FOCS 2011
TechTalks from event: FOCS 2011
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Markov LayoutConsider the following problem of laying out a set of $n$ images that match a query onto the nodes of a $\sqrt{n}\times\sqrt{n}$ grid. We are given a score for each image, as well as the distribution of patterns by which a user's eye scans the nodes of the grid and we wish to maximize the expected total score of images selected by the user. This is a special case of the \emph{Markov layout problem}, in which we are given a Markov chain $M$ together with a set o f objects to be placed at the states of the Markov chain. Each object has a utility to the user if viewed, as well as a stopping probability with which the user ceases to look further at objects. We point out that this layout problem is prototypical in a number of applications in web search and advertising, particularly in the emerging genre of search results pages from major engines. In a different class of applications, the states of the Markov chain are web pages at a publishers website and the objects are advertisements. In this paper we study the approximability of the Markov layout problem. Our main result is an $O(\log n)$ approximation algorithm for the most general version of the problem. The core idea behind the algorithm is to transform an optimization problem over partial permutations into an optimization problem over sets by losing a logarithmic factor in approximation; the latter problem is then shown to be submodular with two matroid constraints, which admits a constantfactor approximation. In contrast, we also show the problem is APXhard via a reduction from {\sc Cubic MaxBisection}. We then study harder variants of the problem in which no \emph{gaps}  states of $M$ with no object placed on them  are allowed. By exploiting the geometry, we obtain an $O(\log^{3/2} n)$ approximation algorithm when the digraph underlying $M$ is a grid and an $O(\log n)$ approximation algorithm when it is a tree. These special cases are especially appropriate for our applications.

Limitations of Randomized Mechanisms for Combinatorial AuctionsThe design of computationally efficient and incentive compatible mechanisms that solve or approximate fundamental resource allocation problems is the main goal of algorithmic mechanism design. A central example in both theory and practice is welfaremaximization in combinatorial auctions. Recently, a randomized mechanism has been discovered for combinatorial auctions that is truthful in expectation and guarantees a $(11/e)$approximation to the optimal social welfare when players have coverage valuations \cite{DRY11}. This approximation ratio is the best possible even for nontruthful algorithms, assuming $P \neq NP$ \cite{KLMM05}. Given the recent sequence of negative results for combinatorial auctions under more restrictive notions of incentive compatibility \cite{DN07,BDFKMPSSU10,Dobzin11}, this development raises a natural question: Are truthfulinexpectation mechanisms compatible with polynomialtime approximation in a way that deterministic or universally truthful mechanisms are not? In particular, can polynomialtime truthfulinexpectation mechanisms guarantee a nearoptimal approximation ratio for more general variants of combinatorial auctions? We prove that this is not the case. Specifically, the result of \cite{DRY11} cannot be extended to combinatorial auctions with submodular valuations in the value oracle model. (Absent strategic considerations, a $(11/e)$approximation is still achievable in this setting \cite{V08}.) More precisely, we prove that there is a constant $\gamma>0$ such that there is no randomized mechanism that is truthfulinexpectation  or even approximately truthfulinexpectation  and guarantees an $m^{\gamma}$approximation to the optimal social welfare for combinatorial auctions with submodular valuations in the value oracle model. We also prove an analogous result for the flexible combinatorial public projects (CPP) problem, where a truthfulinexpectation $(11/e)$approximation for coverage valuations has been recently developed \cite{Dughmi11}. We show that there is no truthfulinexpectation  or even approximately truthfulinexpectation  mechanism that achieves an $m^{\gamma}$approximation to the optimal social welfare for combinatorial public projects with submodular valuations in the value oracle model. Both our results present an unexpected separation between coverage functions and submodular functions, which does not occur for these problems without strategic considerations.

Bayesian Combinatorial Auctions: Expanding Single Buyer Mechanisms to Many BuyersFor Bayesian combinatorial auctions, we present a general framework for reducing the mechanism design problem for many buyers to the mechanism design problem for one buyer. Our generic reduction works for any separable objective (e.g., welfare, revenue, etc) and any space of valuations (e.g. submodular, additive, etc) and any distribution of valuations as long as valuations of different buyers are distributed independently (not necessarily identically). Roughly speaking, we present two generic $n$buyer mechanisms that use $1$buyer mechanisms as black boxes. We show that if we have an $\alpha$approximate $1$buyer mechanism for each buyer\footnote{Note that we can use different $1$buyer mechanisms for different buyers.} then our generic $n$buyer mechanisms are $\frac{1}{2}\alpha$approximation of the optimal $n$buyer mechanism. Furthermore, if we have several copies of each item and no buyer ever needs more than $\frac{1}{k}$ of all copies of each item then our generic $n$buyer mechanisms are $\gamma_k \alpha$approximation of the optimal $n$buyer mechanism where $\gamma_k \ge 1\frac{1}{\sqrt{k+3}}$. Observe that $\gamma_k$ is at least $\frac{1}{2}$ and approaches $1$ as $k$ increases. Applications of our main theorem include the following improvements on results from the literature. For each of the following models we construct a $1$buyer mechanism and then apply our generic expansion: For revenue maximization in combinatorial auctions with hard budget constraints, \cite{BGGM10} presented a $\frac{1}{4}$approximate BIC mechanism for additive/correlated valuations and an $O(1)$approximate\footnote{$O(1)=\frac{1}{96}$} sequential posted pricing mechanism for additive/independent valuations. We improve this to a $\gamma_k$approximate BIC mechanism and a $\gamma_k (1\frac{1}{e})$approximate sequential posted pricing mechanism respectively. For revenue maximization in combinatorial auctions with unit demand buyers, \cite{CHMS10} presented a $\frac{1}{6.75}$approximate sequential posted pricing mechanism. We improve this to a $\frac{1}{2} \gamma_k$ approximate sequential posted pricing mechanism. We also present a $\gamma_k$approximate sequential posted pricing mechanism for unitdemand multiunit auctions(homogeneous) with hardbudget constraints. Furthermore, our sequential posted pricing mechanisms assume no control or prior information about the order in which buyers arrive.

ExtremeValue Theorems for Optimal Multidimensional PricingWe provide a Polynomial Time Approximation Scheme for the {\em multidimensional unitdemand pricing problem}, when the buyer's values are independent (but not necessarily identically distributed.) For all $\epsilon>0$, we obtain a $(1+\epsilon)$factor approximation to the optimal revenue in time polynomial, when the values are sampled from Monotone Hazard Rate (MHR) distributions, quasipolynomial, when sampled from regular distributions, and polynomial in $n^{{\rm poly}(\log r)}$, when sampled from general distributions supported on a set $[u_{min}, r u_{min}]$. We also provide an additive PTAS for all bounded distributions.Our algorithms are based on novel extreme value theorems for MHR and regular distributions, and apply probabilistic techniques to understand the statistical properties of revenue distributions, as well as to reduce the size of the search space of the algorithm. As a byproduct of our techniques, we establish structural properties of optimal solutions. We show that, for all $\epsilon >0$, $g(1/\epsilon)$ distinct prices suffice to obtain a $(1+\epsilon)$factor approximation to the optimal revenue for MHR distributions, where $g(1/\epsilon)$ is a quasilinear function of $1/\epsilon$ that does not depend on the number of items. Similarly, for all $\epsilon>0$ and $n>0$, $g(1/\epsilon \cdot \log n)$ distinct prices suffice for regular distributions, where $n$ is the number of items and $g(\cdot)$ is a polynomial function. Finally, in the i.i.d. MHR case, we show that, as long as the number of items is a sufficiently large function of $1/\epsilon$, a single price suffices to achieve a $(1+\epsilon)$factor approximation.

Efficient computation of approximate pure Nash equilibria in congestion gamesCongestion games constitute an important class of games in which computing an exact or even approximate pure Nash equilibrium is in general {\sf PLS}complete. We present a surprisingly simple polynomialtime algorithm that computes $O(1)$approximate Nash equilibria in these games. In particular, for congestion games with linear latency functions, our algorithm computes $(2+\epsilon)$approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and $1/\epsilon$. It also applies to games with polynomial latency functions with constant maximum degree $d$; there, the approximation guarantee is $d^{O(d)}$. The algorithm essentially identifies a polynomially long sequence of bestresponse moves that lead to an approximate equilibrium; the existence of such short sequences is interesting in itself. These are the first positive algorithmic results for approximate equilibria in nonsymmetric congestion games. We strengthen them further by proving that, for congestion games that deviate from our mild assumptions, computing $\rho$approximate equilibria is {\sf PLS}complete for any polynomialtime computable $\rho$.