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Description

Consider 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 constant-factor approximation. In contrast, we also show the problem is APX-hard via a reduction from {\sc Cubic Max-Bisection}. 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.

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