In this paper, we present a thorough study of maximizing a regularized non-monotone submodular function subject to various constraints, i.e., $\max \{ g(A) - \ell(A) : A \in \mathcal{F} \}$, where $g \colon 2^\Omega \to \mathbb{R}_+$ is a non-monotone submodular function, $\ell \colon 2^\Omega \to \mathbb{R}_+$ is a normalized modular function and $\mathcal{F}$ is the constraint set. Though the objective function $f := g - \ell$ is still submodular, the fact that $f$ could potentially take on negative values prevents the existing methods for submodular maximization from providing a constant approximation ratio for the regularized submodular maximization problem. To overcome the obstacle, we propose several algorithms which can provide a relatively weak approximation guarantee for maximizing regularized non-monotone submodular functions. More specifically, we propose a continuous greedy algorithm for the relaxation of maximizing $g - \ell$ subject to a matroid constraint. Then, the pipage rounding procedure can produce an integral solution $S$ such that $\mathbb{E} [g(S) - \ell(S)] \geq e^{-1}g(OPT) - \ell(OPT) - O(\epsilon)$. Moreover, we present a much faster algorithm for maximizing $g - \ell$ subject to a cardinality constraint, which can output a solution $S$ with $\mathbb{E} [g(S) - \ell(S)] \geq (e^{-1} - \epsilon) g(OPT) - \ell(OPT)$ using $O(\frac{n}{\epsilon^2} \ln \frac 1\epsilon)$ value oracle queries. We also consider the unconstrained maximization problem and give an algorithm which can return a solution $S$ with $\mathbb{E} [g(S) - \ell(S)] \geq e^{-1} g(OPT) - \ell(OPT)$ using $O(n)$ value oracle queries.
翻译:在本文中,我们展示了对受各种制约的常规非分子子模块功能最大化的透彻研究,即:$=max= g(A) -\ell(A) :$\ in\mathcal{F} $,其中$=croom 2\\\to\mathb{R}$是一个非分子子模块功能,$\ell\croom 2\\\\\\oge\to\math{R}(O} 美元是一个常规模块功能,$\\maxxx(S) ====g(A) :$=g=g==call{F} 美元,其中美元=g=croom2\\hbldroom2}(cell\croom2\croom2\\\\\\maxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx