Maximizing Sequence-Submodular Functions and its Application to Online Advertising

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.

💡 Research Summary

The paper introduces a novel class of optimization problems in which the objective depends not only on which actions are taken but also on the order and duration of those actions. To capture this, the authors define sequence‑submodularity and sequence‑monotonicity, extending the classic notions of submodularity and monotonicity from set functions to functions over ordered action‑time pairs. A function (f) is sequence‑submodular if, for any two sequences (A) and (B) with (A) a prefix of (B), the marginal gain of appending a new action‑duration pair (x) to (A) is at least as large as appending (x) to (B). Sequence‑non‑decreasing means that adding any pair never reduces the function value.

Under these two properties, the authors prove that a simple greedy algorithm—repeatedly adding the action‑duration pair with the largest marginal increase—achieves a ((1-1/e)) approximation of the optimal value. The proof follows the classic continuous‑time analysis for submodular maximization, showing that the marginal gain decays exponentially and that the integral of the greedy gains reaches the ((1-1/e)) bound.

The framework is then applied to two concrete problems in online advertising.

1. Online ad allocation: Advertisers bid on keywords and have budgets; the platform decides how long to display each ad for each keyword. The expected revenue from an advertiser is a concave, diminishing‑returns function of the exposure time. The total revenue is shown to be sequence‑submodular and non‑decreasing, so the greedy allocation algorithm attains a ((1-1/e)) guarantee, matching the best known results for this problem.

2. Query rewriting: When a user issues a search query, the system can present several rewritten queries, each with a click‑through probability that decreases with the time it is shown. With a limited total time budget, the goal is to schedule rewritten queries to maximize expected clicks. This objective also fits the sequence‑submodular model, but the marginal decay is stronger, leading to an improved approximation factor of (1-1/e^{,1-1/e}), which surpasses the previously best bound of (1-1/e^{2}).

Overall, the paper establishes that many time‑sensitive decision problems can be modeled as maximizing a sequence‑submodular function, and that the classical greedy approach retains its strong performance guarantees in this richer setting. The results broaden the applicability of submodular optimization techniques to domains where order and duration are intrinsic, offering both theoretical insight and practical algorithms for online advertising and related fields.