Parametric nonlinear discrete optimization over well-described sets and matroid intersections

Abstract

We address optimization of parametric nonlinear functions of the form f(Wx), where ${f : \mathbb {R}^d \rightarrow \mathbb {R}}$ is a nonlinear function, W is a d × n matrix, and feasible x are in some large finite set ${\mathcal {F}}$ of integer points in ${\mathbb {R}^n}$. Generally, such problems are intractable, so we obtain positive algorithmic results by looking at broad natural classes of f, W and ${\mathcal {F}}$. One of our main motivations is multi-objective discrete optimization, where f trades off the linear functions given by the rows of W. Another motivation is that we want to extend as much as possible the known results about polynomial-time linear optimization over trees, assignments, matroids, polymatroids, etc. to nonlinear optimization over such structures. We assume that ${\mathcal {F}}$ is well described (i.e., we can efficiently optimize a linear objective function on ${\mathcal {F}}$; equivalently, we have an efficient separation oracle for the convex hull of ${\mathcal {F}}$). For example, the sets of characteristic vectors of (i) matchings of a graph, and (ii) common bases of a pair of matroids on a common ground set satisfy this property. In this setting, the problem is already known to be intractable (even for a single matroid), for general f (given by a comparison oracle), for (i) d = 1 and binary-encoded W, and for (ii) d = n and W = I. Our main results (a few technicalities and some generality suppressed):

• 1. When ${\mathcal {F}}$ is well described, f is convex (or even quasi-convex), and W has a fixed number of rows and is unary encoded or with entries in a fixed set, we give an efficient deterministic algorithm for maximization.

• 2. When ${\mathcal {F}}$ is well described, f is a norm, and W is binary-encoded, we give an efficient deterministic constant-approximation algorithm for maximization (Note that the approximation factor depends on the norm, and hence implicitly on the number of rows of W, while the running time increases only linearly in the number of rows of W).

• 3. When non-negative ${\mathcal {F}}$ is well described, f is “ray concave” and non-decreasing, and non-negative W has a fixed number of rows and is unary encoded or with entries in a fixed set, we give an efficient deterministic constant-approximation algorithm for minimization.

• 4. When ${\mathcal {F}}$ is the set of characteristic vectors of common independent sets or bases of a pair of rational vectorial matroids on a common ground set, f is arbitrary, and W has a fixed number of rows and is unary encoded, we give an efficient randomized algorithm for optimization.