(Near)-Optimal Algorithms for Sparse Separable Convex Integer Programs

TL;DR

This paper develops near-optimal algorithms for solving sparse separable convex integer programs efficiently, especially when the constraint matrix has small coefficients and low treedepth, extending previous linear case techniques to nonlinear functions.

Contribution

It introduces new algorithms that match lower bounds for sparse convex integer programs with small treedepth and coefficients, advancing beyond linear cases.

Findings

Algorithms match information-theoretic lower bounds in certain regimes.

Designed an $n \, \log n$ time complexity algorithm for dual treedepth cases.

Proposed a new dynamic data structure for integer programming.

Abstract

We study the general integer programming (IP) problem of optimizing a separable convex function over the integer points of a polytope: $\min \{f(\mathbf{x}) \mid A\mathbf{x} = \mathbf{b}, \, \mathbf{l} \leq \mathbf{x} \leq \mathbf{u}, \, \mathbf{x} \in \mathbb{Z}^n\}$. The number of variables $n$ is a variable part of the input, and we consider the regime where the constraint matrix $A$ has small coefficients $\|A\|_\infty$ and small primal or dual treedepth $\mathrm{td}_P(A)$ or $\mathrm{td}_D(A)$, respectively. Equivalently, we consider block-structured matrices, in particular $n$-fold, tree-fold, $2$-stage and multi-stage matrices. We ask about the possibility of near-linear time algorithms in the general case of (non-linear) separable convex functions. The techniques of previous works for the linear case are inherently limited to it; in fact, no strongly-polynomial algorithm may exist due to a simple unconditional information-theoretic lower bound of $n \log \|\mathbf{u}-\mathbf{l}\|_\infty$, where $\mathbf{l}, \mathbf{u}$ are the vectors of lower and upper bounds. Our first result is that with parameters $\mathrm{td}_P(A)$ and $\|A\|_\infty$, this lower bound can be matched (up to dependency on the parameters). Second, with parameters $\mathrm{td}_D(A)$ and $\|A\|_\infty$, the situation is more involved, and we design an algorithm with time complexity $g(\mathrm{td}_D(A), \|A\|_\infty) n \log n \log \|\mathbf{u}-\mathbf{l}\|_\infty$ where $g$ is some computable function. We conjecture that a stronger lower bound is possible in this regime, and our algorithm is in fact optimal. Our algorithms combine ideas from scaling, proximity, and sensitivity of integer programs, together with a new dynamic data structure.