Ranking with Multiple Objectives

In search and advertisement ranking, it is often required to simultaneously maximize multiple objectives. For example, the objectives can correspond to multiple intents of a search query, or in the context of advertising, they can be relevance and revenue. It is important to efficiently find rankings which strike a good balance between such objectives. Motivated by such applications, we formulate a general class of problems where - each result gets a different score corresponding to each objective, - the results of a ranking are aggregated by taking, for each objective, a weighted sum of the scores in the order of the ranking, and - an arbitrary concave function of the aggregates is maximized. Combining the aggregates using a concave function will naturally lead to more balanced outcomes. We give an approximation algorithm in a bicriteria/resource augmentation setting: the algorithm with a slight advantage does as well as the optimum. In particular, if the aggregation step is just the sum of the top k results, then the algorithm outputs k + 1 results which do as well the as the optimal top k results. We show how this approach helps with balancing different objectives via simulations on synthetic data as well as on real data from LinkedIn.