Maximizing Modular plus Non-monotone Submodular Functions

The research problem in this work is the relaxation of maximizing non-negative submodular plus modular with the entire real number domain as its value range over a family of down-closed sets. We seek a feasible point $\mathbf{x}^*$ in the polytope of the given constraint such that $\mathbf{x}^*\in\arg\max_{\mathbf{x}\in\mathcal{P}\subseteq[0,1]^n}F(\mathbf{x})+L(\mathbf{x})$, where $F$, $L$ denote the extensions of the underlying submodular function $f$ and modular function $\ell$. We provide an approximation algorithm named \textsc{Measured Continuous Greedy with Adaptive Weights}, which yields a guarantee $F(\mathbf{x})+L(\mathbf{x})\geq \left(1/e-\mathcal{O}(\epsilon)\right)\cdot f(OPT)+\left(\frac{\beta-e}{e(\beta-1)}-\mathcal{O}(\epsilon)\right)\cdot\ell(OPT)$ under the assumption that the ratio of non-negative part within $\ell(OPT)$ to the absolute value of its negative part is demonstrated by a parameter $\beta\in[0, \infty]$, where $OPT$ is the optimal integral solution for the discrete problem. It is obvious that the factor of $\ell(OPT)$ is $1$ when $\beta=0$, which means the negative part is completely dominant at this time; otherwise the factor is closed to $1/e$ whe $\beta\rightarrow\infty$. Our work first breaks the restriction on the specific value range of the modular function without assuming non-positivity or non-negativity as previous results and quantifies the relative variation of the approximation guarantee for optimal solutions with arbitrary structure. Moreover, we also give an analysis for the inapproximability of the problem we consider. We show a hardness result that there exists no polynomial algorithm whose output $S$ satisfies $f(S)+\ell(S)\geq0.478\cdot f(OPT)+\ell(OPT)$.