Beyond Submodular Maximization via One-Sided Smoothness

The multilinear framework has achieved the breakthrough $1-1/e$ approximation for maximizing a monotone submodular function subject to a matroid constraint. This framework has a continuous optimization part and a rounding part. We extend both parts to a wider array of problems. In particular, we make a conceptual contribution by identifying a family of parameterized functions. As a running example we focus on solving diversity problems $\max f(S)=\frac{1}{2}\sum_{i,j\in A}A_{ij}:S\in\mathcal{M}$, where $\mathcal{M}$ is a matroid. These diversity functions have $A_{ij}\geq 0$ as a measure of dissimilarity of $i,j$, and $A$ has $0$-diagonal. The multilinear framework cannot be directly applied to the multilinear extension of such functions. We introduce a new parameter for functions $F\in{\bf C}^2$ which measures the approximability of the associated problem $\max\{F(x):x\in P\}$, for solvable downwards-closed polytopes $P$. A function $F$ is called one-sided $σ$-smooth if $\frac{1}{2}u^T\nabla^2 F(x) u\leqσ\cdot\frac{||u||_1}{||x||_1}u^T\nabla F(x)$ for all $u,x\geq 0$, $x\neq 0$. We give an $Ω(1/σ)$-approximation for the maximization problem of monotone, normalized one-sided $σ$-smooth $F$ with an additional property: non-positive third order partial derivatives. Using the multilinear framework and new matroid rounding techniques for quadratic objectives, we give an $Ω(1/σ^{3/2})$-approximation for maximizing a $σ$-semi-metric diversity function subject to matroid constraint. This improves upon the previous best bound of $Ω(1/σ^2)$ and we give evidence that it may be tight. For general one-sided smooth functions, we show the continuous process gives an $Ω(1/3^{2σ})$-approximation, independent of $n$. In this setting, by discretizing, we present a poly-time algorithm for multilinear one-sided $σ$-smooth functions.