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 $\sigma$-smooth if $\frac{1}{2}u^T\nabla2 F(x) u\leq\sigma\cdot\frac{||u||1}{||x||_1}u^T\nabla F(x)$ for all $u,x\geq 0$, $x\neq 0$. We give an $\Omega(1/\sigma)$-approximation for the maximization problem of monotone, normalized one-sided $\sigma$-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 $\Omega(1/\sigma^{{3/2})$-approximation} for maximizing a $\sigma$-semi-metric diversity function subject to matroid constraint. This improves upon the previous best bound of $\Omega(1/\sigma^2)$ and we give evidence that it may be tight. For general one-sided smooth functions, we show the continuous process gives an $\Omega(1/3^{{2\sigma})$-approximation,} independent of $n$. In this setting, by discretizing, we present a poly-time algorithm for multilinear one-sided $\sigma$-smooth functions.