arXiv subjects

S. Muthukrishnan

Publications and source records attributed to S. Muthukrishnan.

At least 55 records · Page 3Linked to original sources

arXiv2010

Approximation Schemes for Sequential Posted Pricing in Multi-Unit Auctions

We design algorithms for computing approximately revenue-maximizing {\em sequential posted-pricing mechanisms (SPM)} in $K$-unit auctions, in a standard Bayesian model. A seller has $K$ copies of an item to sell, and there are $n$ buyers, each interested in only one copy, who have some value for the item. The seller must post a price for each buyer, the buyers arrive in a sequence enforced by the seller, and a buyer buys the item if its value exceeds the price posted to it. The seller does not know the values of the buyers, but have Bayesian information about them. An SPM specifies the ordering of buyers and the posted prices, and may be {\em adaptive} or {\em non-adaptive} in its behavior. The goal is to design SPM in polynomial time to maximize expected revenue. We compare against the expected revenue of optimal SPM, and provide a polynomial time approximation scheme (PTAS) for both non-adaptive and adaptive SPMs. This is achieved by two algorithms: an efficient algorithm that gives a $(1-\frac{1}{\sqrt{2πK}})$-approximation (and hence a PTAS for sufficiently large $K$), and another that is a PTAS for constant $K$. The first algorithm yields a non-adaptive SPM that yields its approximation guarantees against an optimal adaptive SPM -- this implies that the {\em adaptivity gap} in SPMs vanishes as $K$ becomes larger.

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arXiv2009

Quasi-Proportional Mechanisms: Prior-free Revenue Maximization

Inspired by Internet ad auction applications, we study the problem of allocating a single item via an auction when bidders place very different values on the item. We formulate this as the problem of prior-free auction and focus on designing a simple mechanism that always allocates the item. Rather than designing sophisticated pricing methods like prior literature, we design better allocation methods. In particular, we propose quasi-proportional allocation methods in which the probability that an item is allocated to a bidder depends (quasi-proportionally) on the bids. We prove that corresponding games for both all-pay and winners-pay quasi-proportional mechanisms admit pure Nash equilibria and this equilibrium is unique. We also give an algorithm to compute this equilibrium in polynomial time. Further, we show that the revenue of the auctioneer is promisingly high compared to the ultimate, i.e., the highest value of any of the bidders, and show bounds on the revenue of equilibria both analytically, as well as using experiments for specific quasi-proportional functions. This is the first known revenue analysis for these natural mechanisms (including the special case of proportional mechanism which is common in network resource allocation problems).

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arXiv2009

Online Stochastic Matching: Beating 1-1/e

We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of $1-1/e$. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the $1 - 1/e$ bound. Our main result is a 0.67-approximation online algorithm for stochastic bipartite matching, breaking this $1 - {1/e}$ barrier. Furthermore, we show that no online algorithm can produce a $1-ε$ approximation for an arbitrarily small $ε$ for this problem. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-)matchings. These two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution.

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arXiv2009

Optimal cache-aware suffix selection

Given string $S[1..N]$ and integer $k$, the {\em suffix selection} problem is to determine the $k$th lexicographically smallest amongst the suffixes $S[i... N]$, $1 \leq i \leq N$. We study the suffix selection problem in the cache-aware model that captures two-level memory inherent in computing systems, for a \emph{cache} of limited size $M$ and block size $B$. The complexity of interest is the number of block transfers. We present an optimal suffix selection algorithm in the cache-aware model, requiring $\Thetah{N/B}$ block transfers, for any string $S$ over an unbounded alphabet (where characters can only be compared), under the common tall-cache assumption (i.e. $M=\Omegah{B^{1+ε}}$, where $ε<1$). Our algorithm beats the bottleneck bound for permuting an input array to the desired output array, which holds for nearly any nontrivial problem in hierarchical memory models.

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arXiv2009

Bid Optimization in Broad-Match Ad auctions

Ad auctions in sponsored search support ``broad match'' that allows an advertiser to target a large number of queries while bidding only on a limited number. While giving more expressiveness to advertisers, this feature makes it challenging to optimize bids to maximize their returns: choosing to bid on a query as a broad match because it provides high profit results in one bidding for related queries which may yield low or even negative profits. We abstract and study the complexity of the {\em bid optimization problem} which is to determine an advertiser's bids on a subset of keywords (possibly using broad match) so that her profit is maximized. In the query language model when the advertiser is allowed to bid on all queries as broad match, we present an linear programming (LP)-based polynomial-time algorithm that gets the optimal profit. In the model in which an advertiser can only bid on keywords, ie., a subset of keywords as an exact or broad match, we show that this problem is not approximable within any reasonable approximation factor unless P=NP. To deal with this hardness result, we present a constant-factor approximation when the optimal profit significantly exceeds the cost. This algorithm is based on rounding a natural LP formulation of the problem. Finally, we study a budgeted variant of the problem, and show that in the query language model, one can find two budget constrained ad campaigns in polynomial time that implement the optimal bidding strategy. Our results are the first to address bid optimization under the broad match feature which is common in ad auctions.

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arXiv2008

General Auction Mechanism for Search Advertising

In sponsored search, a number of advertising slots is available on a search results page, and have to be allocated among a set of advertisers competing to display an ad on the page. This gives rise to a bipartite matching market that is typically cleared by the way of an automated auction. Several auction mechanisms have been proposed, with variants of the Generalized Second Price (GSP) being widely used in practice. A rich body of work on bipartite matching markets builds upon the stable marriage model of Gale and Shapley and the assignment model of Shapley and Shubik. We apply insights from this line of research into the structure of stable outcomes and their incentive properties to advertising auctions. We model advertising auctions in terms of an assignment model with linear utilities, extended with bidder and item specific maximum and minimum prices. Auction mechanisms like the commonly used GSP or the well-known Vickrey-Clarke-Groves (VCG) are interpreted as simply computing a \emph{bidder-optimal stable matching} in this model, for a suitably defined set of bidder preferences. In our model, the existence of a stable matching is guaranteed, and under a non-degeneracy assumption a bidder-optimal stable matching exists as well. We give an algorithm to find such matching in polynomial time, and use it to design truthful mechanism that generalizes GSP, is truthful for profit-maximizing bidders, implements features like bidder-specific minimum prices and position-specific bids, and works for rich mixtures of bidders and preferences.

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arXiv2008

Range Medians

We study a generalization of the classical median finding problem to batched query case: given an array of unsorted $n$ items and $k$ (not necessarily disjoint) intervals in the array, the goal is to determine the median in {\em each} of the intervals in the array. We give an algorithm that uses $O(n\log n + k\log k \log n)$ comparisons and show a lower bound of $Ω(n\log k)$ comparisons for this problem. This is optimal for $k=O(n/\log n)$.

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arXiv2008

Algorithmic Methods for Sponsored Search Advertising

Modern commercial Internet search engines display advertisements along side the search results in response to user queries. Such sponsored search relies on market mechanisms to elicit prices for these advertisements, making use of an auction among advertisers who bid in order to have their ads shown for specific keywords. We present an overview of the current systems for such auctions and also describe the underlying game-theoretic aspects. The game involves three parties--advertisers, the search engine, and search users--and we present example research directions that emphasize the role of each. The algorithms for bidding and pricing in these games use techniques from three mathematical areas: mechanism design, optimization, and statistical estimation. Finally, we present some challenges in sponsored search advertising.

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arXiv2008

Online Ad Slotting With Cancellations

Many advertisers buy advertisements (ads) on the Internet or on traditional media and seek simple, online mechanisms to reserve ad slots in advance. Media publishers represent a vast and varying inventory, and they too seek automatic, online mechanisms for pricing and allocating such reservations. In this paper, we present and study a simple model for auctioning such ad slots in advance. Bidders arrive sequentially and report which slots they are interested in. The seller must decide immediately whether or not to grant a reservation. Our model allows a seller to accept reservations, but possibly cancel the allocations later and pay the bidder a cancellation compensation (bump payment). Our main result is an online mechanism to derive prices and bump payments that is efficient to implement. This mechanism has many desirable properties. It is individually rational; winners have an incentive to be honest and bidding one's true value dominates any lower bid. Our mechanism's efficiency is within a constant fraction of the a posteriori optimally efficient solution. Its revenue is within a constant fraction of the a posteriori revenue of the Vickrey-Clarke-Groves mechanism. Our results make no assumptions about the order of arrival of bids or the value distribution of bidders and still hold if the items for sale are elements of a matroid, a more general setting than slot allocation.

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arXiv2008

A Truthful Mechanism for Offline Ad Slot Scheduling

We consider the "Offline Ad Slot Scheduling" problem, where advertisers must be scheduled to "sponsored search" slots during a given period of time. Advertisers specify a budget constraint, as well as a maximum cost per click, and may not be assigned to more than one slot for a particular search. We give a truthful mechanism under the utility model where bidders try to maximize their clicks, subject to their personal constraints. In addition, we show that the revenue-maximizing mechanism is not truthful, but has a Nash equilibrium whose outcome is identical to our mechanism. As far as we can tell, this is the first treatment of sponsored search that directly incorporates both multiple slots and budget constraints into an analysis of incentives. Our mechanism employs a descending-price auction that maintains a solution to a certain machine scheduling problem whose job lengths depend on the price, and hence is variable over the auction. The price stops when the set of bidders that can afford that price pack exactly into a block of ad slots, at which point the mechanism allocates that block and continues on the remaining slots. To prove our result on the equilibrium of the revenue-maximizing mechanism, we first show that a greedy algorithm suffices to solve the revenue-maximizing linear program; we then use this insight to prove that bidders allocated in the same block of our mechanism have no incentive to deviate from bidding the fixed price of that block.

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arXiv2007

Stochastic Models for Budget Optimization in Search-Based Advertising

Internet search companies sell advertisement slots based on users' search queries via an auction. Advertisers have to determine how to place bids on the keywords of their interest in order to maximize their return for a given budget: this is the budget optimization problem. The solution depends on the distribution of future queries. In this paper, we formulate stochastic versions of the budget optimization problem based on natural probabilistic models of distribution over future queries, and address two questions that arise. [Evaluation] Given a solution, can we evaluate the expected value of the objective function? [Optimization] Can we find a solution that maximizes the objective function in expectation? Our main results are approximation and complexity results for these two problems in our three stochastic models. In particular, our algorithmic results show that simple prefix strategies that bid on all cheap keywords up to some level are either optimal or good approximations for many cases; we show other cases to be NP-hard.

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arXiv2007

Relative-Error CUR Matrix Decompositions

Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of ``components.'' Typically, these components are linear combinations of the rows and columns of the matrix, and are thus difficult to interpret in terms of the original features of the input data. In this paper, we propose and study matrix approximations that are explicitly expressed in terms of a small number of columns and/or rows of the data matrix, and thereby more amenable to interpretation in terms of the original data. Our main algorithmic results are two randomized algorithms which take as input an $m \times n$ matrix $A$ and a rank parameter $k$. In our first algorithm, $C$ is chosen, and we let $A'=CC^+A$, where $C^+$ is the Moore-Penrose generalized inverse of $C$. In our second algorithm $C$, $U$, $R$ are chosen, and we let $A'=CUR$. ($C$ and $R$ are matrices that consist of actual columns and rows, respectively, of $A$, and $U$ is a generalized inverse of their intersection.) For each algorithm, we show that with probability at least $1-δ$: $$ ||A-A'||_F \leq (1+ε) ||A-A_k||_F, $$ where $A_k$ is the ``best'' rank-$k$ approximation provided by truncating the singular value decomposition (SVD) of $A$. The number of columns of $C$ and rows of $R$ is a low-degree polynomial in $k$, $1/ε$, and $\log(1/δ)$. Our two algorithms are the first polynomial time algorithms for such low-rank matrix approximations that come with relative-error guarantees; previously, in some cases, it was not even known whether such matrix decompositions exist. Both of our algorithms are simple, they take time of the order needed to approximately compute the top $k$ singular vectors of $A$, and they use a novel, intuitive sampling method called ``subspace sampling.''

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arXiv2007

Radix Sorting With No Extra Space

It is well known that n integers in the range [1,n^c] can be sorted in O(n) time in the RAM model using radix sorting. More generally, integers in any range [1,U] can be sorted in O(n sqrt{loglog n}) time. However, these algorithms use O(n) words of extra memory. Is this necessary? We present a simple, stable, integer sorting algorithm for words of size O(log n), which works in O(n) time and uses only O(1) words of extra memory on a RAM model. This is the integer sorting case most useful in practice. We extend this result with same bounds to the case when the keys are read-only, which is of theoretical interest. Another interesting question is the case of arbitrary c. Here we present a black-box transformation from any RAM sorting algorithm to a sorting algorithm which uses only O(1) extra space and has the same running time. This settles the complexity of in-place sorting in terms of the complexity of sorting.

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arXiv2007

On the Complexity of Processing Massive, Unordered, Distributed Data

An existing approach for dealing with massive data sets is to stream over the input in few passes and perform computations with sublinear resources. This method does not work for truly massive data where even making a single pass over the data with a processor is prohibitive. Successful log processing systems in practice such as Google's MapReduce and Apache's Hadoop use multiple machines. They efficiently perform a certain class of highly distributable computations defined by local computations that can be applied in any order to the input. Motivated by the success of these systems, we introduce a simple algorithmic model for massive, unordered, distributed (mud) computation. We initiate the study of understanding its computational complexity. Our main result is a positive one: any unordered function that can be computed by a streaming algorithm can also be computed with a mud algorithm, with comparable space and communication complexity. We extend this result to some useful classes of approximate and randomized streaming algorithms. We also give negative results, using communication complexity arguments to prove that extensions to private randomness, promise problems and indeterminate functions are impossible. We believe that the line of research we introduce in this paper has the potential for tremendous impact. The distributed systems that motivate our work successfully process data at an unprecedented scale, distributed over hundreds or even thousands of machines, and perform hundreds of such analyses each day. The mud model (and its generalizations) inspire a set of complexity-theoretic questions that lie at their heart.

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arXiv2006

Budget Optimization in Search-Based Advertising Auctions

Internet search companies sell advertisement slots based on users' search queries via an auction. While there has been a lot of attention on the auction process and its game-theoretic aspects, our focus is on the advertisers. In particular, the advertisers have to solve a complex optimization problem of how to place bids on the keywords of their interest so that they can maximize their return (the number of user clicks on their ads) for a given budget. We model the entire process and study this budget optimization problem. While most variants are NP hard, we show, perhaps surprisingly, that simply randomizing between two uniform strategies that bid equally on all the keywords works well. More precisely, this strategy gets at least 1-1/e fraction of the maximum clicks possible. Such uniform strategies are likely to be practical. We also present inapproximability results, and optimal algorithms for variants of the budget optimization problem.

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arXiv2006

Estimating Aggregate Properties on Probabilistic Streams

The probabilistic-stream model was introduced by Jayram et al. \cite{JKV07}. It is a generalization of the data stream model that is suited to handling ``probabilistic'' data where each item of the stream represents a probability distribution over a set of possible events. Therefore, a probabilistic stream determines a distribution over potentially a very large number of classical "deterministic" streams where each item is deterministically one of the domain values. The probabilistic model is applicable for not only analyzing streams where the input has uncertainties (such as sensor data streams that measure physical processes) but also where the streams are derived from the input data by post-processing, such as tagging or reconciling inconsistent and poor quality data. We present streaming algorithms for computing commonly used aggregates on a probabilistic stream. We present the first known, one pass streaming algorithm for estimating the \AVG, improving results in \cite{JKV07}. We present the first known streaming algorithms for estimating the number of \DISTINCT items on probabilistic streams. Further, we present extensions to other aggregates such as the repeat rate, quantiles, etc. In all cases, our algorithms work with provable accuracy guarantees and within the space constraints of the data stream model.

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arXiv2006

Bidding to the Top: VCG and Equilibria of Position-Based Auctions

Many popular search engines run an auction to determine the placement of advertisements next to search results. Current auctions at Google and Yahoo! let advertisers specify a single amount as their bid in the auction. This bid is interpreted as the maximum amount the advertiser is willing to pay per click on its ad. When search queries arrive, the bids are used to rank the ads linearly on the search result page. The advertisers pay for each user who clicks on their ad, and the amount charged depends on the bids of all the advertisers participating in the auction. In order to be effective, advertisers seek to be as high on the list as their budget permits, subject to the market. We study the problem of ranking ads and associated pricing mechanisms when the advertisers not only specify a bid, but additionally express their preference for positions in the list of ads. In particular, we study "prefix position auctions" where advertiser $i$ can specify that she is interested only in the top $b_i$ positions. We present a simple allocation and pricing mechanism that generalizes the desirable properties of current auctions that do not have position constraints. In addition, we show that our auction has an "envy-free" or "symmetric" Nash equilibrium with the same outcome in allocation and pricing as the well-known truthful Vickrey-Clarke-Groves (VCG) auction. Furthermore, we show that this equilibrium is the best such equilibrium for the advertisers in terms of the profit made by each advertiser. We also discuss other position-based auctions.

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