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

2025-05-28

AI Summary

This paper studies the minimization of separable convex functions over integer feasible sets defined by constraint matrices with small coefficients and bounded primal/dual tree depths. For this class of sparse integer programs, we present the first near-linear-time algorithm for nonlinear separable convex objectives that matches information-theoretic lower bounds. Our method introduces a unified framework integrating scaling techniques, proximity analysis, sensitivity theory, and dynamic data structures. When parameterized by the primal tree depth, the algorithm achieves the optimal time complexity $O(n log |u-l|_infty)$. When parameterized by the dual tree depth, it runs in $O(g n log n log |u-l|_infty)$ time—nearly matching the conjectured optimal bound. These results substantially extend the frontier of efficient solvability for convex integer programming, particularly for structured sparse instances.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Programming
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We study the general integer programming (IP) problem of optimizing a separable convex function over the integer points of a polytope: $min {f(mathbf{x}) mid Amathbf{x} = mathbf{b}, , mathbf{l} leq mathbf{x} leq mathbf{u}, , mathbf{x} in mathbb{Z}^n}$. 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.

Problem

• Optimizing separable convex functions over integer polytopes
• Handling sparse constraint matrices with small coefficients
• Achieving near-linear time algorithms for non-linear cases

Innovation

• Dynamic data structure for integer programs
• Scaling and proximity techniques
• Sensitivity analysis optimization