Maximizing Sequence-Submodular Functions and its Application to Online Advertising

TL;DR

This paper introduces sequence-submodularity and sequence-monotonicity to extend submodular optimization to sequences, providing greedy algorithms with strong performance guarantees for online advertising applications.

Contribution

The paper develops the concepts of sequence-submodularity and sequence-monotonicity, and applies them to design greedy algorithms with provable guarantees for online advertising problems.

Findings

Greedy algorithm achieves 1-1/e approximation for online ad allocation.

The framework improves performance guarantees for query rewriting.

Applications demonstrate the effectiveness of the proposed approach.

Abstract

Motivated by applications in online advertising, we consider a class of maximization problems where the objective is a function of the sequence of actions as well as the running duration of each action. For these problems, we introduce the concepts of \emph{sequence-submodularity} and \emph{sequence-monotonicity} which extend the notions of submodularity and monotonicity from functions defined over sets to functions defined over sequences. We establish that if the objective function is sequence-submodular and sequence-non-decreasing, then there exists a greedy algorithm that achieves $1-1/e$ of the optimal solution. We apply our algorithm and analysis to two applications in online advertising: online ad allocation and query rewriting. We first show that both problems can be formulated as maximizing non-decreasing sequence-submodular functions. We then apply our framework to these two problems, leading to simple greedy approaches with guaranteed performances. In particular, for online ad allocation problem the performance of our algorithm is $1-1/e$, which matches the best known existing performance, and for query rewriting problem the performance of our algorithm is $1- 1/e^{1-1/e}$ which improves upon the best known existing performance in the literature.