Sometimes, Convex Separable Optimization Is Much Harder than Linear Optimization, and Other Surprises

TL;DR

This paper demonstrates that convex separable optimization can be significantly more difficult than linear optimization in certain classes of mixed integer programs, challenging previous assumptions and revealing new complexity insights.

Contribution

It identifies classes of mixed integer programs where convex separable optimization is harder than linear optimization, and provides bounds on mixed Graver basis elements with implications for algorithm design.

Findings

Convex separable optimization is harder than linear in certain classes.

Established bounds on the norm of mixed Graver basis elements.

Proved the non-existence of efficient algorithms based on Graver basis bounds.

Abstract

An influential 1990 paper of Hochbaum and Shanthikumar made it common wisdom that "convex separable optimization is not much harder than linear optimization" [JACM 1990]. We exhibit two fundamental classes of mixed integer (linear) programs that run counter this intuition. Namely those whose constraint matrices have small coefficients and small primal or dual treedepth: While linear optimization is easy [Brand, Kouteck\'y, Ordyniak, AAAI 2021], we prove that separable convex optimization IS much harder. Moreover, in the pure integer and mixed integer linear cases, these two classes have the same parameterized complexity. We show that they yet behave quite differently in the separable convex mixed integer case. Our approach employs the mixed Graver basis introduced by Hemmecke [Math. Prog. 2003]. We give the first non-trivial lower and upper bounds on the norm of mixed Graver basis elements. In previous works involving the integer Graver basis, such upper bounds have consistently resulted in efficient algorithms for integer programming. Curiously, this does not happen in our case. In fact, we even rule out such an algorithm.