Multi-Objective Submodular Maximization by Regret Ratio Minimization with Theoretical Guarantee

Submodular maximization has attracted much attention due to its wide application and attractive property. Previous works mainly considered one single objective function, while there can be multiple ones in practice. As the objectives are usually conflicting, there exists a set of Pareto optimal solutions, attaining different optimal trade-offs among multiple objectives. In this paper, we consider the problem of minimizing the regret ratio in multi-objective submodular maximization, which is to find at most k solutions to approximate the whole Pareto set as well as possible. We propose a new algorithm RRMS by sampling representative weight vectors and solving the corresponding weighted sums of objective functions using some given -approximation algorithm for single-objective submodular maximization. We prove that the regret ratio of the output of RRMS is upper bounded by 1-+O(d-1(dk-d)^1d-1), where d is the number of objectives. This is the first theoretical guarantee for the situation with more than two objectives. When d=2, it reaches the (1-+O(1/k))-guarantee of the only existing algorithm Polytope. Empirical results on the applications of multi-objective weighted maximum coverage and Max-Cut show the superior performance of RRMS over Polytope.