serves request $j$ with option $k$ with probability $\frac{x_{jk}^{\ast}}{1+\epsilon_{x,r}}$. When $\widetilde{P}(r)$ is run on instance $\mathcal{I}_{r}$, with a probability at least $1-\frac{\delta}{2n}$, at most $\frac{t_{r}c_{i}}{m}$ amount of resource $i$ is consumed, and with probability at least $1-\frac{\delta}{2n}$, at least $\frac{t_{r}W_{E}}{m}\frac{1-\epsilon_{x,r}}{1+\epsilon_{x,r}}$ resource $i$ profit is obtained. Thus with a probability at least $1-2n\cdot\frac{\delta}{2n}=1-\delta$, $\widetilde{P}(r)$ achieves an objective value of at least $\frac{t_{r}W_{E}}{m}(1-2\epsilon_{x,r})$. Therefore the optimal objective value $e_{r}$ will also be at least $\frac{t_{r}W_{E}}{m}(1-2\epsilon_{x,r})$.

To prove the upper bound, we consider the primal and dual LPs that define $e_{r}$ in LP 9 and the primal and dual LPs defining the expected instance in LP (10). In the latter, for convenience, we use $mp_{j}\beta_{j}$ as the dual multiplier instead of just $\beta_{j}$.

Note that the set of constraints in the dual of LP (10) is a superset of the set of constraints in the dual of LP (9), making any feasible solution to dual of LP (10) also feasible to dual of LP (9). In particular, the optimal solution to dual of LP (10) given by $\beta_{j}^{*}$’s, $\alpha_{i}^{*}$’s and $\rho_{i}^{*}$’s is feasible for dual of LP (9). Hence the value of $e_{r}$ is upper bounded the objective of dual of LP (9) at $\beta_{j}^{*}$’s, $\alpha_{i}^{*}$’s and $\rho_{i}^{*}$’s. That is we have

[TABLE]

We now upper bound the RHS by applying Chernoff bounds on $\sum_{j\in\mathcal{I}_{r}}\beta_{j}^{*}$. Since the dual LP in LP (10) is a minimization LP, the constraints there imply that $\beta_{j}^{*}\leq w_{\max}$. Applying Chernoff bounds we have,

[TABLE]

where the first inequality holds with probability at least $1-\delta$ and the second inequality uses the fact the optimal value of the expected instance (dual of LP (10)) is $W_{E}$. This proves the required upper bound on $e_{r}$ that $e_{r}\leq\frac{t_{r}W_{E}}{m}(1+\epsilon_{x,r})$ with probability at least $1-\delta$.

Going back to our application of Chernoff bounds above, in order to apply it in the form above, we require that the multiplicative deviation from mean $\sqrt{\frac{4w_{\max}\ln(1/\delta)}{t_{r}\sum_{j}p_{j}\beta_{j}^{*}}}\in[0,2e-1]$. If $\sum_{j}p_{j}\beta_{j}^{*}\geq\frac{\epsilon W_{E}}{m}$, then this requirement would follow. Suppose on the other hand that $\sum_{j}p_{j}\beta_{j}^{*}<\frac{\epsilon W_{E}}{m}$. Since we are happy if the excess over mean is at most $\frac{t_{r}W_{E}}{m}\epsilon_{x,r}$, let us look for a multiplicative error of $\frac{\frac{t_{r}W_{E}\epsilon_{x,r}}{m}}{t_{r}\sum_{j}p_{j}\beta_{j}^{*}}$. Based on the fact that $\sum_{j}p_{j}\beta_{j}^{*}<\frac{\epsilon W_{E}}{m}$ and that $\epsilon_{x,r}>\epsilon$ for all $r$, the multiplicative error can be seen to be at least a constant, and can be made larger than $2e-1$ depending on the constant inside the big O of $\gamma$. We now use the version of Chernoff bounds for multiplicative error larger than $2e-1$, which gives us that a deviation of $\frac{t_{r}W_{E}}{m}\epsilon_{x,r}$ occurs with a probability at most $2^{-\left(1+\frac{\frac{t_{r}W_{E}\epsilon_{x,r}}{m}}{t_{r}\sum_{j}p_{j}\beta_{j}^{*}}\right)\frac{t_{r}\sum_{j}p_{j}\beta_{j}^{*}}{w_{\max}}}$, where the division by $w_{\max}$ is because of the fact that $\beta_{j}^{*}\leq w_{\max}$. Noting that $w_{\max}\leq\gamma W_{E}$, we get that this probability is at most $\delta/n$ which is at most $\delta$.

Lemma 3.3 implies that $W_{E}(1-3\epsilon_{x,r-1})\leq Z_{r}\leq W_{E},\ \forall r\in\{0,1,\dots,l-1\}$. The rest of the proof is similar to that of Section 3.2, and is focused on the second point in the high-level overview we gave in the beginning of this section 3.3. In Section 3.2 we knew $W_{E}$ and obtained a $W_{E}(1-2\epsilon)$ approximation with no resource $i$ consumed beyond $c_{i}$ with probability $1-\epsilon$. Here, instead of $W_{E}$ we have an approximation for $W_{E}$ in the form of $Z_{r}$ which gets increasingly accurate as $r$ increases. We set a target of $\frac{t_{r}Z_{r}}{m}$ for stage $r$, and show that with a probability of at least $1-\delta$ we get a profit of $\frac{t_{r}Z_{r}}{m}(1-\epsilon_{y,r})$ from every resource $i$ and no resource $i$ consumed beyond $\frac{t_{r}c_{i}}{m}(1+\epsilon_{x,r})$ capacity.888Note that we are allowed to consume a bit beyond $\frac{t_{r}c_{i}}{m}$ because our goal is just that over all we don’t consume beyond $c_{i}$, and not that for every stage we respect the $\frac{t_{r}c_{i}}{m}$ constraint. In spite of this $(1+\epsilon_{x,r})$ excess consumption in all stages, since stage $-1$ consumes nothing at all, we will see that no excess consumption occurs at the end.

As in Section 3.2, call stage $r$ of algorithm $A$ a failure if at least one of the following fails:

For all $i$, $\sum_{t=t_{r}+1}^{t_{r+1}}X_{i,t}^{A}\leq\frac{t_{r}c_{i}}{m}(1+\epsilon_{x,r})$. 2. 2.

For all $i$, $\sum_{t=t_{r}+1}^{t_{r+1}}Y_{i,t}^{A}\geq\frac{t_{r}Z_{r}}{m}(1-\epsilon_{y,r})$.

Let $S_{s}^{r}\left(X_{i}\right)=\sum_{t=t_{r}+1}^{t_{r}+s}X_{i,t}$ denote the amount of resource $i$ consumed in the first $s$ steps of stage $r$, and let $S_{s}^{r}\left(Y_{i}\right)=\sum_{t=t_{r}+1}^{t_{r}+s}Y_{i,t}$ denote the resource $i$ profit in the first $s$ steps of stage $r$.

[TABLE]

[TABLE]

If our algorithm $A$ was $P$ \bigg{(}and therefore we can use $\mathop{\operatorname{\bf E}}[X_{i,t}^{\ast}]\leq\frac{c_{i}}{m}$ and $\mathop{\operatorname{\bf E}}[Y_{i,t}^{\ast}]\geq\frac{W_{E}}{m}$ \geq\frac{Z_{r}}{m}\bigg{)}, the total failure probability for each stage $r$ which is the sum of (3.3) and (3.3) for all the $i$’s would have been $n\cdot\left[e^{\frac{-\epsilon_{x,r}^{2}}{4\gamma}\frac{t_{r}}{m}}+e^{\frac{-\epsilon_{y,r}^{2}}{2}\frac{t_{r}Z_{r}}{mw_{\max}}}\right]=n\cdot\left[\frac{\delta}{2n}+\frac{\delta}{2n}\right]=\delta$. The goal is to design an algorithm $A$ for stage $r$ that, unlike $P$, does not know the distribution but also obtains the same $\delta$ failure probability, just as we did in Section 3.2. That is we want to show that the sum of (3.3) and (3.3) over all $i$’s is at most $\delta$:

[TABLE]

For the algorithm $A^{s}P^{t_{r}-s}$, the above quantity can be rewritten as

[TABLE]

which, by using $(1+\epsilon)^{x}\leq 1+\epsilon x$ for $0\leq x\leq 1$, is in turn upper bounded by

[TABLE]

Since for all $t$, the random variables $X_{i,t}^{\ast}$, $X_{i,t}^{A}$, $Y_{i,t}^{\ast}$ and $Y_{i,t}^{A}$ are all independent, and $\mathop{\operatorname{\bf E}}[X_{i,t}^{\ast}]\leq\frac{c_{i}}{m}$, $\mathop{\operatorname{\bf E}}[Y_{i,t}^{\ast}]\geq\frac{W_{E}}{m}$, and $\frac{W_{E}}{w_{\max}}\geq\frac{1}{\gamma}$, the above is in turn upper bounded by

[TABLE]

Let $\mathcal{F}_{r}[A^{s}P^{t_{r}-s}]$ denote the quantity in  (13), which is an upper bound on failure probability of the hybrid algorithm $A^{s}P^{t_{r}-s}$ for stage $r$. We just showed that $\mathcal{F}_{r}[P^{t_{r}}]\leq\delta$. We now prove that for all $s\in\{0,1,\dots,t_{r}-1\}$, $\mathcal{F}_{r}[A^{s+1}P^{t_{r}-s-1}]\leq\mathcal{F}_{r}[A^{s}P^{t_{r}-s}]$, thus proving that $\mathcal{F}_{r}[A^{t_{r}}]\leq\delta$, i.e., running the algorithm $A$ for all the $t_{r}$ steps of stage $r$ results in a failure with probability at most $\delta$.

Assuming that for all $s<p$, the algorithm $A$ has been defined for the first $s+1$ steps in such a way that $\mathcal{F}_{r}[A^{s+1}P^{t_{r}-s-1}]\leq\mathcal{F}_{r}[A^{s}P^{t_{r}-s}]$, we now define $A$ for the $p+1$-th step of stage $r$ in a way that will ensure that $\mathcal{F}_{r}[A^{p+1}P^{t_{r}-p-1}]\leq\mathcal{F}_{r}[A^{p}P^{t_{r}-p}]$. We have

[TABLE]

Define

[TABLE]

Define the step $p+1$ of algorithm $A$ as picking the following option $k^{*}$ for request $j$:

[TABLE]

By the above definition of step $p+1$ of algorithm $A$ (for stage $r$), it follows that for any two algorithms with the first $p$ steps being identical, and the last $t_{r}-p-1$ steps following the Hypothetical-Oblivious algorithm $P$, algorithm $A$’s $p+1$-th step is the one that minimizes expression (14). In particular it follows that expression (14) is upper bounded by the same expression where the $p+1$-the step is according to $X_{i,t_{r}+p+1}^{\ast}$ and $Y_{i,t_{r}+p+1}^{\ast}$, i.e., we replace $X_{i,t_{r}+p+1}^{A}$ by $X_{i,t_{r}+p+1}^{\ast}$ and $Y_{i,t_{r}+p+1}^{A}$ by $Y_{i,t_{r}+p+1}^{\ast}$. Therefore we have

[TABLE]

This completes the proof of Theorem 2.2.

3.4 Approximate Estimations

Our Algorithm 2 in Section 3.3 required periodically computing the optimal solution to an offline instance. Similarly, our Algorithm 1 in Section 3.2 requires the value of $W_{E}$ to be given. Suppose we could only approximately estimate these quantities, do our results carry through approximately? That is, suppose the solution to the offline instance is guaranteed to be at least $\frac{1}{\alpha}$ of the optimal, and the stand-in that we are given for $W_{E}$ is guaranteed to be at least $\frac{1}{\alpha}$ of $W_{E}$. Both our Theorem 2.2 and Theorem 3.2 go through with just the $W_{E}$ replaced by $W_{E}/\alpha$. Every step of the proof of the exact version goes through in this approximate version, and so we skip the formal proof for this statement.

3.5 Adversarial Stochastic Input

In this section, we relax the assumption that requests are drawn i.i.d. every time step. Namely, the distribution for each time step need not be identical, but an adversary gets to decide which distribution to sample a request from. The adversary could even use how the algorithm has performed in the first $t-1$ steps in picking the distribution for a given time step $t$. The relevance of this model for the real world is that for settings like display ads, the distribution fluctuates over the day. In general a day is divided into many chunks and within a chunk, the distribution is assumed to remain i.i.d. This is exactly captured by this model.

We give algorithms that give guarantees against three different benchmarks in the three models below. The benchmarks get successively stronger, and hence the information sought by the algorithm also increases successively.

3.5.1 ASI Model 1

In this model, the guarantee we give is against the worst distribution over all time steps picked by the adversary. More formally, let $W_{E}(t)$ denote the optimal profit for the expected instance of distribution of time step $t$. Our benchmark will be $W_{E}=\min_{t}W_{E}(t)$. Given just the single number $W_{E}$, our Algorithm 1 in Section 3.2 will guarantee a revenue of $W_{E}(1-2\epsilon)$ with a probability of at least $1-\epsilon$ assuming $\gamma=O(\frac{\epsilon^{2}}{\log(n/\epsilon)})$, just like the guarantee in Theorem 3.2.

Algorithm 1 works for this ASI model because, the proof did not use the similarity of the distributions beyond the fact that $\mathop{\operatorname{\bf E}}[X_{i,t}^{\ast}|X_{i,t^{\prime}}^{\ast}\forall t^{\prime}<t]\leq\frac{c_{i}}{m}$ for all values of $X_{i,t^{\prime}}^{\ast}$, and $\mathop{\operatorname{\bf E}}[Y_{i,t}^{\ast}|Y_{i,t^{\prime}}^{\ast}\forall t^{\prime}<t]\geq\frac{W_{E}}{m}$ for all values of $Y_{i,t^{\prime}}^{\ast}$ (Here $X_{i,t}^{\ast}$ and $Y_{i,t}^{\ast}$ denote the random variables for resource consumption and profit at time $t$ following from allocation according the optimal solution to the expected instance of the distribution used in stage $t$). In other words, distributions being identical and independent is not crucial, but the fact that the expected instances of these distributions have a minimum profit guarantee in spite of all the dependencies between the distributions is sufficient. Both of these inequalities remain true in this model of ASI also, and thus it easy to verify that Algorithm 1 works for this model.

3.5.2 ASI Model 2

In this model, which is otherwise identical to model 1, our benchmark is stronger, namely, $W_{E}=\frac{\sum_{t=1}^{m}W_{E}(t)}{m}$: this is clearly a much stronger benchmark than $\min_{t}W_{E}(t)$. Correspondingly, our algorithm requires more information than in model 1: we ask for $W_{E}(t)$ for every $t$, at the beginning of the algorithm.

A slight modification of our Algorithm 1 in Section 3.2 will give a revenue of $\frac{\sum_{t=1}^{m}W_{E}(t)}{m}(1-2\epsilon)$ with probability at least $1-\epsilon$, i.e., $W_{E}(1-2\epsilon)$ w.p. at least $(1-\epsilon)$. Among the two potential functions $\phi_{i,s}$ and $\psi_{i,s}$, we modify $\psi_{i,s}$ in the most natural way to account for the fact that distributions change every step.

Define

[TABLE]

Note that when $W_{E}(t)=W_{E}$ for all $t$, then we get precisely the $\psi_{i,s}$ defined in Section 3.2 for Algorithm 1. We present our algorithm below in Algorithm 3.

Algorithm 3 works for this ASI model much for the same reason why Algorithm 1 worked for ASI model 2: all the proof needs is that $\mathop{\operatorname{\bf E}}[X_{i,t}^{\ast}|X_{i,t^{\prime}}^{\ast}\forall t^{\prime}<t]\leq\frac{c_{i}}{m}$ for all values of $X_{i,t^{\prime}}^{\ast}$, and $\mathop{\operatorname{\bf E}}[Y_{i,t}^{\ast}|Y_{i,t^{\prime}}^{\ast}\forall t^{\prime}<t]=\frac{W_{E}(t)}{m}$ for all values of $Y_{i,t^{\prime}}^{\ast}$ (Here $X_{i,t}^{\ast}$ and $Y_{i,t}^{\ast}$ denote the random variables for resource consumption and profit at time $t$ following from allocation according the optimal solution to the expected instance of the distribution used in stage $t$).

We skip the proof for the profit guarantee of $W_{E}(1-2\epsilon)$ since it is almost identical to the proof in Section 3.2 for Algorithm 1.

3.5.3 ASI Model 3

In this model, which is otherwise identical to models 1 and 2, our benchmark is even stronger: namely, the optimal profit of the expected instance with all the time varying distributions (explicitly spelled out in LP (15)). This benchmark $W_{E}$ is the strongest benchmark possible. Correspondingly, our algorithm requires more information than in model 2: we ask for $W_{E,i}(t)$ for every $i$ and $t$, and $c_{i}(t)$ for every $i$ and $t$ at the beginning of the algorithm, where $W_{E,i}(t)$ and $c_{i}(t)$ are the amount of type $i$ profit obtained and type $i$ resource consumed by the optimal solution to the expected instance in LP (15) at step $t$. Namely, $W_{E,i}(t)=\sum_{j,k}p_{j,t}w_{ijk}x_{j,k,t}^{*}$, and $c_{i}(t)=\sum_{j,k}p_{j,t}a_{ijk}x_{j,k,t}^{*}$, where $x_{j,k,t}^{*}$’s are the optimal solution to LP (15).

A slight modification of our Algorithm 1 in Section 3.2 will give a revenue of $W_{E}(1-2\epsilon)$ with probability at least $1-\epsilon$. We modify the two potential functions $\phi_{i,s}$ and $\psi_{i,s}$ in the most natural way to account for the fact that distributions change every step. Let $W_{E,i}=\sum_{t=1}^{m}W_{E,i}(t)$, and thus, our benchmark $W_{E}$ is simply $\min_{i}W_{E,i}$. Note also that $\sum_{t=1}^{m}c_{i}(t)$, call it $c_{i}^{*}$, is at most $c_{i}$ by the feasibility of the optimal solution to LP (15).

Define

[TABLE]

We present our algorithm below in Algorithm 4.

Algorithm 4 works for this ASI model much for the same reason why Algorithm 3 worked for ASI model 2: all the proof needs is that $\mathop{\operatorname{\bf E}}[X_{i,t}^{\ast}|X_{i,t^{\prime}}^{\ast}\forall t^{\prime}<t]=\frac{c_{i}(t)}{m}$ for all values of $X_{i,t^{\prime}}^{\ast}$, and $\mathop{\operatorname{\bf E}}[Y_{i,t}^{\ast}|Y_{i,t^{\prime}}^{\ast}\forall t^{\prime}<t]=\frac{W_{E,i}(t)}{m}$ for all values of $Y_{i,t^{\prime}}^{\ast}$ (Here $X_{i,t}^{\ast}$ and $Y_{i,t}^{\ast}$ denote the random variables for resource consumption and profit at time $t$ following from allocation according the optimal solution to the expected instance captured by LP (15)).

We skip the proof for the profit guarantee of $W_{E}(1-2\epsilon)$ since it is almost identical to the proof in Section 3.2 for Algorithm 1.

4 Proof of Near-Optimality of Online Algorithm for Resource

Allocation

In this section, we construct a family of instances of the resource allocation problem in the i.i.d. setting for which $\gamma=\omega(\epsilon^{2}/\log n)$ will rule out a competitive ratio of $1-O(\epsilon)$. The construction closely follows the construction by Agrawal et al. (2014) for proving a similar result in the random-permutation model.

The instance has $n=2^{z}$ resources with $B$ units of each resource, and $Bz(2+1/\alpha)+\sqrt{Bz}$ requests where $\alpha<1$ is some scalar. Each request has only one “option”, i.e., each request can either be dropped, or if served, consumes the same number of units of a specific subset of resources (which we construct below). This means that a request is simply a scalar times a binary string of length $2^{z}$, with the ones (or the scalars) representing the coordinates of resources that are consumed by this request, if served.

The requests are classified into $z$ categories. Each category in expectation consists of $m/z=B(2+1/\alpha)+\sqrt{B/z}$ requests. A category, indexed by $i$, is composed of two different binary vectors $v_{i}$ and $w_{i}$ (each of length $2^{z}$). The easiest way to visualize these vectors is to construct two $2^{z}\times z$ $0-1$ matrices, with each matrix consisting of all possible binary strings of length $z$, written one string in a row. The first matrix lists the strings in ascending order and the second matrix in descending order. The $i$-th column of the first matrix multiplied by the scalar $\alpha$ is the vector $v_{i}$ and the $i$-th column of the second matrix is the vector $w_{i}$. There are two properties of these vectors that are useful for us:

The vectors $v_{i}/\alpha$ and $w_{i}$ are complements of one another 2. 2.

Any matrix of $z$ columns, with column $i$ being either $v_{i}/\alpha$ or $w_{i}$ has exactly one row with all ones in it.

We are ready to construct the i.i.d. instance now. Each request is drawn from the following distribution. A given request could be, for each $i$, of type:

$v_{i}$ and profit 4$\alpha$ with probability $\frac{B}{\alpha zm}$ 2. 2.

$w_{i}$ and profit 3 with probability $\frac{B}{zm}$ 3. 3.

$w_{i}$ and profit 2 with probability $\sqrt{\frac{B}{zm}}$ 4. 4.

$w_{i}$ and profit 1 with probability $\frac{B}{zm}$ 5. 5.

Zero vector with zero profit with probability $1-\frac{2B}{zm}-\sqrt{\frac{B}{zm}}-\frac{B}{\alpha zm}$

We use the following notation for request types: a $(2,w_{i})$ request stands for a $w_{i}$ type request of profit $2$. Observe that the expected instance has an optimal profit of $\mbox{\rm OPT}=7B$. This is obtained by picking for each $i$, the $\frac{B}{\alpha z}$ vectors of type $v_{i}$ and profit 4$\alpha$, along with $\frac{B}{z}$ vectors of type $w_{i}$ with profit 3. Note that this exhausts every unit of every item, and thus, combined with the fact that the most profitable requests have been served, the value of $7B$ is indeed the optimal value. This means, any algorithm that obtains a $1-\epsilon$ competitive ratio must have an expected profit of at least $7B-7\epsilon B$.

Let $r_{i}(w)$ and $r_{i}(v)$ be the random variables denoting the number of vectors of type $w_{i}$ and $v_{i}$ picked by some $1-\epsilon$ competitive algorithm $ALG$. Let $a_{i}(v)$ denote the total number of vectors of type $v_{i}$ that arrived in this instance.

Lemma 4.1

For some constant $k$, the $r_{i}(w)$’s satisfy

[TABLE]

Proof: Let $Y$ denote the set of indices $i$ for which $r_{i}(w)>B/z$. One way to upper bound the total number of vectors of type $v$ picked by $ALG$ is the following. Split the set of indices into $Y$ and $X=[z]\setminus Y$. The number of $v$’s from $Y$ is, by chosen notation, $\sum_{i\in Y}r_{i}(v)$. The number of $v$’s from $X$, we show, is at most $\frac{B-\sum_{i\in Y}r_{i}(w)}{\alpha}$. Note that since there are only $B$ copies of every item, it follows that $\alpha[\sum_{i}r_{i}(v)]\leq B$, and $\sum_{i}r_{i}(w)\leq B$. Further, by property 2 of $v_{i}$’s and $w_{i}$’s, we have that $\alpha[\sum_{i\in X}r_{i}(v)]+\sum_{i\in Y}r_{i}(w)\leq B$. This means that the number of $v$’s from $X$ is at most $\frac{B-\sum_{i\in Y}r_{i}(w)}{\alpha}$.

Let $P=\sum_{i\in Y}(r_{i}(w)-B/z)$, and $M=\sum_{i\in X}(B/z-r_{i}(w))$. Showing $\mathop{\operatorname{\bf E}}[P+M]\leq 7\epsilon B+4\sqrt{\alpha kBz}$ proves the lemma. By an abuse of notation, let $ALG$ also be the profit obtained by the algorithm $ALG$ and let best$w_{i}(t)$ denote the most profitable $t$ requests of type $w_{i}$ in a given instance. Note that $4B+\sum_{i=1}^{z}$best$w_{i}(B/z)\leq 7B=\mbox{\rm OPT}$. We upper-bound $\mathop{\operatorname{\bf E}}[ALG]$ as:

[TABLE]

The inequality that follows from CLT uses the fact that for a random variable $X\sim(m,c/m)$ ($X$ is binomially distributed with success probability of $c/m$), whenever $c=\omega(1)$, and $c\leq m$, we have that $E[X|X\geq c]=c+k^{\prime}\sqrt{c}$, for some constant $k^{\prime}$. In this case, we have $\frac{B}{\alpha z}$ in place of $c$. For example, if $n=\log(m)$ (and thus $z=\log n=\log\log m$), as long as $B=\omega(\log\log m)$ and $B\leq m$, the CLT inequality will hold. Note that $\alpha$ could have been any constant and this argument still holds.

We are now ready to prove Theorem 2.3, which we restate here for convenience. Theorem 2.3. There exist instances with $\gamma=\frac{\epsilon^{2}}{\log(n)}$ such that no algorithm, even with complete knowledge of the distribution, can get a $1-o(\epsilon)$ approximation factor.

Proof: We first give the overview of the proof before providing a detailed argument.

Overview.

Lemma 4.1 says that $r_{i}(w)$ has to be almost always close to $B/z$ for all $i$. In particular, the probability that $\sum_{i}|r_{i}(w)-B/z|\leq 4\left(7\epsilon B+4\sqrt{\alpha kBz}\right)$ is at least $3/4$. In this proof we show, in an argument similar to the one in Agrawal et al. (2014), that if this has to be true, one has to lose a revenue of $\Omega(\sqrt{Bz})-4(7\epsilon B+4\sqrt{\alpha kBz})$. Since $\alpha$ can be set to any arbitrary constant, this means that we lose a revenue of $\Omega(\sqrt{Bz})-28\epsilon B$. Since OPT is $7B$, to get a $1-\epsilon$ approximation, we require that $\Omega(\sqrt{Bz})-28\epsilon B\leq 7\epsilon B$. Thus, we need $B\geq\Omega(\frac{\log m}{\epsilon^{2}})$. In other words, we require $\gamma=\frac{1}{B}\leq O(\frac{\epsilon^{2}}{\log m})$.

In Detail

We now proceed to prove the claim that a revenue loss of $\Omega(\sqrt{Bz})-4(7\epsilon B+\sqrt{\alpha kBz})$ is inevitable. We just showed that with a probability of at least 3/4, $\sum_{i}|r_{i}(w)-B/z|\leq 4\left(7\epsilon B+4\sqrt{\alpha kBz}\right)$. For now we assume that $r_{i}(w)$ should be exactly $B/z$ and later account for the probability $1/4$ leeway and also the $4\left(7\epsilon B+4\sqrt{\alpha kBz}\right)$ error that is allowed by Lemma 4.1. With this assumption, we show that for each $i$ there is a loss of $\Omega(\sqrt{B/z})$.

For each $i$ let $o_{i}$ denote the number of $(1,w_{i})$ requests that the algorithm served in total. With a constant probability the number of $3$’s and $2$’s (of type $w_{i}$) exceed $B/z$. If $o_{i}=\Omega(\sqrt{B/z})$ there is a loss of at least $\Omega(\sqrt{B/z})$ because of picking ones instead of 2’s or 3’s. This establishes the $\Omega(\sqrt{B/z})$ loss that we wanted to prove, for this case.

Suppose $o_{i}<\Omega(\sqrt{Bz})$. For each $i$, let $R_{i}$ be the set of requests of type $w_{i}$ with profit either 1 or 3. For every $i$, with a constant probability $2B/z-2\sqrt{B/z}\leq|R_{i}|\leq 2B/z+2\sqrt{B/z}$. Conditional on the set $R_{i}$ we make the following two observations:

• •

the types of requests in $R_{i}$ are independent random variables that take value $1$ or $3$ with equal probability.

• •

the order of requests in $R_{i}$ is a uniformly random permutation of $R_{i}$

Now consider any $(2,w_{i})$ request, say $t$-th request, of profit 2. With a constant probability this request can be served without violating any capacity constraints, and thus, the algorithm has to decide whether or not to serve this request. In at least 1/2 of the random permutations of $R_{i}$, the number of bids from set $R_{i}$ before the bid $t$ is less than $B/z$. Conditional on this event, the profits of requests in $R_{i}$ before $t$, with a constant probability could:

take values such that there are enough $(3,w_{i})$ requests after $t$ to make the total number of $w_{i}$ requests picked by the algorithm to be at least $B/z$; 2. 2.

take values such that even if all the $(3,w_{i})$ requests after $t$ were picked, the total number of $w_{i}$ requests picked is at most $B/z-\sqrt{B/z}$ with a constant probability.

In the first kind of instances (where number of $(3,w_{i})$ requests are more than $B/z$) retaining $(2,w_{i})$ causes a loss of 1 as we could have picked a 3 instead. In the second kind, skipping $(2,w_{i})$ causes a loss of $1$ since we could have picked that $2$ instead of a $1$. Thus there is an inevitable constant probability loss of 1 per $(2,w_{i})$ request. Thus in expectation, there is a $\Omega(\sqrt{B/z})$ loss.

Thus whether $o_{i}=\sqrt{B/z}$ or $o_{i}<\sqrt{B/z}$, we have established a loss of $\Omega(\sqrt{B/z})$ per $i$ and thus a total expected loss of $\Omega(\sqrt{Bz})$. This is under the assumption that $r_{i}(w)$ is exactly $B/z$. There is a leeway of $4\left(7\epsilon B+4\sqrt{\alpha kBz}\right)$ granted by Lemma 4.1. Even after that leeway, since $\alpha$ can be made an arbitrarily small constant and Lemma 4.1 still holds, we have the loss at $\Omega(\sqrt{Bz})-28\epsilon B$. Now after the leeway, the statement $\sum_{i}|r_{i}(w)-B/z|\leq 4\left(7\epsilon B+4\sqrt{\alpha kBz}\right)$ has to hold only with probability $3/4$. But even this puts the loss at $\Omega(\sqrt{Bz})-21\epsilon B$

Therefore, $E[ALG]\leq\mbox{\rm OPT}-\Omega(\sqrt{Bz})-21\epsilon B.$ Since $\mbox{\rm OPT}=7B$, we have $E[ALG]\leq\mbox{\rm OPT}(1-\Omega(\sqrt{z/B})-21\epsilon)$, and in order to get $1-O(\epsilon)$ approximation we need $\Omega(\sqrt{z/B}-21\epsilon)\leq O(\epsilon)$, implying that $B\geq\Omega(z/\epsilon^{2})=\Omega(\log m/\epsilon^{2}).$

5 Greedy Algorithm for Adwords

In this section, we give a simple proof of Theorem 2.4, which we restate below for convenience. Theorem 2.4. The greedy algorithm achieves an approximation factor of $1-1/e$ for the Adwords problem in the i.i.d. unknown distributions model for all $\gamma$, i.e., $0\leq\gamma\leq 1$.

As noted in Section 2.2 where the Adwords problem was introduced, the budget constraints are not hard, i.e., when a query $j$ arrives, with a bid amount $b_{ij}>$ remaining budget of $i$, we are still allowed to allot that query to advertiser $i$, but we only earn a revenue of the remaining budget of $i$, and not the total value $b_{ij}$.

Goel and Mehta (2008) prove that the greedy algorithm gives a $(1-1/e)$ approximation to the adwords problem when the queries arrive in a random permutation or in i.i.d., but under an assumption which almost gets down to $\gamma$ tending to zero, i.e., bids being much smaller than budgets. We give a much simpler proof for a $(1-1/e)$ approximation by greedy algorithm for the i.i.d. unknown distributions case, and our proof works for all $\gamma$.

Let $p_{j}$ be the probability of query $j$ appearing in any given impression. Let $y_{j}=mp_{j}$. Let $x_{ij}$ denote the offline fractional optimal solution for the expected instance. Let $w_{i}(t)$ denote the amount of money spent by advertiser $i$ at time step $t$, i.e., for the $t$-th query in the greedy algorithm (to be described below). Let $f_{i}(0)=\sum_{j}b_{ij}x_{ij}y_{j}$. Let $f_{i}(t)=f_{i}(0)-\sum_{r=1}^{t}w_{i}(r)$. Let $f(t)=\sum_{i=1}^{n}f_{i}(t)$. Note that $f_{i}(0)$ is the amount spent by $i$ in the offline fractional optimal solution to the expected instance.

Consider the greedy algorithm which allocates the query $j$ arriving at time $t$ to the advertiser who has the maximum effective bid for that query, i.e., $\underset{i}{\mathrm{argmax}}\min\{b_{ij},B_{i}-\sum_{r=1}^{t-1}w_{i}(r)\}$. We prove that this algorithm obtains a revenue of $(1-1/e)\sum_{i,j}b_{ij}x_{ij}y_{j}$ and thus gives the desired $1-1/e$ competitive ratio against the fractional optimal solution to the expected instance. Consider a hypothetical algorithm that allocates queries to advertisers according to the $x_{ij}$’s. We prove that this hypothetical algorithm obtains an expected revenue of $(1-1/e)\sum_{i,j}b_{ij}x_{ij}y_{j}$, and argue that the greedy algorithm only performs better. Let $w^{h}_{i}(t)$ and $f^{h}_{i}(t)$ denote the quantities analogous to $w_{i}(t)$ and $f_{i}(t)$ for the hypothetical algorithm, with the initial value $f^{h}_{i}(0)=f_{i}(0)=\sum_{j}b_{ij}x_{ij}y_{j}$. Let $f^{h}(t)=\sum_{i=1}^{n}f^{h}_{i}(t)$. Let EXCEED${}_{i}(t)$ denote the set of all $j$ such that $b_{ij}$ is strictly greater than the remaining budget at the beginning of time step $t$, namely $b_{ij}>B_{i}-\sum_{r=1}^{t-1}w^{h}_{i}(r).$

Lemma 5.1

$\mathop{\operatorname{\bf E}}[w^{h}_{i}(t)|f^{h}_{i}(t-1)]\geq\frac{f^{h}_{i}(t-1)}{m}$**

Proof: The expected amount amount of money spent at time step $t$, is given by

[TABLE]

If $\underset{j\in\text{EXCEED}_{i}(t)}{\sum}x_{ij}y_{j}\geq 1$, then by (16),

[TABLE]

Suppose on the other hand $\underset{j\in\text{EXCEED}_{i}(t)}{\sum}x_{ij}y_{j}<1.$ We can write $\mathop{\operatorname{\bf E}}[w^{h}_{i}(t)|f^{h}_{i}(t-1)]$ as

[TABLE]

Since $b_{ij}\leq B_{i}$, and $\underset{j\in\text{EXCEED}_{i}(t)}{\sum}x_{ij}y_{j}<1$, (17) can be simplified to

[TABLE]

Lemma 5.2

The hypothetical algorithm satisfies the following: $\mathop{\operatorname{\bf E}}[f^{h}(t)|f^{h}(t-1)]\leq f^{h}(t-1)(1-1/m)$

Proof: From the definition of $f^{h}_{i}(t)$, we have

[TABLE]

where the inequality is due to Lemma 5.1. Summing over all $i$ gives the Lemma.

Lemma 5.3

$\mathop{\operatorname{\bf E}}[\mathrm{GREEDY}]\geq(1-1/e)\sum_{i,j}b_{ij}x_{ij}y_{j}$**

Proof: Lemma 5.2 proves that for the hypothetical algorithm, the value of the difference $f^{h}(t-1)-\mathop{\operatorname{\bf E}}[f^{h}(t)|f^{h}(t-1)]$, which is the expected amount spent at time $t$ by all the advertisers together, conditioned on $f^{h}(t-1)$, is at least $\frac{f^{h}(t-1)}{m}$. But by definition, conditioned on the amount of money spent in first $t-1$ steps, the greedy algorithm earns the maximum revenue at time step $t$ . Thus, for the greedy algorithm too, the statement of the lemma 5.2 must hold, namely, $\mathop{\operatorname{\bf E}}[f(t)|f(t-1)]\leq f(t-1)(1-1/m)$. This means that $\mathop{\operatorname{\bf E}}[f(m)]\leq f(0)(1-1/m)^{m}\leq f(0)(1/e)$. Thus the expected revenue earned is

[TABLE]

and this proves the lemma.

Lemma 5.3 proves Theorem 2.4.

6 Fast Approximation Algorithm for Large Mixed Packing & Covering

Integer Programs

In this section, we consider the mixed packing-covering problem stated in Section 2.4. and prove Theorem 2.5. We restate the integer program for the mixed covering-packing problem here.

[TABLE]

The goal is to check if there is a feasible solution to this IP. We solve a gap version of this problem. Distinguish between the two cases with a high probability, say $1-\delta$:

• •

YES: There is a feasible solution.

• •

NO: There is no feasible solution even with a slack, namely, even if all of the $c_{i}$’s are multiplied by $1+3\epsilon(1+\epsilon)$ and all of the $d_{i}$’s are multiplied by $1-3\epsilon(1+\epsilon)$.

We use $1+3\epsilon(1+\epsilon)$ and $1-3\epsilon(1+\epsilon)$ for slack instead of just $1+\epsilon$ and $1-\epsilon$ purely to reduce notational clutter in what follows (mainly for the $\mathrm{NO}$ case).

Like in the online problem, we refer to the quantities indexed by $j$ as requests, $a_{ijk}$ as resource $i$ consumption, and $w_{ijk}$ as resource $i$ profit, and the quantities indexed by $k$ as options. There are a total of $m$ requests, $n$ resources, and $K$ options, and the “zero” option denoted by $\bot$. Recall that the parameter $\gamma$ for this problem is defined by $\gamma=\max\left(\left\{\frac{a_{ijk}}{c_{i}}\right\}_{i,j,k}\cup\left\{\frac{w_{ijk}}{d_{i}}\right\}_{i,j,k}\right)$. Our algorithm needs the value of $m$, $n$ and $\gamma$ (an upper bound on the value of $\gamma$ also suffices).

High-level overview.

We solve this offline problem in an online manner via random sampling. We sample $T=\Theta(\frac{\gamma m\log(n/\delta)}{\epsilon^{2}})$ requests $j$ from the set of possible requests uniformly at random with replacement, and then design an algorithm that allocates resources online for these requests. At the end of serving $T$ requests we check if the obtained solution proportionally satisfies the constraints of IP (6). If yes, we declare YES as the answer and declare NO otherwise. At the core of the solution is the online sampling algorithm we use, which is identical to the techniques used to develop the online algorithm in Sections 3.2 and 3.3. We describe our algorithm in Algorithm 5

The main theorem of this section is Theorem 2.5, which we restate here: Theorem 2.5. For any $\epsilon>0$, Algorithm 5 solves the gap version of the mixed covering-packing problem with $\Theta(\frac{\gamma m\log(n/\delta)}{\epsilon^{2}})$ oracle calls.

Detailed Description and Proof

The proof is in two parts. The first part proves that our algorithm indeed answers YES when the actual answer is YES with a probability at least $1-\delta$. The second part is the identical statement for the NO case.

The YES case

We begin with the case where the true answer is YES. Let $x_{jk}^{\ast}$ denote some feasible solution to the LP relaxation of IP  (6). In a similar spirit to Sections 3.1, 3.2 and 3.3, we define the algorithm $P$ as follows. It samples a total of $T=\Theta(\frac{\gamma m\log(n/\delta)}{\epsilon^{2}})$ requests uniformly at random, with replacement, from the total pool of $m$ requests. When request $j$ is sampled, $P$ serves $j$ using option $k$ with probability $x_{jk}^{\ast}$. Thus, if we denote by $X_{i,t}^{\ast}$ the consumption of resource $i$ in step $t$ of $P$, then we have $\mathop{\operatorname{\bf E}}[X_{i,t}^{\ast}]=\sum_{j=1}^{m}\frac{1}{m}\sum_{k}a_{ijk}{x_{jk}^{\ast}}\leq\frac{c_{i}}{m}$. This inequality follows from $x_{jk}^{\ast}$ being a feasible solution to LP relaxation of (6). Similarly let $Y_{i,t}^{\ast}$ denote the resource $i$ profit in step $t$ of $P$. We have $\mathop{\operatorname{\bf E}}[Y_{i,t}^{\ast}]\geq\frac{d_{i}}{m}$. We now write the probability that our condition for YES is violated for some algorithm $A$.

[TABLE]

[TABLE]

If our algorithm $A$ was $P$ \bigg{(}and therefore we can use $\mathop{\operatorname{\bf E}}[X_{i,t}^{\ast}]\leq\frac{c_{i}}{m}$ and \mathop{\operatorname{\bf E}}[Y_{i,t}^{\ast}]\geq\frac{d_{i}}{m}\bigg{)}, the total failure probability in the YES case, which is the sum of (6) and (6) for all the $i$’s would have been at most $\delta$, if $T=\Theta(\frac{\gamma m\log(n/\delta)}{\epsilon^{2}})$ for an appropriate constant inside $\Theta$. The goal is to design an algorithm $A$ that, unlike $P$, does not first solve LP relaxation of IP (6) and then use $x_{jk}^{\ast}$’s to allocate resources, but allocates online and also obtains the same $\delta$ failure probability, just as we did in Sections 3.2 and 3.3. That is we want to show that the sum of (6) and (6) over all $i$’s is at most $\delta$:

[TABLE]

For the algorithm $A^{s}P^{T-s}$, the above quantity can be rewritten as

[TABLE]

which, by using $(1+\epsilon)^{x}\leq 1+\epsilon x$ for $0\leq x\leq 1$, is in turn upper bounded by

[TABLE]

Since for all $t$, the random variables $X_{i,t}^{\ast}$, $X_{i,t}^{A}$, $Y_{i,t}^{\ast}$ and $Y_{i,t}^{A}$ are all independent, and $\mathop{\operatorname{\bf E}}[X_{i,t}^{\ast}]\leq\frac{c_{i}}{m}$, $\mathop{\operatorname{\bf E}}[Y_{i,t}^{\ast}]\geq\frac{d_{i}}{m}$, the above is in turn upper bounded by

[TABLE]

Let $\mathcal{F}[A^{s}P^{T-s}]$ denote the quantity in  (21), which is an upper bound on failure probability of the hybrid algorithm $A^{s}P^{T-s}$. We just said that $\mathcal{F}[P^{T}]\leq\delta$. We now prove that for all $s\in\{0,1,\dots,T-1\}$, $\mathcal{F}[A^{s+1}P^{T-s-1}]\leq\mathcal{F}[A^{s}P^{T-s}]$, thus proving that $\mathcal{F}[A^{T}]\leq\delta$, i.e., running the algorithm $A$ for all the $T$ steps of stage $r$ results in a failure with probability at most $\delta$.

Assuming that for all $s<p$, the algorithm $A$ has been defined for the first $s+1$ steps in such a way that $\mathcal{F}[A^{s+1}P^{T-s-1}]\leq\mathcal{F}[A^{s}P^{T-s}]$, we now define $A$ for the $p+1$-th step of stage $r$ in a way that will ensure that $\mathcal{F}[A^{p+1}P^{T-p-1}]\leq\mathcal{F}[A^{p}P^{T-p}]$. We have

[TABLE]

Define

[TABLE]

Define the step $p+1$ of algorithm $A$ as picking the following option $k$ for request $j$:

[TABLE]

By the above definition of step $p+1$ of algorithm $A$, it follows that for any two algorithms with the first $p$ steps being identical, and the last $T-p-1$ steps following the Hypothetical-Oblivious algorithm $P$, algorithm $A$’s $p+1$-th step is the one that minimizes expression (22). In particular it follows that expression (22) is upper bounded by the same expression where the $p+1$-the step is according to $X_{i,p+1}^{\ast}$ and $Y_{i,p+1}^{\ast}$, i.e., we replace $X_{i,p+1}^{A}$ by $X_{i,p+1}^{\ast}$ and $Y_{i,p+1}^{A}$ by $Y_{i,p+1}^{\ast}$. Therefore we have

[TABLE]

The NO case

We now proceed to prove that when the real answer is $\mathrm{NO}$, our algorithm says $\mathrm{NO}$ with a probability at least $1-\delta$. To prove this result (formally stated in Lemma 6.3), we use as a tool the fact that when the integer program in (6) is in the $\mathrm{NO}$ case where even a slack of $3\epsilon(1+\epsilon)$ will not make it feasible, then even the LP relaxation of (6) will be infeasible with a slack of $2\epsilon$. We prove this statement now by proving its contrapositive in Lemma 6.1.

Lemma 6.1

If the LP relaxation of (6) is feasible with a slack of $s$, then the integer program in (6) is feasible with a slack of $s(1+\epsilon)+\epsilon$.

Proof: To prove this, we write the LP relaxation of the integer program in (6) slightly differently below.

The optimal value $\lambda^{*}$ of the primal LP in (23) represents the slack in the YES/NO problem. I.e., if $\lambda^{*}=0$, then we have zero slack and hence are in the YES case. Else, we are in the NO case with a slack equal to $\lambda^{*}$. Given this, all we have to show is that when the LP in (23) has an optimal value of $\lambda^{*}$, then the corresponding integer program’s optimal solution is at most $\lambda^{*}(1+\epsilon)+\epsilon$. To see this is true, let $x_{j,k}^{*}$ denote the optimal solution to primal LP (23). Consider the integral solution that does a randomized rounding of the $x_{j,k}^{*}$’s and allocates according to these rounded integers, and let $X_{jk}$ be the corresponding $\{0,1\}$ random variable. Let random variable $X_{ij}=\sum_{k}\frac{a_{ijk}X_{jk}}{c_{i}}$. By the definition of LP (23), we have $\mathop{\operatorname{\bf E}}[\sum_{j}X_{ij}]\leq 1+\lambda^{*}$. Noting that each of the $X_{ij}$’s is at most $\gamma$, by Chernoff bounds it follows that $\mbox{\bf Pr}[\sum_{j}X_{ij}\geq(1+\lambda^{*})(1+\epsilon)]\leq e^{-\frac{\epsilon^{2}}{4\gamma}}$, which given that $\gamma=O(\frac{\epsilon^{2}}{\log(n/\epsilon)})$, is at most $\frac{\epsilon}{2n}$ (the probability derivation is just like the derivation in Section 3.1). Likewise, if we define by $Y_{ij}$ the random variable $\sum_{k}\frac{w_{ijk}X_{jk}}{d_{i}}$, then by the definition of LP (23), we have $\mathop{\operatorname{\bf E}}[\sum_{j}Y_{ij}]\geq 1-\lambda^{*}$. By Chernoff bounds, we get an identical argument, we get $\mbox{\bf Pr}[\sum_{j}Y_{ij}\leq(1-\lambda^{*})(1-\epsilon)]\leq\frac{\epsilon}{2n}$. By doing a union bound over the $2n$ Chernoff bounds, we have that the randomized rounding integer solution is feasible with an optimal value of at most $\lambda^{*}+\epsilon+\lambda^{*}\epsilon$ with probability at least $1-\epsilon$. This means that there exists an integer solution of value $\lambda^{*}+\epsilon+\lambda^{*}\epsilon$, and this proves the lemma.

Corollary 6.2

For a $\mathrm{NO}$ instance with a slack of $3\epsilon(1+\epsilon)$, the LP relaxation of the instance is still infeasible with a slack of $2\epsilon$. In particular, this implies that the optimal value of the primal $\lambda^{*}$ in (23) is at least $2\epsilon$, and likewise the optimal dual value in (23) is $\sum_{i}(\rho_{i}^{*}-\alpha_{i}^{*})-\sum_{j}\beta_{j}^{*}\geq 2\epsilon$.

Lemma 6.3

For a $\mathrm{NO}$ instance, if $T\geq\Theta(\frac{\gamma m\log(n/\delta)}{\epsilon^{2}})$, then

[TABLE]

Proof: Let $R$ denote the set of requests sampled. Consider the following LP.

If the primal in LP (24) has an optimal objective value at least $\frac{T\epsilon}{m}$, then by definition of our algorithm 5, we would have declared $\mathrm{NO}$, i.e., if the sampled LP itself had a slack of $\epsilon$ (scaled by $\frac{T}{m}$), then no integral allocation based on those samples can obtain a smaller slack. We now show that by picking $T=\Theta(\frac{\gamma m\ln(n/\delta)}{\epsilon^{2}})$, the above LP (24) will have its optimal objective value at least $\frac{T\epsilon}{m}$, with a probability at least $1-\delta$. This makes our algorithm answer $\mathrm{NO}$ with a probability at least $1-\delta$.

Now, the primal of LP (24) has an optimal value equal to that of the dual which in turn is lower bounded by the value of dual at any feasible solution. One such feasible solution is $\alpha^{*},\beta^{*},\rho^{*}$, which is the optimal solution to the full version of the dual in LP (24), namely the one written in LP (23) where $R=[m]$, $T=m$. This is because the set of constraints in the full version of the dual is clearly a superset of the constraints in the dual of LP (24). Thus, the optimal value of the primal of LP (24) is lower bounded by value of dual at $\alpha^{*},\beta^{*},\rho^{*}$, which is

[TABLE]

For proceeding further in lower bounding (25), we apply Chernoff bounds to $\sum_{j\in R}\beta^{*}_{j}$. The fact that the dual of the full version of LP (24) is a maximization LP, coupled with the constraints there in imply that $\beta_{j}^{*}\leq\gamma$. Further, let $\tau^{*}$ denote the optimal value of the full version of LP (24), i.e., $\sum_{i}(\rho_{i}^{*}-\alpha_{i}^{*})-\sum_{j}\beta_{j}^{*}=\tau^{*}$. Now, the constraint $\sum_{i}(\alpha_{i}^{*}+\rho_{i}^{*})\leq 1$ coupled with the fact that $\tau^{*}\geq 0$ implies $\sum_{j}\beta_{j}^{*}\leq 1$. We are now ready to lower bound the quantity in (25). We have the optimal solution to primal of LP (24)

[TABLE]

where the second inequality is a “with probability at least $1-\delta$” inequality, i.e., we apply Chernoff bounds for $\sum_{j\in S}\beta^{*}_{j}$, along with the observation that each $\beta^{*}_{j}\in[0,\gamma]$. The third inequality follows from $\sum_{j}\beta_{j}^{*}\leq 1$ and $\sum_{i}(\rho_{i}^{*}-\alpha_{i}^{*})-\sum_{j}\beta_{j}^{*}=\tau^{*}$. Setting $T=\Theta(\frac{\gamma m\ln(n/\delta)}{\epsilon^{2}})$ with a appropriate constant inside the $\Theta$, coupled with the fact that $\tau^{*}\geq 2\epsilon$ in the NO case (see Corollary 6.2), it is easy to verify that the quantity in (26) is at least $\frac{T\epsilon}{m}$.

Going back to our application of Chernoff bounds above, in order to apply it in the form above, we require that the multiplicative deviation from mean $\sqrt{\frac{4\gamma m\ln(1/\delta)}{T\sum_{j}\beta_{j}^{*}}}\in[0,2e-1]$. If $\sum_{j}\beta_{j}^{*}\geq\epsilon^{2}$, then this requirement would follow. Suppose on the other hand that $\sum_{j}\beta_{j}^{*}<\epsilon^{2}$. Since we are happy if the excess over mean is at most $\frac{T\epsilon}{m}$, let us look for a multiplicative error of $\frac{\frac{T\epsilon}{m}}{\frac{T\sum_{j}\beta_{j}^{*}}{m}}$. Based on the fact that $\sum_{j}\beta_{j}^{*}<\epsilon^{2}$ the multiplicative error can be seen to be at least $1/\epsilon$ which is larger than $2e-1$ when $\epsilon<\frac{1}{2e-1}$. We now use the version of Chernoff bounds for multiplicative error larger than $2e-1$, which gives us that a deviation of $\frac{T\epsilon}{m}$ occurs with a probability at most $2^{-\left(1+\frac{\frac{T\epsilon}{m}}{\frac{T\sum_{j}\beta_{j}^{*}}{m}}\right)\frac{T\sum_{j}\beta_{j}^{*}}{m\gamma}}$, where the division by $\gamma$ is because of the fact that $\beta_{j}^{*}\leq\gamma$. This probability is at most $(\frac{\delta}{n})^{1/\epsilon}$, which is at most $\delta$.

The proofs for the YES and NO cases together prove Theorem 2.5.

7 Special Cases of the Resource Allocation

Framework

We now list the problems that are special cases of the resource allocation framework and have been previously considered. The Adwords and Display ads special cases were already discussed in Section 2.2.

7.1 Network Routing and Load Balancing

Consider a graph (either undirected or directed) with edge capacities. Requests arrive online; a request $j$ consists of a source-sink pair, $(s_{j},t_{j})$ and a bandwidth $\rho_{j}$. In order to satisfy a request, a capacity of $\rho_{j}$ must be allocated to it on every edge along some path from $s_{j}$ to $t_{j}$ in the graph. In the throughput maximization version, the objective is to maximize the number of satisfied requests while not allocating more bandwidth than the available capacity for each edge (Different requests could have different values on them, and one could also consider maximizing the total value of the satisfied requests). Our Algorithm 2 for resource allocation framework directly applies here and the approximation guarantee there directly carries over.  Kamath et al. (1996) consider a different version of this problem where requests according to a Poisson process with unknown arrival rates. Each request has an associated holding time that is assumed to be exponentially distributed, and once a request has been served, the bandwidth it uses up gets freed after its holding time (this is unlike our setting where once a certain amount of resource capacity has been consumed, it remains unavailable to all future requests). Some aspects of the distribution are assumed to be known, namely, that the algorithm knows the average rate of profit generated by all the incoming circuits, the average holding time, and also the target offline optimal offline solution that the online algorithm is aiming to approximate (again this is unlike our setting where no aspect of the request distribution is known to the algorithm, and there could even be some adversarial aspects like in the ASI model). When each request consumes at most $\gamma$ fraction of any edge’s bandwidth, Kamath et al. (1996) give an online algorithm that achieves an expected profit of $(1-\epsilon)$ times the optimal offline solution when $\gamma=O(\frac{\epsilon^{2}}{\log n})$.

7.2 Combinatorial Auctions

Suppose we have $n$ items for sale, with $c_{i}$ copies of item $i$. Bidders arrive online, and bidder $j$ has a utility function $U_{j}:2^{[n]}\rightarrow\mbox{\rm\bf R}$. If we posted prices $p_{i}$ for each item $i$, then bidder $j$ buys a bundle $S$ that maximizes $U_{j}(S)-\sum_{i\in S}p_{i}$. We assume that bidders can compute such a bundle. The goal is to maximize social welfare, the total utility of all the bidders, subject to the supply constraint that there are only $c_{i}$ copies of item $i$. Firstly, incentive constraints aside, this problem can be written as an LP in the resource allocation framework. The items are the resources and agents arriving online are the requests. All the different subsets of items form the set of options. The utility $U_{j}(S)$ represents the profit $w_{j,S}$ of serving agent $j$ through option $S$, i.e. subset $S$. If an item $i\in S$, then $a_{i,j,S}=1$ for all $j$ and zero otherwise. Incentive constraints aside, our algorithm for resource allocation at step $s$, will choose the option $k^{*}$ (or equivalently the bundle $S$) as specified in point 8 of Algorithm 2, i.e., minimize the potential function. That is, if step $s$ falls in stage $r$,

[TABLE]

(note that unlike Algorithm 2 there is no subscripting for $w_{j,k}$). This can be equivalently written as

[TABLE]

Now, maximizing the above expression at step $s$ is the same as picking the $k$ to maximize $w_{j,k}-\sum_{i}p_{i}(s)a_{ijk}$, where $p_{i}(s)=\frac{\phi_{i,s-1}^{r}}{\psi_{s-1}^{r}}$. Thus, if we post a price of $p_{i}(s)$ on item $i$ for bidder number $s$, he will do exactly what the algorithm would have done otherwise. Suppose that the bidders are i.i.d samples from some distribution (or they arrive as in the adversarial stochastic input model). We can use Theorem 2.2 to get an incentive compatible posted price auction999Here we assume that each agent reveals his true utility function after he makes his purchase. This information is necessary to compute the prices to be charged for future agents. with a competitive ratio of $1-O(\epsilon)$ whenever $\gamma=\min_{i}\{c_{i}\}\geq\Omega\left(\frac{\log(n/\epsilon)}{\epsilon^{2}}\right).$ Further if an analog of Theorem 2.2 also holds in the random permutation model then we get a similar result for combinatorial auctions in the offline case: we simply consider the bidders one by one in a random order.