Dan-Xuan Liu

Chao Qian, Dan-Xuan Liu, Zhi-Hua Zhou

Given a ground set of items, the result diversification problem aims to select a subset with high "quality" and "diversity" while satisfying some constraints. It arises in various real-world artificial intelligence applications, such as web-based search, document summarization and feature selection, and also has applications in other areas, e.g., computational geometry, databases, finance and operations research. Previous algorithms are mainly based on greedy or local search. In this paper, we propose to reformulate the result diversification problem as a bi-objective maximization problem, and solve it by a multi-objective evolutionary algorithm (EA), i.e., the GSEMO. We theoretically prove that the GSEMO can achieve the (asymptotically) optimal theoretical guarantees under both static and dynamic environments. For cardinality constraints, the GSEMO can achieve the optimal polynomial-time approximation ratio, $1/2$. For more general matroid constraints, the GSEMO can achieve an asymptotically optimal polynomial-time approximation ratio, $1/2-ε/(4n)$, where $ε>0$ and $n$ is the size of the ground set of items. Furthermore, when the objective function (i.e., a linear combination of quality and diversity) changes dynamically, the GSEMO can maintain this approximation ratio in polynomial running time, addressing the open question proposed by Borodin. This also theoretically shows the superiority of EAs over local search for solving dynamic optimization problems for the first time, and discloses the robustness of the mutation operator of EAs against dynamic changes. Experiments on the applications of web-based search, multi-label feature selection and document summarization show the superior performance of the GSEMO over the state-of-the-art algorithms (i.e., the greedy algorithm and local search) under both static and dynamic environments.

Chao Qian, Dan-Xuan Liu, Chao Feng et al.

Evolutionary algorithms (EAs) are general-purpose optimization algorithms, inspired by natural evolution. Recent theoretical studies have shown that EAs can achieve good approximation guarantees for solving the problem classes of submodular optimization, which have a wide range of applications, such as maximum coverage, sparse regression, influence maximization, document summarization and sensor placement, just to name a few. Though they have provided some theoretical explanation for the general-purpose nature of EAs, the considered submodular objective functions are defined only over sets or multisets. To complement this line of research, this paper studies the problem class of maximizing monotone submodular functions over sequences, where the objective function depends on the order of items. We prove that for each kind of previously studied monotone submodular objective functions over sequences, i.e., prefix monotone submodular functions, weakly monotone and strongly submodular functions, and DAG monotone submodular functions, a simple multi-objective EA, i.e., GSEMO, can always reach or improve the best known approximation guarantee after running polynomial time in expectation. Note that these best-known approximation guarantees can be obtained only by different greedy-style algorithms before. Empirical studies on various applications, e.g., accomplishing tasks, maximizing information gain, search-and-tracking and recommender systems, show the excellent performance of the GSEMO.